%!PS-Adobe-3.0 %%Title: (Microsoft Word - clarkbook.new) %%Creator: (Microsoft Word: LaserWriter 8 8.3.4) %%CreationDate: (9:25 PM Thursday, May 15, 1997) %%For: (peter) %%Pages: 46 %%DocumentFonts: Times-Bold Symbol Times-Roman Times-Italic Times-BoldItalic %%DocumentNeededFonts: Times-Bold Symbol Times-Roman Times-Italic Times-BoldItalic %%DocumentSuppliedFonts: %%DocumentData: Clean7Bit %%PageOrder: Ascend %%Orientation: Portrait %%DocumentMedia: Default 612 792 0 () () %ADO_ImageableArea: 31 31 583 761 %%EndComments userdict begin/dscInfo 5 dict dup begin /Title(Microsoft Word - clarkbook.new)def /Creator(Microsoft Word: LaserWriter 8 8.3.4)def /CreationDate(9:25 PM Thursday, May 15, 1997)def /For(peter)def /Pages 46 def end def end save /version23-manualfeedpatch where { pop false } { true }ifelse % we don't do an explicit 'get' since product and version MAY % be in systemdict or statusdict - this technique gets the lookup % without failure statusdict begin product (LaserWriter) eq % true if LaserWriter version cvr 23.0 eq % true if version 23 end and % only install this patch if both are true and % true only if patch is not installed and is for this printer % save object and boolean on stack dup { exch restore }if % either true OR saveobject false dup { /version23-manualfeedpatch true def /oldversion23-showpage /showpage load def /showpage % this showpage will wait extra time if manualfeed is true {% statusdict /manualfeed known {% manualfeed known in statusdict statusdict /manualfeed get {% if true then we loop for 5 seconds usertime 5000 add % target usertime { % loop dup usertime sub 0 lt { exit }if }loop pop % pop the usertime off the stac }if }if oldversion23-showpage }bind def }if not{ restore }if /md 221 dict def md begin/currentpacking where {pop /sc_oldpacking currentpacking def true setpacking}if %%BeginFile: adobe_psp_basic %%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. All Rights Reserved. /bd{bind def}bind def /xdf{exch def}bd /xs{exch store}bd /ld{load def}bd /Z{0 def}bd /T/true /F/false /:L/lineto /lw/setlinewidth /:M/moveto /rl/rlineto /rm/rmoveto /:C/curveto /:T/translate /:K/closepath /:mf/makefont /gS/gsave /gR/grestore /np/newpath 14{ld}repeat /$m matrix def /av 83 def /por true def /normland false def /psb-nosave{}bd /pse-nosave{}bd /us Z /psb{/us save store}bd /pse{us restore}bd /level2 /languagelevel where { pop languagelevel 2 ge }{ false }ifelse def /featurecleanup { stopped cleartomark countdictstack exch sub dup 0 gt { {end}repeat }{ pop }ifelse }bd /noload Z /startnoload { {/noload save store}if }bd /endnoload { {noload restore}if }bd level2 startnoload /setjob { statusdict/jobname 3 -1 roll put }bd /setcopies { userdict/#copies 3 -1 roll put }bd level2 endnoload level2 not startnoload /setjob { 1 dict begin/JobName xdf currentdict end setuserparams }bd /setcopies { 1 dict begin/NumCopies xdf currentdict end setpagedevice }bd level2 not endnoload /pm Z /mT Z /sD Z /realshowpage Z /initializepage { /pm save store mT concat }bd /endp { pm restore showpage }def /$c/DeviceRGB def /rectclip where { pop/rC/rectclip ld }{ /rC { np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl :K clip np }bd }ifelse /rectfill where { pop/rF/rectfill ld }{ /rF { gS np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl fill gR }bd }ifelse /rectstroke where { pop/rS/rectstroke ld }{ /rS { gS np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl :K stroke gR }bd }ifelse %%EndFile %%BeginFile: adobe_psp_colorspace_level1 %%Copyright: Copyright 1991-1993 Adobe Systems Incorporated. All Rights Reserved. /G/setgray ld /:F1/setgray ld /:F/setrgbcolor ld /:F4/setcmykcolor where { pop /setcmykcolor ld }{ { 3 { dup 3 -1 roll add dup 1 gt{pop 1}if 1 exch sub 4 1 roll }repeat pop setrgbcolor }bd }ifelse /:Fx { counttomark {0{G}0{:F}{:F4}} exch get exec pop }bd /:rg{/DeviceRGB :ss}bd /:sc{$cs :ss}bd /:dc{/$cs xdf}bd /:sgl{}def /:dr{}bd /:fCRD{pop}bd /:ckcs{}bd /:ss{/$c xdf}bd /$cs Z %%EndFile %%BeginFile: adobe_psp_uniform_graphics %%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. All Rights Reserved. /@a { np :M 0 rl :L 0 exch rl 0 rl :L fill }bd /@b { np :M 0 rl 0 exch rl :L 0 rl 0 exch rl fill }bd /arct where { pop }{ /arct { arcto pop pop pop pop }bd }ifelse /x1 Z /x2 Z /y1 Z /y2 Z /rad Z /@q { /rad xs /y2 xs /x2 xs /y1 xs /x1 xs np x2 x1 add 2 div y1 :M x2 y1 x2 y2 rad arct x2 y2 x1 y2 rad arct x1 y2 x1 y1 rad arct x1 y1 x2 y1 rad arct fill }bd /@s { /rad xs /y2 xs /x2 xs /y1 xs /x1 xs np x2 x1 add 2 div y1 :M x2 y1 x2 y2 rad arct x2 y2 x1 y2 rad arct x1 y2 x1 y1 rad arct x1 y1 x2 y1 rad arct :K stroke }bd /@i { np 0 360 arc fill }bd /@j { gS np :T scale 0 0 .5 0 360 arc fill gR }bd /@e { np 0 360 arc :K stroke }bd /@f { np $m currentmatrix pop :T scale 0 0 .5 0 360 arc :K $m setmatrix stroke }bd /@k { gS np :T 0 0 :M 0 0 5 2 roll arc fill gR }bd /@l { gS np :T 0 0 :M scale 0 0 .5 5 -2 roll arc fill gR }bd /@m { np arc stroke }bd /@n { np $m currentmatrix pop :T scale 0 0 .5 5 -2 roll arc $m setmatrix stroke }bd %%EndFile %%BeginFile: adobe_psp_basic_text %%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. All Rights Reserved. /S/show ld /A{ 0.0 exch ashow }bd /R{ 0.0 exch 32 exch widthshow }bd /W{ 0.0 3 1 roll widthshow }bd /J{ 0.0 32 4 2 roll 0.0 exch awidthshow }bd /V{ 0.0 4 1 roll 0.0 exch awidthshow }bd /fcflg true def /fc{ fcflg{ vmstatus exch sub 50000 lt{ (%%[ Warning: Running out of memory ]%%\r)print flush/fcflg false store }if pop }if }bd /$f[1 0 0 -1 0 0]def /:ff{$f :mf}bd /MacEncoding StandardEncoding 256 array copy def MacEncoding 39/quotesingle put MacEncoding 96/grave put /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis /dagger/degree/cent/sterling/section/bullet/paragraph/germandbls /registered/copyright/trademark/acute/dieresis/notequal/AE/Oslash /infinity/plusminus/lessequal/greaterequal/yen/mu/partialdiff/summation /product/pi/integral/ordfeminine/ordmasculine/Omega/ae/oslash /questiondown/exclamdown/logicalnot/radical/florin/approxequal/Delta/guillemotleft /guillemotright/ellipsis/space/Agrave/Atilde/Otilde/OE/oe /endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide/lozenge /ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright/fi/fl /daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand /Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex/Idieresis/Igrave /Oacute/Ocircumflex/apple/Ograve/Uacute/Ucircumflex/Ugrave/dotlessi/circumflex/tilde /macron/breve/dotaccent/ring/cedilla/hungarumlaut/ogonek/caron MacEncoding 128 128 getinterval astore pop level2 startnoload /copyfontdict { findfont dup length dict begin { 1 index/FID ne{def}{pop pop}ifelse }forall }bd level2 endnoload level2 not startnoload /copyfontdict { findfont dup length dict copy begin }bd level2 not endnoload md/fontname known not{ /fontname/customfont def }if /Encoding Z /:mre { copyfontdict /Encoding MacEncoding def fontname currentdict end definefont :ff def }bd /:bsr { copyfontdict /Encoding Encoding 256 array copy def Encoding dup }bd /pd{put dup}bd /:esr { pop pop fontname currentdict end definefont :ff def }bd /scf { scalefont def }bd /scf-non { $m scale :mf setfont }bd /ps Z /fz{/ps xs}bd /sf/setfont ld /cF/currentfont ld /mbf { /makeblendedfont where { pop makeblendedfont /ABlend exch definefont }{ pop }ifelse def }def %%EndFile %%BeginFile: adobe_psp_derived_styles %%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. 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%%EndFeature }featurecleanup countdictstack[{ %%BeginFeature: *PageRegion LetterSmall lettersmall %%EndFeature }featurecleanup (peter)setjob /mT[1 0 0 -1 31 761]def /sD 16 dict def 300 level2{1 dict dup/WaitTimeout 4 -1 roll put setuserparams}{statusdict/waittimeout 3 -1 roll put}ifelse %%IncludeFont: Times-Bold %%IncludeFont: Symbol %%IncludeFont: Times-Roman %%IncludeFont: Times-Italic %%IncludeFont: Times-BoldItalic /f0_1/Times-Bold :mre /f0_14 f0_1 14 scf /f0_12 f0_1 12 scf /f0_8 f0_1 8 scf /f0_7 f0_1 7 scf /f1_1/Symbol :bsr 240/apple pd :esr /f1_12 f1_1 12 scf /f1_10 f1_1 10 scf /f1_9 f1_1 9 scf /f1_7 f1_1 7 scf /f2_1 f1_1 def /f2_12 f2_1 12 scf /f3_1/Times-Roman :mre /f3_24 f3_1 24 scf /f3_18 f3_1 18 scf /f3_14 f3_1 14 scf /f3_13 f3_1 13 scf /f3_12 f3_1 12 scf /f3_11 f3_1 11 scf /f3_10 f3_1 10 scf /f3_9 f3_1 9 scf /f3_8 f3_1 8 scf /f3_7 f3_1 7 scf /f3_5 f3_1 5 scf /f4_1/Times-Italic :mre /f4_12 f4_1 12 scf /f4_10 f4_1 10 scf /f4_9 f4_1 9 scf /f4_8 f4_1 8 scf /f4_7 f4_1 7 scf /f5_1 f1_1 :mi /f5_12 f5_1 12 scf /f5_10 f5_1 10 scf /f5_9 f5_1 9 scf /f5_8 f5_1 8 scf /f5_7 f5_1 7 scf /f6_1 f3_1 :v def /f7_1 f4_1 :v def /f8_1/Times-BoldItalic :mre /f8_12 f8_1 12 scf /Courier findfont[10 0 0 -10 0 0]:mf setfont %%EndSetup %%Page: 1 1 %%BeginPageSetup initializepage (peter; page: 1 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 85 86 :M f0_14 sf (Causal Inference in the Presence of Latent Variables and Selection)S 260 104 :M (Bias)S 286 101 :M f0_8 sf (1)S 278 128 :M f3_12 sf (by)S 146 152 :M (Peter Spirtes, Christopher Meek, and Thomas Richardson)S 59 661 :M ( )S 59 658.48 -.48 .48 203.48 658 .48 59 658 @a 77 673 :M f3_10 sf (1)S 82 676 :M .648 .065( We wish to thank Clark Glymour and Greg Cooper for many helpful conversations. This research)J 59 687 :M (was supported in part by ONR contract Grant #: N00014-93-1-0568)S endp %%Page: 2 2 %%BeginPageSetup initializepage (peter; page: 2 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 263 701 24 24 rC 281 722 :M f3_12 sf (2)S gR gS 0 0 552 730 rC 59 69 :M f0_14 sf (I)S 64 69 :M (.)S 67 69 :M ( )S 95 69 :M (Introduction)S 77 92 :M f3_12 sf 2.148 .215(Whenever the use of non-experimental data for discovering causal relations or)J 59 110 :M .329 .033(predicting the outcomes of experiments or interventions is contemplated, two difficulties)J 59 128 :M 1.862 .186(are routinely faced. One is the problem of latent variables, or confounders: factors)J 59 146 :M .453 .045(influencing two or more measured variables may not themselves have been measured or)J 59 164 :M .921 .092(recorded. The other is the problem of sample selection bias: values of the variables or)J 59 182 :M 1.26 .126(features under study may themselves influence whether a unit is included in the data)J 59 200 :M (sample.)S 77 224 :M .22 .022(Latent variables produce an association between measured variables that is not due to)J 59 242 :M 2.108 .211(the influence of any measured variable on any other. It is well known that where)J 59 260 :M .851 .085(unrecognized latent common causes occur, regression methods, for example, no matter)J 59 278 :M 1.238 .124(whether linear or nonlinear, give incorrect estimates of influence. When two or more)J 59 296 :M .394 .039(variables under study both influence membership in a sample or inclusion in a database,)J 59 314 :M .202 .02(an association between the variables occurs in the sample that is not due to any influence)J 59 332 :M 1.59 .159(of a measured variable on other measured variables, nor to an unmeasured common)J 59 350 :M (cause.)S 77 374 :M .304 .03(Difficult as these problems are separately, they can both occur in the same sample or)J 59 392 :M .853 .085(database, as in the following example. Suppose a survey of college students is done to)J 59 410 :M .046 .005(determine whether there is a link between )J f4_12 sf .013(intelligence)A 319 410 :M f3_12 sf .079 .008( and )J 343 410 :M f4_12 sf .078 .008(sex drive)J f3_12 sf .051 .005(. Let)J f4_12 sf .055 .006( student status )J 480 410 :M f3_12 sf (be)S 59 428 :M .675 .067(a binary variable that takes on the value 1 when someone is a college student. Suppose)J 59 446 :M f4_12 sf .051(age)A f3_12 sf .118 .012( causes )J f4_12 sf .2 .02(sex drive)J f3_12 sf .093 .009(, and )J 183 446 :M f4_12 sf .038(age)A f3_12 sf .048 .005( and )J f4_12 sf .034(intelligence)A 282 446 :M f3_12 sf .156 .016( also causes )J 343 446 :M f4_12 sf .095 .01(student status)J 409 446 :M f3_12 sf .146 .015( . Hence whether)J 59 464 :M .305 .03(or not one is in the sample is influenced by the two variables in the study, and there may)J 59 482 :M .616 .062(be a statistical dependency between )J 239 482 :M f4_12 sf (intelligence)S 295 482 :M f3_12 sf .416 .042( and )J f4_12 sf 1.009 .101(sex drive)J f3_12 sf .743 .074( in the sample even when)J 59 500 :M .934 .093(no such dependency exists in the population. \(This example will be discussed in more)J 59 518 :M 1.276 .128(detail in section II)J 152 518 :M .95 .095(. Here )J f4_12 sf .459(age)A f3_12 sf 1.368 .137( is obviously a proxy for a combination of physical and)J 59 536 :M 1.463 .146(mental states associated with age.\) The combination of a latent variable \()J 436 536 :M f4_12 sf .563(age)A f3_12 sf 1.223 .122(\) and a)J 59 554 :M 2.564 .256(selection variable \()J f4_12 sf 3.208 .321(student status)J 234 554 :M f3_12 sf 3.458 .346(\) would produce a dependency which \(naively)J 59 572 :M .146 .015(interpreted\) would make it appear as if there is a causal connection of some kind between)J 59 590 :M f4_12 sf (intelligence)S 115 590 :M f3_12 sf ( and )S f4_12 sf (sex drive)S 181 590 :M f3_12 sf (, even though none exists.)S 77 614 :M 1.067 .107(A reasonable attitude toward most uncontrolled convenience samples \(and a lot of)J 59 632 :M .797 .08(\322experimental\323 samples as well\) is that they may be liable to both difficulties. For that)J 59 650 :M .273 .027(reason many statistical writers have implicitly or explicitly concluded that reliable causal)J 59 668 :M .511 .051(and predictive inference is impossible in such cases, no matter whether by human or by)J 59 686 :M .346 .035(machine. We think it is more fruitful to consider whether, under conditions only slightly)J endp %%Page: 3 3 %%BeginPageSetup initializepage (peter; page: 3 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 263 701 24 24 rC 281 722 :M f3_12 sf (3)S gR gS 0 0 552 730 rC 59 56 :M f3_12 sf .292 .029(stronger than those used for causal inference from experimentally controlled data, causal)J 59 74 :M .039 .004(inferences can sometimes reliably be made. This paper uses Bayesian network models for)J 59 92 :M 1.501 .15(that investigation. Bayesian networks, or directed acyclic graph \(DAG\) models have)J 59 110 :M .753 .075(proved very useful in representing both causal and statistical hypotheses. The nodes of)J 59 128 :M .475 .048(the graph represent vertices, directed edges represent direct influences, and the topology)J 59 146 :M .204 .02(of the graph encodes statistical constraints. We will consider features of such models that)J 59 164 :M .16 .016(can be determined from data under assumptions that are related to those routinely applied)J 59 182 :M (in experimental situations:)S 77 206 :M f1_12 sf S 82 206 :M 10 1( )J 95 206 :M f3_12 sf .206 .021(The Markov condition for DAGs interpreted as causal hypotheses. An instance of)J 95 224 :M 1.251 .125(the Causal Markov Assumption is the foundation of the theory of randomized)J 95 242 :M 1.05 .105(experiments. It is also the foundation for the practice of constructing Bayesian)J 95 260 :M .984 .098(networks to be used for diagnosis or classification by eliciting )J f4_12 sf .282(causal)A f3_12 sf 1.259 .126( relations)J 95 278 :M (from experts.)S 77 302 :M f1_12 sf S 82 302 :M 8 .8( )J 95 302 :M f3_12 sf 1.417 .142(An assumption that the population selected by sampling criteria has the same)J 95 320 :M 1.189 .119(causal structure \(although because of sample selection bias not necessarily the)J 95 338 :M .32 .032(same statistical properties\) as the population about which causal inferences are to)J 95 356 :M .261 .026(be made. This assumption, which we call the Population Inference Assumption is)J 95 374 :M .664 .066(of course essential whenever experimental results on a sample are used to guide)J 95 392 :M (policy on a larger population.)S 77 416 :M f1_12 sf S 82 416 :M 8 .8( )J 95 416 :M f3_12 sf 1.461 .146(A version of the Causal Faithfulness Assumption, which says essentially that)J 95 434 :M 1.514 .151(observed independence and conditional independence relations are due to the)J 95 452 :M 2.592 .259(topology of the causal graph rather than to special parameter values. The)J 95 470 :M .608 .061(assumption is used, implicitly or in other terms through the behavioral sciences,)J 95 488 :M (for example in econometrics to test \322exogeneity\323. See Epstein \(1987\).)S 77 512 :M .904 .09(In order to deal with the problems raised by latent variables and selection bias, we)J 59 530 :M .239 .024(will use,)J f4_12 sf .245 .024( partial ancestor graphs)J 218 530 :M f3_12 sf .295 .03( \(or PAGs\), to represent a class of DAGs. For a DAG )J f4_12 sf (G)S 59 548 :M f3_12 sf .551 .055(which may have both latent and selection variables, the PAG that represents )J f4_12 sf (G)S 447 548 :M f3_12 sf .512 .051( contains)J 59 566 :M 2.399 .24(information about both the conditional independencies entailed by )J f4_12 sf (G)S 425 566 :M f3_12 sf 2.811 .281(, and partial)J 59 584 :M .149 .015(information about the ancestor relations in )J 268 584 :M f4_12 sf (G)S 277 584 :M f3_12 sf .169 .017(. We will briefly describe how PAGs can be)J 59 602 :M .157 .016(used to search for latent variable DAG models, to perform efficient classifications, and to)J 59 620 :M 1.899 .19(predict the effects of interventions in causal systems. The advantages of the PAGs)J 59 638 :M (representation include:)S 77 662 :M f1_12 sf S 82 662 :M 9 .9( )J 95 662 :M f3_12 sf 1.109 .111(The space of PAGs is smaller than the space of DAGs they represent, making)J 95 680 :M 1.322 .132(search over PAGs more feasible than a search over DAGs. For a given set of)J endp %%Page: 4 4 %%BeginPageSetup initializepage (peter; page: 4 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 263 701 24 24 rC 281 722 :M f3_12 sf (4)S gR gS 0 0 552 730 rC 95 56 :M f3_12 sf 1.773 .177(measured variables, the set of PAGs is finite, whereas the set of DAGs that)J 95 74 :M (contain the measured variables but may also contain latent variables is infinite.)S 77 98 :M f1_12 sf S 82 98 :M 5 .5( )J 95 98 :M f3_12 sf 3.287 .329(In some cases where large sample \(or even population\) data would not)J 95 116 :M .582 .058(discriminate between different DAGs, the same data would select a single PAG,)J 95 134 :M .789 .079(which could be used to answer some \(but not all\) questions about the effects of)J 95 152 :M (intervening upon an existing causal structure.)S 77 176 :M f1_12 sf S 82 176 :M 10 1( )J 95 176 :M f3_12 sf .08 .008(Quite apart from any causal interpretation of PAGs, in some cases there is a PAG)J 95 194 :M 2.21 .221(that is a more parsimonious representation of a distribution than any DAG)J 95 212 :M 1.469 .147(containing the same variables. Hence a PAG may be used to obtain unbiased)J 95 230 :M 2.3 .23(estimates of population parameters that have lower variance than estimates)J 95 248 :M (obtained from any DAG without latent variables.)S 77 272 :M 1.496 .15(Using PAGs, we characterize the causal information that can \(and in some cases)J 59 290 :M 1.731 .173(cannot\) be obtained from independence and conditional independence relations in a)J 59 308 :M 1.18 .118(population subject to both sample selection bias and latent variables. Given an oracle)J 59 326 :M 1.234 .123(\(such as a family of statistical tests\) for judging population conditional independence)J 59 344 :M 2.12 .212(relations among a set of recorded variables, we provide an asymptotically reliable)J 59 362 :M 1.424 .142(algorithm for constructing PAGs, under the set of assumptions described above \(and)J 59 380 :M .553 .055(described again more precisely in sections I)J 274 380 :M .73 .073( and II)J 307 380 :M .587 .059(\). The algorithm is exponential in the)J 59 398 :M .313 .031(worst case, but feasible for sparse graphs with up to 100 variables. We will also describe)J 59 416 :M (the results of a simulation study on the reliability of the algorithm.)S 59 471 :M f0_14 sf (I)S 64 471 :M (I)S 69 471 :M (.)S 72 471 :M ( )S 95 471 :M (Representation of Selection Bias)S 77 494 :M f3_12 sf .706 .071(We distinguish two different reasons why a sample distribution may differ from the)J 59 512 :M 2.977 .298(population distribution from which it is drawn. The first is simply the familiar)J 59 530 :M 2.427 .243(phenomenon of sample variation, or as we shall say, )J 348 530 :M f0_12 sf 1.697 .17(sample bias)J f4_12 sf (:)S 415 530 :M f3_12 sf 2.304 .23( the frequency)J 59 548 :M .105 .011(distribution of a finite random sample of variables in a set )J 342 548 :M f0_12 sf (V)S 351 548 :M f3_12 sf .103 .01( does not in general perfectly)J 59 566 :M .009 .001(represent the probability distribution over )J f0_12 sf (V)S 271 566 :M f3_12 sf .012 .001( from which the sample is drawn. The second)J 59 584 :M 1.512 .151(reason is that causal relationships between variables in )J 345 584 :M f0_12 sf 1.318 .132(V, )J f3_12 sf 1.982 .198(on the one hand)J f0_12 sf .351(,)A f3_12 sf 1.644 .164( and the)J 59 602 :M 1.142 .114(mechanism by which individuals in the sample are selected from a population, on the)J 59 620 :M .69 .069(other hand, may lead to differences between the expected parameter values in a sample)J 59 638 :M .396 .04(and the population parameter values. In this case we will say that the differences are due)J 59 656 :M .855 .085(to )J f0_12 sf 2.65 .265(selection bias)J 146 656 :M f3_12 sf 2.224 .222(. Sampling bias tends to be remedied by drawing larger samples;)J 59 674 :M (selection bias does not.)S endp %%Page: 5 5 %%BeginPageSetup initializepage (peter; page: 5 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 263 701 24 24 rC 281 722 :M f3_12 sf (5)S gR gS 0 0 552 730 rC 77 56 :M f3_12 sf 2.108 .211(We will not consider the problems of sample bias in this paper \(except in the)J 59 74 :M .237 .024(simulation studies\); we will always assume that we are dealing with an idealized selected)J 59 92 :M (subpopulation of infinite size, but one which may be selection biased.)S 77 116 :M .499 .05(For the purposes of representing selection bias, following Cooper \(1995\) we assume)J 59 134 :M .883 .088(that for each )J 126 134 :M f0_12 sf (measured)S 176 134 :M f3_12 sf .877 .088( random variable )J f4_12 sf .359(A)A f3_12 sf .779 .078(, there is a binary random variable )J 446 134 :M f4_12 sf (S)S f4_8 sf 0 3 rm (A)S 0 -3 rm 457 134 :M f3_12 sf .938 .094( that is)J 59 152 :M .272 .027(equal to one if the value of )J 194 152 :M f4_12 sf .104(A)A f3_12 sf .242 .024( has been recorded, and is equal to zero otherwise. \(We will)J 59 170 :M .416 .042(say that a variable is measured if its value is recorded for any member of the sample.\) If)J 59 188 :M f0_12 sf (V)S 68 188 :M f3_12 sf .56 .056( is a set of variables, we will always suppose that )J 316 188 :M f0_12 sf (V)S 325 188 :M f3_12 sf .518 .052( can be partitioned into three sets:)J 59 206 :M .332 .033(the set )J 94 206 :M f0_12 sf .135(O)A f3_12 sf .261 .026( \(standing for observed\) of measured variables, the set )J f0_12 sf (S)S 377 206 :M f3_12 sf .249 .025( \(standing for selection\))J 59 224 :M .076 .008(of selection variables for )J 182 224 :M f0_12 sf (O)S f3_12 sf .074 .007(, and the remaining variables )J 334 224 :M f0_12 sf (L)S f3_12 sf .067 .007( \(standing for latent\). Although)J 59 242 :M .085 .009(this representation allows for the possibility that some units have missing values for some)J 59 260 :M .219 .022(variables and not others, the algorithms for causal inference that we will describe assume)J 59 278 :M .328 .033(that we are using only the data for the subset of the sample in which all of the units have)J 59 296 :M 2.805 .281(no missing data for any of the measured variables \(i.e. )J f0_12 sf (S)S 374 296 :M f3_12 sf 4.079 .408( = )J 396 296 :M f0_12 sf .97(1)A f3_12 sf 2.776 .278(\). Since in some)J 59 314 :M 1.452 .145(circumstances this reduces the usable sample dramatically \(or even to zero\) it would)J 59 332 :M .304 .03(obviously be desirable to make use of the full sample; how to do this is an open research)J 59 350 :M (problem.)S 77 374 :M .446 .045(In the marginal distribution over a subset )J f0_12 sf (X)S 291 374 :M f3_12 sf .205 .02( of )J f0_12 sf .287(O)A f3_12 sf .546 .055( in a selected subpopulation, the set)J 59 392 :M (of selection variables )S 164 392 :M f0_12 sf (S)S 171 392 :M f3_12 sf ( has been conditioned on, since its value is always equal to )S 456 392 :M f0_12 sf (1)S f3_12 sf ( in the)S 59 410 :M .208 .021(selected subpopulation. Hence for disjoint subsets )J f0_12 sf (X)S 313 410 :M f3_12 sf .362 .036(, )J 320 410 :M f0_12 sf (Y)S 329 410 :M f3_12 sf .201 .02(, and )J f0_12 sf .159(Z)A f3_12 sf .132 .013( of )J f0_12 sf .185(O)A f3_12 sf .333 .033(, we will assume that)J 59 428 :M .444 .044(we cannot determine whether )J 207 428 :M f0_12 sf (X)S 216 428 :M f3_12 sf .744 .074( )J 219 415 16 16 rC -1 -1 226 428 1 1 225 418 @b -1 -1 230 428 1 1 229 418 @b 222 429 -1 1 233 428 1 222 428 @a gR gS 0 0 552 730 rC 235 428 :M f3_12 sf .744 .074( )J 239 428 :M f0_12 sf .726(Z)A f3_12 sf .346 .035( | )J 257 428 :M f0_12 sf (Y)S 266 428 :M f3_12 sf .496 .05(, but that we can determine whether )J 446 428 :M f0_12 sf (X)S 455 428 :M f3_12 sf .744 .074( )J 459 415 16 16 rC -1 -1 466 428 1 1 465 418 @b -1 -1 470 428 1 1 469 418 @b 462 429 -1 1 473 428 1 462 428 @a gR gS 0 0 552 730 rC 475 428 :M f3_12 sf .247 .025( )J f0_12 sf .726(Z)A f3_12 sf .445 .045( |)J 59 446 :M <28>S 63 446 :M f0_12 sf (Y)S 72 446 :M f3_12 sf .383 .038( )J f1_12 sf 1.295A f3_12 sf .894 .089( \()J 94 446 :M f0_12 sf (S)S 101 446 :M f3_12 sf .612 .061( = )J f0_12 sf .661(1)A f3_12 sf 1.415 .142(\)\). \()J 142 446 :M f0_12 sf (X)S 151 433 16 16 rC -1 -1 158 446 1 1 157 436 @b -1 -1 162 446 1 1 161 436 @b 154 447 -1 1 165 446 1 154 446 @a gR gS 0 0 552 730 rC 167 446 :M f0_12 sf .49(Z)A f3_12 sf .275 .028( |)J f0_12 sf (Y)S 190 446 :M f3_12 sf 1.039 .104( means )J 230 446 :M f0_12 sf (X)S 239 446 :M f3_12 sf .914 .091( is independent of )J f0_12 sf .487(Z)A f3_12 sf .808 .081( given all values of )J f0_12 sf (Y)S 451 446 :M f3_12 sf 1.125 .112(. If )J 471 446 :M f0_12 sf (Y)S 480 446 :M f3_12 sf 1.125 .113( is)J 59 464 :M .591 .059(empty, we simply write )J f0_12 sf (X)S 188 451 16 16 rC -1 -1 195 464 1 1 194 454 @b -1 -1 199 464 1 1 198 454 @b 191 465 -1 1 202 464 1 191 464 @a gR gS 0 0 552 730 rC 204 464 :M f0_12 sf .367(Z)A f3_12 sf .623 .062(. If the only member of )J f0_12 sf (X)S 341 464 :M f3_12 sf .385 .039( is )J f4_12 sf .484(X)A f3_12 sf 1.047 .105(, then we write)J f0_12 sf .18 .018( )J f4_12 sf .484(X)A f3_12 sf .198 .02( )J 453 451 16 16 rC -1 -1 460 464 1 1 459 454 @b -1 -1 464 464 1 1 463 454 @b 456 465 -1 1 467 464 1 456 464 @a gR gS 0 0 552 730 rC 469 464 :M f0_12 sf .363(Z)A f3_12 sf .204 .02( |)J f0_12 sf (Y)S 59 482 :M f3_12 sf .481 .048(instead of {)J 117 482 :M f4_12 sf (X)S f3_12 sf (})S 130 469 16 16 rC -1 -1 137 482 1 1 136 472 @b -1 -1 141 482 1 1 140 472 @b 133 483 -1 1 144 482 1 133 482 @a gR gS 0 0 552 730 rC 146 482 :M f0_12 sf .476(Z)A f3_12 sf .217 .022( | )J f0_12 sf (Y)S 172 482 :M f3_12 sf .678 .068(. )J 179 482 :M f0_12 sf (X)S 188 482 :M f3_12 sf .745 .075( )J 192 469 16 16 rC -1 -1 199 482 1 1 198 472 @b -1 -1 203 482 1 1 202 472 @b 195 483 -1 1 206 482 1 195 482 @a gR gS 0 0 552 730 rC 208 482 :M f3_12 sf .745 .075( )J 212 482 :M f0_12 sf .558(Z)A f3_12 sf .36 .036( | \()J f0_12 sf .739 .074(Y )J 246 482 :M f1_12 sf .424A f3_12 sf .292 .029( \()J 263 482 :M f0_12 sf (S)S 270 482 :M f3_12 sf .261 .026( = )J f0_12 sf .282(1)A f3_12 sf .75 .075(\)\) means )J 337 482 :M f0_12 sf (X)S 346 482 :M f3_12 sf .514 .051( is independent of )J 438 482 :M f0_12 sf .227(Z)A f3_12 sf .436 .044( given all)J 59 500 :M .422 .042(values of )J f0_12 sf (Y)S 116 500 :M f3_12 sf .545 .055(, and the value )J 192 500 :M f0_12 sf (S)S 199 500 :M f3_12 sf .682 .068( = )J 214 500 :M f0_12 sf .201(1)A f3_12 sf .49 .049(.\) There may be cases in which all of the variables in )J f0_12 sf (S)S 59 518 :M f3_12 sf .73 .073(always take on the same value; this corresponds to the case where there are no missing)J 59 536 :M .453 .045(values in the sample. In such cases we will represent the selection with a single variable)J 59 554 :M f4_12 sf (S)S f3_12 sf (.)S 77 578 :M .496 .05(The three causal DAGs for a given population shown in Figure 1 illustrate a number)J 59 596 :M .791 .079(of different ways in which selection variables can be related to non-selection variables.)J 59 614 :M .263 .026(The causal DAG in \(i\) would occur, for example if the members of the population whose)J 59 632 :M f4_12 sf (X)S f3_12 sf .086 .009( values were recorded and the members of the population whose )J 380 632 :M f4_12 sf (Y)S 387 632 :M f3_12 sf .083 .008( values were recorded)J 59 650 :M .719 .072(were randomly selected by flips of a pair of independent coins. The DAG in \(ii\) would)J 59 668 :M .374 .037(occur if the flip of a single coin was used to choose units which would have both their )J 485 668 :M f4_12 sf (X)S 59 686 :M f3_12 sf .84 .084(and )J 81 686 :M f4_12 sf (Y)S 88 686 :M f3_12 sf .627 .063( values recorded \(i.e. )J f4_12 sf .23(S)A f4_8 sf 0 3 rm (X)S 0 -3 rm 206 686 :M f3_12 sf .993 .099( = )J 221 686 :M f4_12 sf .252(S)A f4_8 sf 0 3 rm .187(Y)A 0 -3 rm f3_12 sf .694 .069( and there are no missing values in the sample\). The)J endp %%Page: 6 6 %%BeginPageSetup initializepage (peter; page: 6 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 263 701 24 24 rC 281 722 :M f3_12 sf (6)S gR gS 0 0 552 730 rC 59 56 :M f3_12 sf .242 .024(DAG in \(iii\) would occur if, for example, )J f4_12 sf .107(X)A f3_12 sf .255 .026( is years of education, and people with higher)J 59 74 :M f4_12 sf (X)S f3_12 sf .067 .007( values respond to a questionnaire about their education--and thus appear in the sample--)J 59 92 :M .986 .099(more often than people with lower )J 237 92 :M f4_12 sf .418(X)A f3_12 sf .96 .096( values. We do not preclude the possibility that a)J 59 110 :M .186 .019(variable )J f4_12 sf (Y)S 108 110 :M f3_12 sf .31 .031( can be cause of )J 191 110 :M f4_12 sf (S)S f4_8 sf 0 3 rm (X)S 0 -3 rm 202 110 :M f3_12 sf .311 .031( for some variable )J f4_12 sf .149(X)A f3_12 sf .061 .006( )J 304 110 :M f1_12 sf S 311 110 :M f3_12 sf S f4_12 sf (Y)S 321 110 :M f3_12 sf .258 .026(, nor do we preclude the possibility)J 59 128 :M (that )S 80 128 :M f4_12 sf (S)S f4_8 sf 0 3 rm (X)S 0 -3 rm 91 128 :M f3_12 sf ( can be a cause as well as an effect, respectively, of one or more different variables.)S 372 138 90 12 rC 373 147 :M f4_12 sf (X Y)S gR gS 107 138 90 12 rC 108 147 :M f4_12 sf (X Y)S gR gS 242 138 90 12 rC 243 147 :M f4_12 sf (X Y)S gR gS 372 183 90 12 rC 373 192 :M f4_12 sf (S S)S gR gS 379 188 10 12 rC 380 197 :M f4_10 sf (X)S gR gS 452 189 10 12 rC 453 198 :M f4_10 sf (Y)S gR gS 106 137 357 84 rC np 379 180 :M 376 168 :L 379 168 :L 382 168 :L 379 180 :L eofill -1 -1 380 169 1 1 379 155 @b np 441 142 :M 429 145 :L 429 142 :L 429 139 :L 441 142 :L eofill 394 143 -1 1 430 142 1 394 142 @a 109 181 90 12 rC 110 190 :M f4_12 sf (S S)S gR gS 116 186 10 12 rC 117 195 :M f4_10 sf (X)S gR gS 189 187 10 12 rC 190 196 :M f4_10 sf (Y)S gR gS 282 181 21 12 rC 283 190 :M f4_12 sf (S)S gR gS 151 208 293 12 rC 152 217 :M f3_12 sf (\(i\) \(ii\) \(iii\))S gR gS 0 0 552 730 rC 263 242 :M f0_12 sf (Figure )S 300 242 :M (1)S 77 266 :M f3_12 sf .636 .064(The causal DAG that represents the )J 257 266 :M f4_12 sf (intelligence)S 313 266 :M f3_12 sf .421 .042( and )J f4_12 sf 1.076 .108(sex drive)J 382 266 :M f3_12 sf .598 .06( example described in)J 59 284 :M (section )S 96 284 :M (I)S 100 284 :M ( is shown in )S 161 284 :M (Figure 2.)S 184 294 172 78 rC 184 303 :M ( )S 187 303 :M ( )S 190 303 :M f4_12 sf (I)S 194 303 :M (Q)S 202 303 :M f3_12 sf ( )S 205 303 :M ( )S 208 303 :M ( )S 211 303 :M ( )S 214 303 :M ( )S 217 303 :M ( )S 220 303 :M ( )S 223 303 :M ( )S 226 303 :M ( )S 229 303 :M ( )S 232 303 :M ( )S 235 303 :M ( )S 238 303 :M f4_12 sf (A)S 246 303 :M (g)S 252 303 :M (e)S 184 363 :M (S)S 191 363 :M (t)S 194 363 :M (u)S 200 363 :M (d)S 206 363 :M (e)S 211 363 :M (n)S 217 363 :M (t)S 220 363 :M ( )S 223 363 :M (S)S 230 363 :M (t)S 233 363 :M (a)S 238 363 :M (t)S 241 363 :M (u)S 247 363 :M (s)S 252 363 :M ( )S 255 363 :M ( )S 258 363 :M ( )S 261 363 :M ( )S 264 363 :M ( )S 267 363 :M ( )S 270 363 :M ( )S 273 363 :M ( )S 276 363 :M ( )S 279 363 :M ( )S 282 363 :M (s)S 287 363 :M (e)S 292 363 :M (x)S 298 363 :M ( )S 301 363 :M (d)S 307 363 :M (r)S 311 363 :M (i)S 314 363 :M (v)S 320 363 :M (e)S gR gS 175 293 181 81 rC -1 -1 198 342 1 1 197 307 @b np 200 340 :M 195 340 :L 197 343 :L 200 340 :L eofill 195 341 -1 1 201 340 1 195 340 @a 195 341 -1 1 198 343 1 195 340 @a -1 -1 198 344 1 1 200 340 @b -1 -1 222 342 1 1 248 307 @b np 223 342 :M 219 339 :L 219 343 :L 223 342 :L eofill 219 340 -1 1 224 342 1 219 339 @a -1 -1 220 344 1 1 219 339 @b -1 -1 220 344 1 1 223 342 @b 248 308 -1 1 304 349 1 248 307 @a np 304 347 :M 301 350 :L 305 351 :L 304 347 :L eofill -1 -1 302 351 1 1 304 347 @b 301 351 -1 1 306 351 1 301 350 @a 304 348 -1 1 306 351 1 304 347 @a 233.5 293.5 36 14 rS 86 21 219.5 362 @f gR gS 0 0 552 730 rC 262 410 :M f0_12 sf (Figure )S 299 410 :M (2)S 77 434 :M f3_12 sf .654 .065(The causal inferences that we make rest on three different assumptions. The Causal)J 59 452 :M 1.437 .144(Markov and Causal Faithfulness Assumptions are described in the Introduction. The)J 59 470 :M .442 .044(Population Inference Assumption described below is the third assumption we make that)J 59 488 :M .224 .022(guarantees the asymptotic correctness of the causal inference procedures described in the)J 59 506 :M (following sections.)S 77 530 :M 2.038 .204(Consider the case where one is interested in causal inferences about the whole)J 59 548 :M 1.057 .106(population from the selected subpopulation. The notion of a causal graph, as we have)J 59 566 :M .388 .039(defined it is relative to a set of variables and a population. Hence the causal graph of the)J 59 584 :M .496 .05(whole population and the causal graph of the selected subpopulation can conceivably be)J 59 602 :M .364 .036(different. For example, if a drug has an effect on people with black hair, but no effect on)J 59 620 :M 2.435 .244(people with brown hair, then there is an edge from drug to outcome in the first)J 59 638 :M .169 .017(subpopulation, but not in the second. Because of this, in order to draw causal conclusions)J 59 656 :M .556 .056(about either the whole population or the unselected subpopulation \(e.g. the black haired)J endp %%Page: 7 7 %%BeginPageSetup initializepage (peter; page: 7 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 263 701 24 24 rC 281 722 :M f3_12 sf (7)S gR gS 0 0 552 730 rC 59 56 :M f3_12 sf .104 .01(subpopulation\) from the causal graph of the selected subpopulation \(e.g. the brown haired)J 59 74 :M (subpopulation\), we will make the following assumption:)S 77 98 :M f0_12 sf .251 .025(Population Inference Assumption:)J f3_12 sf .045 .005( If )J 268 98 :M f0_12 sf (V)S 277 98 :M f3_12 sf .222 .022( is a causally sufficient set of variables, then)J 59 116 :M .253 .025(the causal DAG relative to )J f0_12 sf (V)S 201 116 :M f3_12 sf .26 .026( in the population is identical with the causal DAGs relative)J 59 134 :M (to )S f0_12 sf (V)S 80 134 :M f3_12 sf ( in the selected subpopulation and the unselected subpopulation.)S 77 158 :M .192 .019(This is the sort of assumption that is routinely made when, for example, the results of)J 59 176 :M .649 .065(drug trials conducted in Cleveland are generalized to the rest of the country. Of course,)J 59 194 :M .425 .042(there may be examples where the assumption is less plausible. For example, a drug may)J 59 212 :M (have no effect on outcome in men, but have an effect on women.)S 77 236 :M .366 .037(There are some subtleties about the application of these assumptions to different sets)J 59 254 :M .068 .007(of variables and different populations which are explained in more detail in the Appendix,)J 59 272 :M (but are not needed in order to understand the rest of the paper.)S 59 303 :M f0_14 sf (I)S 64 303 :M (I)S 69 303 :M (I)S 74 303 :M (.)S 77 303 :M ( )S 95 303 :M (Using Partial Ancestral Graphs)S 77 326 :M f3_12 sf 1.193 .119(Let us consider several different sets of conditional independence and dependence)J 59 344 :M .614 .061(relations, and what they can tell us about the causal DAGs that generated them, under a)J 59 362 :M (variety of different assumptions.)S 77 386 :M -.004(Given a causal graph )A 181 386 :M f4_12 sf (G)S 190 386 :M f3_12 sf -.005( over a set of variables )A f0_12 sf (V)S 310 386 :M f3_12 sf -.005(, we will say there is no selection bias)A 59 404 :M 1.185 .119(if and only if for any three disjoint sets of variables )J 328 404 :M f0_12 sf (X)S 337 404 :M f3_12 sf .606 .061(, )J f0_12 sf (Y)S 353 404 :M f3_12 sf 1.422 .142(, and )J 383 404 :M f0_12 sf .631(Z)A f3_12 sf 1.12 .112( included in )J 456 404 :M f0_12 sf (V)S 465 404 :M f3_12 sf <5C>S f0_12 sf (S)S 475 404 :M f3_12 sf 1.616 .162(, )J 483 404 :M f4_12 sf (G)S 59 422 :M f3_12 sf .494 .049(entails )J f0_12 sf (X)S 103 422 :M f3_12 sf 1.115 .111( )J 108 409 16 16 rC -1 -1 115 422 1 1 114 412 @b -1 -1 119 422 1 1 118 412 @b 111 423 -1 1 122 422 1 111 422 @a gR gS 0 0 552 730 rC 124 422 :M f3_12 sf .284 .028( )J f0_12 sf .835(Z)A f3_12 sf .538 .054( | \()J f0_12 sf (Y)S 159 422 :M f3_12 sf .316 .032( )J f1_12 sf 1.069A f0_12 sf .316 .032( )J f3_12 sf <28>S 180 422 :M f0_12 sf .929 .093(S = 1)J 208 422 :M f3_12 sf .832 .083(\)\) if and only if )J 290 422 :M f4_12 sf (G)S 299 422 :M f3_12 sf .624 .062( entails )J f0_12 sf (X)S 347 422 :M f3_12 sf 1.115 .111( )J 351 409 16 16 rC -1 -1 358 422 1 1 357 412 @b -1 -1 362 422 1 1 361 412 @b 354 423 -1 1 365 422 1 354 422 @a gR gS 0 0 552 730 rC 367 422 :M f3_12 sf .47 .047( )J f0_12 sf 1.379(Z)A f3_12 sf .658 .066( | )J 390 422 :M f0_12 sf (Y)S 399 422 :M f3_12 sf .697 .07(. This happens, for)J 59 440 :M .714 .071(example, when the variables in )J 217 440 :M f0_12 sf (S)S 224 440 :M f3_12 sf .729 .073( are causally unconnected to any other variables in )J 480 440 :M f0_12 sf (V)S 489 440 :M f3_12 sf (.)S 59 458 :M 3.622 .362(\(Note that this does not in general entail that the )J 348 458 :M f4_12 sf (distributions)S 409 458 :M f3_12 sf 3.125 .313( in\312the selected)J 59 476 :M .886 .089(subpopulation and the population are the same; it just entails that the same conditional)J 59 494 :M .222 .022(independence relations holds in both.\) In that case, when we depict a DAG in a figure we)J 59 512 :M (will omit the variables in )S 182 512 :M f0_12 sf (S)S 189 512 :M f3_12 sf (, and edges that have an endpoint in )S 367 512 :M f0_12 sf (S)S 374 512 :M f3_12 sf (.)S 77 536 :M 1.155 .115(For a given DAG )J 170 536 :M f4_12 sf (G)S 179 536 :M f3_12 sf 1.086 .109(, and a partition of the variable set )J 359 536 :M f0_12 sf (V)S 368 536 :M f3_12 sf .858 .086( of )J f4_12 sf (G)S 396 536 :M f3_12 sf 1.059 .106( into observed \()J 476 536 :M f0_12 sf (O)S f3_12 sf (\),)S 59 554 :M .356 .036(selection \()J 110 554 :M f0_12 sf (S)S 117 554 :M f3_12 sf .398 .04(\), and latent \()J f0_12 sf .224(L)A f3_12 sf .446 .045(\) variables, we will write )J f4_12 sf (G)S 326 554 :M f3_12 sf <28>S 330 554 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 349 554 :M f3_12 sf .077(,)A f0_12 sf .205(L)A f3_12 sf .437 .044(\). We assume that the only)J 59 572 :M 1.337 .134(conditional independence relations that can be tested are those among variables in )J 483 572 :M f0_12 sf (O)S 59 590 :M f3_12 sf 1.548 .155(conditional on any subset of )J f0_12 sf .825(O)A f3_12 sf 1.088 .109( when )J f0_12 sf (S)S 261 590 :M f3_12 sf 2.139 .214( = )J 279 590 :M f0_12 sf .815(1)A f3_12 sf 1.637 .164(; we will call this the set of )J 437 590 :M f0_12 sf (observable)S 59 608 :M f3_12 sf .199 .02(conditional independence relations. If )J 245 608 :M f0_12 sf (X)S 254 608 :M f3_12 sf .133 .013(, )J f0_12 sf (Y)S 269 608 :M f3_12 sf .202 .02(, and )J f0_12 sf .16(Z)A f3_12 sf .27 .027( are included in )J f0_12 sf .186(O)A f3_12 sf .202 .02(, and )J f0_12 sf (X)S 427 608 :M f3_12 sf .389 .039( )J 430 595 16 16 rC -1 -1 437 608 1 1 436 598 @b -1 -1 441 608 1 1 440 598 @b 433 609 -1 1 444 608 1 433 608 @a gR gS 0 0 552 730 rC 446 608 :M f3_12 sf .389 .039( )J 450 608 :M f0_12 sf .214(Z)A f3_12 sf .138 .014( | \()J f0_12 sf (Y)S 480 608 :M f3_12 sf .087 .009( )J f1_12 sf S 59 626 :M f3_12 sf <28>S 63 626 :M f0_12 sf .294 .029(S = 1)J 90 626 :M f3_12 sf .263 .026(\)\), then we say it is an )J 202 626 :M f0_12 sf (observed)S 248 626 :M f3_12 sf .217 .022( conditional independence relation. Let )J f0_12 sf .072(Cond)A f3_12 sf .097 .01( be a)J 59 644 :M .314 .031(set of conditional independence relations among the variables in )J 376 644 :M f0_12 sf .233(O)A f3_12 sf .306 .031(. A DAG )J f4_12 sf (G)S 442 644 :M f3_12 sf <28>S 446 644 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 465 644 :M f3_12 sf .062(,)A f0_12 sf .166(L)A f3_12 sf .239 .024(\) is)J 59 662 :M .052 .005(in )J f0_12 sf .035(O-Equiv)A f3_12 sf <28>S 119 662 :M f0_12 sf .075(Cond)A f3_12 sf .143 .014(\) just when )J f4_12 sf (G)S 213 662 :M f3_12 sf <28>S 217 662 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 236 662 :M f3_12 sf (,)S f0_12 sf .097(L)A f3_12 sf .168 .017(\) entails that )J f0_12 sf (X)S 319 662 :M f3_12 sf .332 .033( )J 322 649 16 16 rC -1 -1 329 662 1 1 328 652 @b -1 -1 333 662 1 1 332 652 @b 325 663 -1 1 336 662 1 325 662 @a gR gS 0 0 552 730 rC 338 662 :M f3_12 sf .332 .033( )J 342 662 :M f0_12 sf .26(Z)A f3_12 sf .175 .018( | \()J 362 662 :M f0_12 sf (Y)S 371 662 :M f3_12 sf .332 .033( )J 375 662 :M f1_12 sf .189A f0_12 sf .056 .006( )J f3_12 sf <28>S 391 662 :M f0_12 sf .276 .028(S = 1)J 418 662 :M f3_12 sf .237 .024(\)\) if and only if)J endp %%Page: 8 8 %%BeginPageSetup initializepage (peter; page: 8 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 263 701 24 24 rC 281 722 :M f3_12 sf (8)S gR gS 0 0 552 730 rC 59 56 :M f0_12 sf (X)S 68 56 :M f3_12 sf .304 .03( )J 71 43 16 16 rC -1 -1 78 56 1 1 77 46 @b -1 -1 82 56 1 1 81 46 @b 74 57 -1 1 85 56 1 74 56 @a gR gS 0 0 552 730 rC 87 56 :M f3_12 sf .089 .009( )J f0_12 sf .26(Z)A f3_12 sf .119 .012( | )J f0_12 sf (Y)S 116 56 :M f3_12 sf .129 .013( is in )J f0_12 sf .127(Cond)A f3_12 sf .123 .012(. If )J f4_12 sf .229<47D5>A 201 56 :M f3_12 sf <28>S 205 56 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf <53D5>S 228 56 :M f3_12 sf (,)S f0_12 sf <4CD5>S 243 56 :M f3_12 sf .178 .018(\) entails that )J f0_12 sf (X)S 315 56 :M f3_12 sf .304 .03( )J 319 43 16 16 rC -1 -1 326 56 1 1 325 46 @b -1 -1 330 56 1 1 329 46 @b 322 57 -1 1 333 56 1 322 56 @a gR gS 0 0 552 730 rC 335 56 :M f3_12 sf .078 .008( )J f0_12 sf .228(Z)A f3_12 sf .147 .015( | \()J f0_12 sf (Y)S 368 56 :M f3_12 sf .086 .009( )J f1_12 sf .292A f0_12 sf .086 .009( )J f3_12 sf <28>S 387 56 :M f0_12 sf .243 .024(S\325 = 1)J 418 56 :M f3_12 sf .217 .022(\)\) if and only if)J 59 74 :M f4_12 sf (G)S 68 74 :M f3_12 sf <28>S 72 74 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 91 74 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (\) entails that )S f0_12 sf (X)S 173 74 :M f3_12 sf ( )S 176 61 16 16 rC -1 -1 183 74 1 1 182 64 @b -1 -1 187 74 1 1 186 64 @b 179 75 -1 1 190 74 1 179 74 @a gR gS 0 0 552 730 rC 192 74 :M f3_12 sf ( )S f0_12 sf (Z)S f3_12 sf ( | \()S f0_12 sf (Y)S 224 74 :M f3_12 sf ( )S f1_12 sf S f0_12 sf ( )S f3_12 sf <28>S 243 74 :M f0_12 sf (S = 1)S 269 74 :M f3_12 sf (\)\), then )S 307 74 :M f4_12 sf <47D5>S 320 74 :M f3_12 sf <28>S 324 74 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf <53D5>S 347 74 :M f3_12 sf (,)S f0_12 sf <4CD5>S 362 74 :M f3_12 sf (\) is in )S f0_12 sf (O-Equiv)S f3_12 sf <28>S 440 74 :M f4_12 sf (G)S 449 74 :M f3_12 sf (\).)S 77 98 :M 2.033 .203(Imagine now that a researcher does not know the correct causal DAG, but can)J 59 116 :M .265 .026(determine whether an observed conditional independence relation is in )J f0_12 sf .087(Cond)A f3_12 sf .233 .023(, perhaps by)J 59 134 :M 1.071 .107(performing statistical tests of conditional independence on the selected subpopulation.)J 59 152 :M 1.009 .101(\(As we will see later, because many of the conditional independencies in )J f0_12 sf .387(Cond)A f3_12 sf 1.024 .102( entail)J 59 170 :M .355 .036(other members of )J 149 170 :M f0_12 sf .142(Cond)A f3_12 sf .324 .032(, only a small fraction of the membership of )J 396 170 :M f0_12 sf .081(Cond)A f3_12 sf .233 .023( actually need)J 59 188 :M .55 .055(be tested.\) From this information alone, and the Causal Markov Assumption, the Causal)J 59 206 :M .789 .079(Faithfulness Assumption, and the Population Inference Assumption, the most he or she)J 59 224 :M .983 .098(could conclude is that the true causal DAG is some member of )J f0_12 sf .368(O-Equiv)A f3_12 sf <28>S 431 224 :M f0_12 sf .203(Cond)A f3_12 sf .567 .057(\). This)J 59 242 :M .772 .077(information by itself is not very interesting, unless the members of )J 396 242 :M f0_12 sf (O-Equiv)S f3_12 sf <28>S 444 242 :M f0_12 sf .186(Cond)A f3_12 sf .36 .036(\) all)J 59 260 :M .09 .009(share some important features. The examples below show that sometimes the members of)J 59 278 :M f0_12 sf (O-Equiv)S f3_12 sf <28>S 107 278 :M f0_12 sf (Cond)S f3_12 sf (\) do share important features.)S 77 302 :M .718 .072(Our strategy for finding PAGs even when there may be latent variables or selection)J 59 320 :M .465 .047(bias, is a generalization of the strategy without selection bias described in Spirtes, et al.,)J 59 338 :M .417 .042(1993. )J 90 338 :M f4_12 sf .236(A)A f3_12 sf .247 .025( is an )J f0_12 sf .175(ancestor)A f3_12 sf .215 .021( of )J f4_12 sf .236(B)A f3_12 sf .376 .038( in DAG )J f4_12 sf (G)S 249 338 :M f3_12 sf .435 .044( when there is a directed path from )J 424 338 :M f4_12 sf .347(A)A f3_12 sf .302 .03( to )J f4_12 sf .347(B)A f3_12 sf .36 .036(, or )J f4_12 sf .347(A)A f3_12 sf .42 .042( =)J 59 356 :M f4_12 sf .617(B)A f3_12 sf 1.372 .137(. We will construct from )J 198 356 :M f0_12 sf .467(Cond)A f3_12 sf .983 .098( a graphical object called a )J f0_12 sf .385(partial)A 405 356 :M f4_12 sf .141 .014( )J f0_12 sf 1.73 .173(ancestral graph)J 59 374 :M f3_12 sf (\(PAG\))S 91 371 :M f3_8 sf (2)S 95 374 :M f3_12 sf 2.747 .275(, using the Causal Markov, Causal Faithfulness, and Population Inference)J 59 392 :M .077 .008(Assumptions. The PAG represents information about which variables are or not ancestors)J 59 410 :M .62 .062(of other variables in all of the DAGs in )J f0_12 sf .267(O-Equiv)A f3_12 sf <28>S 308 410 :M f0_12 sf .394(Cond)A f3_12 sf .454 .045(\). If )J f4_12 sf .412(A)A f3_12 sf .697 .07( is an ancestor of )J 456 410 :M f4_12 sf .478(B)A f3_12 sf .685 .068( in all)J 59 428 :M .27 .027(DAGs in )J f0_12 sf .103(O-Equiv)A f3_12 sf <28>S 154 428 :M f0_12 sf .164(Cond)A f3_12 sf .396 .04(\), then although from )J f0_12 sf .164(Cond)A f3_12 sf .43 .043( we cannot tell exactly which DAG)J 59 446 :M .092 .009(in )J f0_12 sf .061(O-Equiv)A f3_12 sf <28>S 119 446 :M f0_12 sf .175(Cond)A f3_12 sf .374 .037(\) is the true causal DAG, because we know that all of the DAGs in )J 479 446 :M f0_12 sf (O-)S 59 464 :M (Equiv)S 90 464 :M f3_12 sf <28>S 94 464 :M f0_12 sf .349(Cond)A f3_12 sf .804 .08(\) contain a directed path from )J 276 464 :M f4_12 sf .466(A)A f3_12 sf .406 .041( to )J f4_12 sf .466(B)A f3_12 sf 1.042 .104( we can reliably conclude that in the)J 59 482 :M 1.933 .193(true causal DAG )J 152 482 :M f4_12 sf .913(A)A f3_12 sf 1.864 .186( is a \(possibly indirect\) cause of )J 335 482 :M f4_12 sf .587(B)A f3_12 sf 1.596 .16(. This strategy is represented)J 59 500 :M 1.773 .177(schematically in Figure 3. In the following examples we will apply this strategy to)J 59 518 :M .76 .076(particular sets of observed conditional independence relations. We will also show what)J 59 536 :M (features of DAGs can be reliably inferred, and what features cannot.)S 59 664 :M ( )S 59 661.48 -.48 .48 203.48 661 .48 59 661 @a 77 673 :M f3_8 sf (2)S 81 676 :M f3_10 sf .476 .048( A similar object was called a partially oriented inducing path graph \(POIPG\) in Spirtes et al. 1993.)J 59 687 :M (PAGs in effect represent a kind of equivalence class of what Wermuth et al.\(1994\) call summary graphs.)S endp %%Page: 9 9 %%BeginPageSetup initializepage (peter; page: 9 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 263 701 24 24 rC 281 722 :M f3_12 sf (9)S gR gS 214 41 34 13 rC 214 50 :M f0_12 sf (C)S 223 50 :M (o)S 230 50 :M (n)S 237 50 :M (d)S gR gS 77 193 404 44 rC 77 211 :M f4_12 sf (G)S 86 211 :M f3_12 sf ( )S 89 214 :M f3_7 sf (1)S 93 211 :M f3_12 sf <28>S 97 211 :M f0_12 sf (O)S 106 211 :M f3_12 sf (,)S 110 211 :M f0_12 sf (S)S 118 214 :M f0_7 sf (1)S 122 211 :M f3_12 sf (,)S 126 211 :M f0_12 sf (L)S 134 214 :M f0_7 sf (1)S 138 211 :M f3_12 sf <29>S 142 211 :M ( )S 145 211 :M f4_12 sf (G)S 154 211 :M f3_12 sf ( )S 157 214 :M f3_7 sf (2)S 161 211 :M f3_12 sf <28>S 165 211 :M f0_12 sf (O)S 174 211 :M f3_12 sf (,)S 178 211 :M f0_12 sf (S)S 186 214 :M f0_7 sf (2)S 190 211 :M f3_12 sf (,)S 194 211 :M f0_12 sf (L)S 202 214 :M f0_7 sf (2)S 206 211 :M f3_12 sf <29>S 210 211 :M ( )S 213 211 :M f4_12 sf (G)S 222 211 :M f3_12 sf ( )S 225 214 :M f3_7 sf (3)S 229 211 :M f3_12 sf <28>S 233 211 :M f0_12 sf (O)S 242 211 :M f3_12 sf (,)S 246 211 :M f0_12 sf (S)S 254 214 :M f0_7 sf (3)S 258 211 :M f3_12 sf (,)S 262 211 :M f0_12 sf (L)S 270 214 :M f0_7 sf (3)S 274 211 :M f3_12 sf <29>S 278 211 :M ( )S 281 211 :M f4_12 sf (G)S 290 211 :M f3_12 sf ( )S 293 214 :M f3_7 sf (4)S 297 211 :M f3_12 sf <28>S 301 211 :M f0_12 sf (O)S 310 211 :M f3_12 sf (,)S 314 211 :M f0_12 sf (S)S 322 214 :M f0_7 sf (4)S 326 211 :M f3_12 sf (,)S 330 211 :M f0_12 sf (L)S 338 214 :M f0_7 sf (1)S 342 211 :M f3_12 sf <29>S 346 211 :M ( )S 349 211 :M ( )S 352 211 :M ( )S 355 211 :M ( )S 358 211 :M ( )S 361 211 :M f3_24 sf (.)S 369 211 :M (.)S 377 211 :M (.)S 385 211 :M (.)S 393 211 :M f0_12 sf (O)S 402 211 :M (-)S 407 211 :M (E)S 415 211 :M (q)S 422 211 :M (u)S 429 211 :M (i)S 433 211 :M (v)S 440 211 :M f3_12 sf <28>S 444 211 :M f0_12 sf (C)S 453 211 :M (o)S 460 211 :M (n)S 467 211 :M (d)S 474 211 :M f3_12 sf <29>S gR gS 205 120 39 13 rC 205 129 :M f3_12 sf ( )S 208 129 :M ( )S 211 129 :M ( )S 214 129 :M (P)S 221 129 :M (A)S 229 129 :M (G)S gR gS 314 137 56 37 rC 317 146 :M f3_12 sf (r)S 321 146 :M (e)S 326 146 :M (p)S 332 146 :M (r)S 336 146 :M (e)S 341 146 :M (s)S 346 146 :M (e)S 351 146 :M (n)S 357 146 :M (t)S 360 146 :M (s)S 317 158 :M (f)S 321 158 :M (e)S 326 158 :M (a)S 331 158 :M (t)S 334 158 :M (u)S 340 158 :M (r)S 344 158 :M (e)S 349 158 :M (s)S 354 158 :M ( )S 357 158 :M (i)S 360 158 :M (n)S 321 170 :M (c)S 326 170 :M (o)S 332 170 :M (m)S 341 170 :M (m)S 350 170 :M (o)S 356 170 :M (n)S gR gS 77 41 404 196 rC np 226 117 :M 223 105 :L 226 105 :L 229 105 :L 226 117 :L 226 117 :L eofill 223 106 -1 1 227 117 1 223 105 @a 223 106 -1 1 227 105 1 223 105 @a 225 106 -1 1 230 105 1 225 105 @a -1 -1 226 118 1 1 229 105 @b -1 -1 227 107 1 1 226 55 @b 261 73 162 37 rC 275 82 :M f3_12 sf (C)S 283 82 :M (a)S 288 82 :M (u)S 294 82 :M (s)S 299 82 :M (a)S 304 82 :M (l)S 307 82 :M ( )S 310 82 :M (M)S 320 82 :M (a)S 325 82 :M (r)S 329 82 :M (k)S 335 82 :M (o)S 341 82 :M (v)S 347 82 :M ( )S 350 82 :M (A)S 358 82 :M (s)S 363 82 :M (s)S 368 82 :M (u)S 374 82 :M (m)S 383 82 :M (p)S 389 82 :M (t)S 392 82 :M (i)S 395 82 :M (o)S 401 82 :M (n)S 265 94 :M (C)S 273 94 :M (a)S 278 94 :M (u)S 284 94 :M (s)S 289 94 :M (a)S 294 94 :M (l)S 297 94 :M ( )S 300 94 :M (F)S 307 94 :M (a)S 312 94 :M (i)S 315 94 :M (t)S 318 94 :M (h)S 324 94 :M (f)S 328 94 :M (u)S 334 94 :M (l)S 337 94 :M (n)S 343 94 :M (e)S 348 94 :M (s)S 353 94 :M (s)S 358 94 :M ( )S 361 94 :M (A)S 369 94 :M (s)S 374 94 :M (s)S 379 94 :M (u)S 385 94 :M (m)S 394 94 :M (p)S 400 94 :M (t)S 403 94 :M (i)S 406 94 :M (o)S 412 94 :M (n)S 262 106 :M (P)S 269 106 :M (o)S 275 106 :M (p)S 281 106 :M (u)S 287 106 :M (l)S 290 106 :M (a)S 295 106 :M (t)S 298 106 :M (i)S 301 106 :M (o)S 307 106 :M (n)S 313 106 :M ( )S 316 106 :M (I)S 320 106 :M (n)S 326 106 :M (f)S 330 106 :M (e)S 335 106 :M (r)S 339 106 :M (e)S 344 106 :M (n)S 350 106 :M (c)S 355 106 :M (e)S 360 106 :M ( )S 363 106 :M (A)S 371 106 :M (s)S 376 106 :M (s)S 381 106 :M (u)S 387 106 :M (m)S 396 106 :M (p)S 402 106 :M (t)S 405 106 :M (i)S 408 106 :M (o)S 414 106 :M (n)S gR gS 77 41 404 196 rC 224 139 -1 1 263 191 1 224 138 @a -1 -1 193 192 1 1 226 139 @b -1 -1 127 191 1 1 225 138 @b 226 139 -1 1 320 192 1 226 138 @a 296 42 49 13 rC 296 51 :M f3_12 sf (O)S 304 51 :M (b)S 310 51 :M (s)S 315 51 :M (e)S 320 51 :M (r)S 324 51 :M (v)S 330 51 :M (e)S 335 51 :M (d)S gR gS 314 52 16 19 rC 314 67 :M f3_18 sf (+)S gR gS 0 0 552 730 rC 263 258 :M f0_12 sf (Figure )S 300 258 :M (3)S 77 282 :M f3_12 sf .222 .022(The formal definition of a PAG is given below. There are three kinds of endpoints an)J 59 300 :M .417 .042(edge in a PAG can have: \322-\323, \322o\323, or \322>\323. These can be combined to form the following)J 59 318 :M .119 .012(four kinds of edges: )J 159 318 :M f4_12 sf .059(A)A f3_12 sf ( )S f1_12 sf S 181 318 :M f3_12 sf ( )S f4_12 sf .1(B)A f3_12 sf .069 .007(, )J f4_12 sf .1(A)A f3_12 sf ( )S f1_12 sf S 220 318 :M f3_12 sf .179 .018( )J 224 318 :M f4_12 sf .063(B)A f3_12 sf .043 .004(, )J f4_12 sf .063(A)A f3_12 sf .065 .006( o)J f1_12 sf S 265 318 :M f3_12 sf .05 .005( )J f4_12 sf .134(B)A f3_12 sf .145 .014(, or )J 295 318 :M f4_12 sf .054(A)A f3_12 sf .055 .005( o)J f1_12 sf .088A f3_12 sf .055 .005(o )J f4_12 sf .054(B)A f3_12 sf .123 .012(. Let \322*\323 be a meta-symbol that)J 59 336 :M (stands for any of the three kinds of endpoints. More formally:)S 77 360 :M (A PAG )S 116 360 :M f5_12 sf (p)S 123 360 :M f3_12 sf ( )S f0_12 sf (represents)S f3_12 sf ( a DAG )S f4_12 sf (G)S 228 360 :M f3_12 sf <28>S 232 360 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (,)S f0_12 sf (S)S 262 360 :M f3_12 sf (\) if and only:)S 77 384 :M (1)S 83 384 :M (.)S 86 384 :M ( )S 95 384 :M (The set of variables in )S 205 384 :M f5_12 sf (p)S 212 384 :M f3_12 sf ( is )S f0_12 sf (O)S f3_12 sf (.)S 77 406 :M (2)S 83 406 :M (.)S 86 406 :M 6 .6( )J 95 406 :M .147 .015(If there is any edge between )J f4_12 sf .07(A)A f3_12 sf .092 .009( and )J 265 406 :M f4_12 sf .1(B)A f3_12 sf .087 .009( in )J f5_12 sf (p)S 294 406 :M f3_12 sf .139 .014(, it is one of the following kinds: )J 456 406 :M f4_12 sf .066(A)A f3_12 sf ( )S f1_12 sf S 478 406 :M f3_12 sf ( )S f4_12 sf .11(B)A f3_12 sf (,)S 95 423 :M f4_12 sf (A)S f3_12 sf ( )S f1_12 sf <6FAE>S f3_12 sf ( )S f4_12 sf (B)S f3_12 sf (, )S f4_12 sf (A)S f3_12 sf ( )S f1_12 sf <6FBE6F>S f3_12 sf ( )S f4_12 sf (B)S f3_12 sf (, or )S 203 423 :M f4_12 sf (A)S f3_12 sf ( )S f1_12 sf S 226 423 :M f3_12 sf ( )S f4_12 sf (B)S f3_12 sf (.)S 77 448 :M (3)S 83 448 :M (.)S 86 448 :M ( )S 95 448 :M (There is at most one edge between any pair of vertices in )S 372 448 :M f5_12 sf (p)S 379 448 :M f3_12 sf (.)S 77 470 :M (4)S 83 470 :M (.)S 86 470 :M 6 .6( )J 95 470 :M f4_12 sf .16(A)A f3_12 sf .204 .02( and )J f4_12 sf .16(B)A f3_12 sf .291 .029( are adjacent in )J f5_12 sf (p)S 217 470 :M f3_12 sf .313 .031( if and only if for every subset )J 369 470 :M f0_12 sf .292(Z)A f3_12 sf .254 .025( of )J 394 470 :M f0_12 sf .076(O)A f3_12 sf .037(\\{)A f4_12 sf .06(A)A f3_12 sf (,)S f4_12 sf .06(B)A f3_12 sf .06 .006(} )J f4_12 sf (G)S 447 470 :M f3_12 sf .313 .031( does not)J 95 487 :M (entail that )S 146 487 :M f4_12 sf (A)S f3_12 sf ( and )S f4_12 sf (B)S f3_12 sf ( are independent conditional on )S f0_12 sf (Z )S f1_12 sf S f3_12 sf ( )S f0_12 sf (S)S 367 487 :M f3_12 sf (.)S 77 510 :M (5)S 83 510 :M (.)S 86 510 :M 6 .6( )J 95 510 :M .627 .063(An edge between )J f4_12 sf .245(A)A f3_12 sf .312 .031( and )J f4_12 sf .245(B)A f3_12 sf .223 .022( in )J 238 510 :M f5_12 sf (p)S 245 510 :M f3_12 sf .518 .052( is oriented as )J f4_12 sf .307(A)A f3_12 sf .125 .013( )J 327 510 :M f1_12 sf S 339 510 :M f3_12 sf .667 .067( )J 343 510 :M f4_12 sf .326(B)A f3_12 sf .441 .044( only if )J 390 510 :M f4_12 sf .247(A)A f3_12 sf .412 .041( is an ancestor of )J f4_12 sf (B)S 95 523 :M f3_12 sf (but not )S 132 523 :M f0_12 sf (S)S 139 523 :M f3_12 sf ( in every DAG in )S f0_12 sf (O-Equiv)S f3_12 sf <28>S 273 523 :M f4_12 sf (G)S 282 523 :M f3_12 sf (\).)S 77 545 :M (6)S 83 545 :M (.)S 86 545 :M 6 .6( )J 95 545 :M .522 .052(An edge between )J f4_12 sf .204(A)A f3_12 sf .26 .026( and )J f4_12 sf .204(B)A f3_12 sf .178 .018( in )J f5_12 sf (p)S 244 545 :M f3_12 sf .423 .042( is oriented as )J 316 545 :M f4_12 sf .151(A)A f3_12 sf .155 .015( *)J f1_12 sf S 344 545 :M f3_12 sf .582 .058( )J 348 545 :M f4_12 sf .284(B)A f3_12 sf .384 .038( only if )J 395 545 :M f4_12 sf .182(B)A f3_12 sf .37 .037( is not an ancestor)J 95 558 :M (of )S 108 558 :M f4_12 sf (A)S f3_12 sf ( or )S 134 558 :M f0_12 sf (S)S 141 558 :M f3_12 sf ( in every DAG in )S f0_12 sf (O-Equiv)S f3_12 sf <28>S 275 558 :M f4_12 sf (G)S 284 558 :M f3_12 sf (\).)S 77 580 :M (7)S 83 580 :M (.)S 86 580 :M 6 .6( )J 95 580 :M f4_12 sf .081(A)A f3_12 sf .082 .008( *)J f1_12 sf S 123 0 6 730 rC 123 580 :M f3_12 sf 12 f6_1 :p 3.312 :m .312 .031( )J 126 580 :M 6.623 :m .297 .03( )J gR gS 0 0 552 730 rC 123 580 :M f3_12 sf 12 f6_1 :p 9.312 :m .283 .028(* )J 129 0 3 730 rC 129 580 :M 6.623 :m .297 .03( )J gR gS 132 0 7 730 rC 132 580 :M f4_12 sf 12 f7_1 :p 6.623 :m .297 .03( )J 136 580 :M 6.623 :m .297 .03( )J gR gS 0 0 552 730 rC 132 580 :M f4_12 sf 12 f7_1 :p 7.33 :m (B)S 132 0 7 730 rC 132 580 :M 6.623 :m .297 .03( )J 136 580 :M 6.623 :m .297 .03( )J gR gS 139 0 6 730 rC 139 580 :M f3_12 sf 12 f6_1 :p 3.312 :m .312 .031( )J 142 580 :M 6.623 :m .297 .03( )J gR gS 0 0 552 730 rC 139 580 :M f3_12 sf 12 f6_1 :p 6 :m (*)S 139 0 6 730 rC 139 580 :M 3.312 :m .312 .031( )J 142 580 :M 6.623 :m .297 .03( )J gR gS 0 0 552 730 rC 145 580 :M f1_12 sf .178A f3_12 sf .121 .012(* )J 167 580 :M f4_12 sf .167(C)A f3_12 sf .133 .013( in )J f5_12 sf (p)S 197 580 :M f1_12 sf .312 .031( )J 201 580 :M f3_12 sf .158 .016(only if in every DAG in )J f0_12 sf .071(O-Equiv)A f3_12 sf <28>S 368 580 :M f4_12 sf (G)S 377 580 :M f3_12 sf .222 .022(\) either )J 415 580 :M f4_12 sf .091(B)A f3_12 sf .192 .019( is an ancestor)J 95 593 :M .077 .008(of )J f4_12 sf .062(C)A f3_12 sf .059 .006(, or )J f4_12 sf .057(A)A f3_12 sf .061 .006(, or )J 162 593 :M f0_12 sf (S)S 169 593 :M f3_12 sf .068 .007(. \(Suppose that )J 244 593 :M f4_12 sf (A)S f3_12 sf .049 .005( and )J f4_12 sf (B)S f3_12 sf .078 .008( are adjacent, and )J f4_12 sf (B)S f3_12 sf .051 .005( and )J 399 593 :M f4_12 sf (C)S f3_12 sf .06 .006( are adjacent, and)J 95 605 :M f4_12 sf .42(A)A f3_12 sf .534 .053( and )J f4_12 sf .458(C)A f3_12 sf .833 .083( are not adjacent, and the edges in the PAG are not both into )J 444 605 :M f4_12 sf .38(B)A f3_12 sf .686 .069(, i.e. the)J 95 621 :M 1.472 .147(PAG does not contain )J 213 621 :M f4_12 sf .574(A)A f3_12 sf .588 .059( *)J f1_12 sf S 243 621 :M f3_12 sf .641 .064( )J f4_12 sf 1.723(B)A f3_12 sf .641 .064( )J f1_12 sf S 277 621 :M f3_12 sf 2.007 .201(* )J 289 621 :M f4_12 sf .676(C)A f3_12 sf 1.382 .138(. Then the underlining of )J 431 621 :M f4_12 sf .571(B)A f3_12 sf 1.326 .133( should be)J 95 638 :M (assumed to be present, although we do not explicitly put the underlining in )S 457 638 :M f5_12 sf (p)S 464 638 :M f3_12 sf (.\))S 77 663 :M .477 .048(A \322o\323 on the end of an edge places no restriction on the ancestor relationships. Note)J 59 681 :M .298 .03(that more than one PAG can represent a DAG )J 287 681 :M f4_12 sf (G)S 296 681 :M f3_12 sf <28>S 300 681 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (,)S f0_12 sf (S)S 330 681 :M f3_12 sf .312 .031(\). Two such PAGs )J 424 681 :M f5_12 sf (p)S 431 684 :M f3_7 sf (1)S 435 681 :M f3_12 sf .273 .027( and )J f5_12 sf (p)S 466 684 :M f3_7 sf (2)S 470 681 :M f3_12 sf .312 .031( that)J endp %%Page: 10 10 %%BeginPageSetup initializepage (peter; page: 10 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (10)S gR gS 0 0 552 730 rC 59 56 :M f3_12 sf .533 .053(represent )J f4_12 sf (G)S 116 56 :M f3_12 sf <28>S 120 56 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (,)S f0_12 sf (S)S 150 56 :M f3_12 sf .776 .078(\) have the same adjacencies, but are differently oriented in the sense)J 59 74 :M .669 .067(that )J 81 74 :M f5_12 sf (p)S 88 77 :M f3_7 sf (1)S 92 74 :M f3_12 sf .669 .067( may contain a \322o\323 where )J 223 74 :M f5_12 sf (p)S 230 77 :M f3_7 sf (2)S 234 74 :M f3_12 sf .643 .064( contains a \322>\323 or a \322)J f1_12 sf (-)S 347 74 :M f3_12 sf .551 .055(\323, or vice-versa. Examples of)J 59 92 :M (PAGs are shown in the following subsections.)S 95 116 :M f8_12 sf (A)S 103 116 :M (.)S 106 116 :M ( )S 131 116 :M (Example 1)S 77 140 :M f3_12 sf .037 .004(We will start out with a very simple example, in which the set of observed conditional)J 59 158 :M 1.072 .107(independence relations is not very informative. \(For simplicity, in all of the following)J 59 176 :M .434 .043(examples we assume that all of the variables in )J 294 176 :M f0_12 sf (S)S 301 176 :M f3_12 sf .452 .045( take on the same value, and hence can)J 59 194 :M .554 .055(be represented by a single variables )J f4_12 sf .185(S)A f3_12 sf .49 .049(.\) For example, suppose first that the set )J f0_12 sf .216(Cond)A f0_8 sf 0 3 rm .124(1)A 0 -3 rm f3_13 sf .091 .009( )J f3_12 sf .309(of)A 59 212 :M .554 .055(observed conditional independence relations is empty, i.e. )J f0_12 sf .178(Cond)A f0_8 sf 0 3 rm .102(1)A 0 -3 rm f3_13 sf .075 .007( )J f3_12 sf .226 .023(= )J 393 212 :M f1_12 sf S 403 212 :M f3_12 sf .67 .067(. We now want to)J 59 230 :M 2.047 .205(find out what DAGs are in )J 208 230 :M f0_12 sf (O-Equiv)S f3_12 sf <28>S 256 230 :M f0_12 sf .666(Cond)A f0_8 sf 0 3 rm .38(1)A 0 -3 rm f3_12 sf 1.06 .106(\). Let )J 323 230 :M f0_12 sf (V)S 332 230 :M f3_12 sf 1.912 .191( be a set of causally sufficient)J 59 248 :M .314 .031(variables. Suppose that we assume or know from background knowledge that )J 440 248 :M f0_12 sf .348(O)A f3_12 sf .301 .03( = {)J 469 248 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf (})S 59 266 :M .814 .081(is causally sufficient and there is no selection bias. \(In general it is not possible to test)J 59 284 :M 1.648 .165(these assumption from observational data alone.\) Under these assumptions there are)J 59 302 :M 1.138 .114(exactly two DAGs that entail )J f0_12 sf .44(Cond)A f0_8 sf 0 3 rm .252(1)A 0 -3 rm f3_12 sf .801 .08(, labeled \(i\) and \(ii\) in )J 360 302 :M .996 .1(Figure 4. In general, when)J 59 320 :M .059 .006(there are no latent variables and no selection bias, there is an edge between )J 424 320 :M f4_12 sf (A)S f3_12 sf .05 .005( and )J f4_12 sf (B)S f3_12 sf .063 .006( if and)J 59 338 :M .591 .059(only if for any subset )J 169 338 :M f0_12 sf (X)S 178 338 :M f3_12 sf .266 .027( of )J f0_12 sf .373(O)A f3_12 sf .182(\\{)A f4_12 sf .293(A)A f3_12 sf .12(,)A f4_12 sf .293(B)A f3_12 sf .392 .039(}, )J 243 338 :M f4_12 sf (G)S 252 338 :M f3_12 sf <28>S 256 338 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 275 338 :M f3_12 sf .105(,)A f0_12 sf .279(L)A f3_12 sf .493 .049(\) entails that )J 351 338 :M f4_12 sf .407(A)A f3_12 sf .54 .054( and )J 382 338 :M f4_12 sf .175(B)A f3_12 sf .485 .049( are dependent given)J 59 356 :M f0_12 sf (X)S 68 356 :M f3_12 sf (.)S 77 383 419 187 rC 305.5 463.5 114 79 rS 307 465 111 76 rC 307 474 :M ( )S 310 474 :M ( )S 313 474 :M ( )S 316 474 :M ( )S 319 474 :M ( )S 322 474 :M ( )S 325 474 :M ( )S 328 474 :M ( )S 331 474 :M ( )S 334 474 :M ( )S 337 474 :M ( )S 340 474 :M ( )S 343 474 :M ( )S 346 474 :M ( )S 349 474 :M ( )S 352 474 :M f4_12 sf ( )S 355 474 :M (T)S 307 498 :M ( )S 310 498 :M ( )S 313 498 :M ( )S 316 498 :M ( )S 319 498 :M ( )S 322 498 :M ( )S 325 498 :M ( )S 328 498 :M ( )S 331 498 :M ( )S 334 498 :M ( )S 337 498 :M ( )S 340 498 :M ( )S 343 498 :M ( )S 346 498 :M ( )S 349 498 :M ( )S 352 498 :M ( )S 355 498 :M ( )S 358 498 :M (U)S 307 522 :M ( )S 310 522 :M ( )S 313 522 :M ( )S 316 522 :M ( )S 319 522 :M (A)S 327 522 :M ( )S 330 522 :M ( )S 333 522 :M ( )S 336 522 :M ( )S 339 522 :M ( )S 342 522 :M ( )S 345 522 :M ( )S 348 522 :M ( )S 351 522 :M ( )S 354 522 :M ( )S 357 522 :M ( )S 360 522 :M ( )S 363 522 :M ( )S 366 522 :M ( )S 369 522 :M ( )S 372 522 :M ( )S 375 522 :M ( )S 378 522 :M ( )S 381 522 :M ( )S 384 522 :M ( )S 387 522 :M ( )S 390 522 :M (B)S 307 534 :M ( )S 310 534 :M ( )S 313 534 :M ( )S 316 534 :M ( )S 319 534 :M ( )S 322 534 :M ( )S 325 534 :M ( )S 328 534 :M ( )S 331 534 :M ( )S 334 534 :M ( )S 337 534 :M ( )S 340 534 :M ( )S 343 534 :M ( )S 346 534 :M ( )S 349 534 :M ( )S 352 534 :M f3_12 sf <28>S 356 534 :M (v)S 362 534 :M (i)S 365 534 :M <29>S gR gS 77 383 419 187 rC 191.5 463.5 114 79 rS 193 465 111 76 rC 193 474 :M f3_12 sf ( )S 196 474 :M ( )S 199 474 :M ( )S 202 474 :M ( )S 205 474 :M ( )S 208 474 :M ( )S 211 474 :M ( )S 214 474 :M ( )S 217 474 :M ( )S 220 474 :M ( )S 223 474 :M ( )S 226 474 :M ( )S 229 474 :M ( )S 232 474 :M ( )S 235 474 :M ( )S 238 474 :M ( )S 241 474 :M ( )S 244 474 :M ( )S 247 474 :M ( )S 250 474 :M ( )S 253 474 :M ( )S 256 474 :M ( )S 259 474 :M ( )S 262 474 :M ( )S 265 474 :M f4_12 sf (T)S 193 510 :M f3_12 sf ( )S 196 510 :M ( )S 199 510 :M ( )S 202 510 :M ( )S 205 510 :M f4_12 sf (A)S 213 510 :M ( )S 216 510 :M ( )S 219 510 :M ( )S 222 510 :M ( )S 225 510 :M ( )S 228 510 :M ( )S 231 510 :M ( )S 234 510 :M ( )S 237 510 :M ( )S 240 510 :M (S)S 247 510 :M ( )S 250 510 :M ( )S 253 510 :M ( )S 256 510 :M ( )S 259 510 :M ( )S 262 510 :M ( )S 265 510 :M ( )S 268 510 :M ( )S 271 510 :M ( )S 274 510 :M ( )S 277 510 :M (B)S 193 534 :M f3_12 sf ( )S 196 534 :M ( )S 199 534 :M ( )S 202 534 :M ( )S 205 534 :M ( )S 208 534 :M ( )S 211 534 :M ( )S 214 534 :M ( )S 217 534 :M ( )S 220 534 :M ( )S 223 534 :M ( )S 226 534 :M ( )S 229 534 :M ( )S 232 534 :M ( )S 235 534 :M ( )S 238 534 :M ( )S 241 534 :M <28>S 245 534 :M (v)S 251 534 :M <29>S gR gS 77 383 419 187 rC 77.5 463.5 114 79 rS 79 465 111 76 rC 79 498 :M f3_12 sf ( )S 82 498 :M ( )S 85 498 :M ( )S 88 498 :M ( )S 91 498 :M f4_12 sf (A)S 99 498 :M ( )S 102 498 :M ( )S 105 498 :M ( )S 108 498 :M ( )S 111 498 :M ( )S 114 498 :M ( )S 117 498 :M ( )S 120 498 :M ( )S 123 498 :M ( )S 126 498 :M (S)S 133 498 :M ( )S 136 498 :M ( )S 139 498 :M ( )S 142 498 :M ( )S 145 498 :M ( )S 148 498 :M ( )S 151 498 :M ( )S 154 498 :M ( )S 157 498 :M ( )S 160 498 :M ( )S 163 498 :M (B)S 79 534 :M f3_12 sf ( )S 82 534 :M ( )S 85 534 :M ( )S 88 534 :M ( )S 91 534 :M ( )S 94 534 :M ( )S 97 534 :M ( )S 100 534 :M ( )S 103 534 :M ( )S 106 534 :M ( )S 109 534 :M ( )S 112 534 :M ( )S 115 534 :M ( )S 118 534 :M ( )S 121 534 :M ( )S 124 534 :M ( )S 127 534 :M <28>S 131 534 :M (i)S 134 534 :M (v)S 140 534 :M <29>S gR gS 77 383 419 187 rC 306.5 383.5 114 79 rS 308 385 111 76 rC 308 406 :M f3_12 sf ( )S 311 406 :M ( )S 314 406 :M ( )S 317 406 :M ( )S 320 406 :M ( )S 323 406 :M ( )S 326 406 :M ( )S 329 406 :M ( )S 332 406 :M ( )S 335 406 :M ( )S 338 406 :M ( )S 341 406 :M ( )S 344 406 :M ( )S 347 406 :M ( )S 350 406 :M ( )S 353 406 :M ( )S 356 406 :M ( )S 359 406 :M ( )S 362 406 :M f4_12 sf (T)S 308 442 :M f3_12 sf ( )S 311 442 :M ( )S 314 442 :M ( )S 317 442 :M f4_12 sf (A)S 325 442 :M ( )S 328 442 :M ( )S 331 442 :M ( )S 334 442 :M ( )S 337 442 :M ( )S 340 442 :M ( )S 343 442 :M ( )S 346 442 :M ( )S 349 442 :M ( )S 352 442 :M ( )S 355 442 :M ( )S 358 442 :M ( )S 361 442 :M ( )S 364 442 :M ( )S 367 442 :M ( )S 370 442 :M ( )S 373 442 :M ( )S 376 442 :M ( )S 379 442 :M ( )S 382 442 :M ( )S 385 442 :M ( )S 388 442 :M ( )S 391 442 :M ( )S 394 442 :M (B)S 308 454 :M f3_12 sf ( )S 311 454 :M ( )S 314 454 :M ( )S 317 454 :M ( )S 320 454 :M ( )S 323 454 :M ( )S 326 454 :M ( )S 329 454 :M ( )S 332 454 :M ( )S 335 454 :M ( )S 338 454 :M ( )S 341 454 :M ( )S 344 454 :M ( )S 347 454 :M ( )S 350 454 :M ( )S 353 454 :M ( )S 356 454 :M <28>S 360 454 :M (i)S 363 454 :M (i)S 366 454 :M (i)S 369 454 :M <29>S gR gS 77 383 419 187 rC 192.5 383.5 114 79 rS 194 385 111 76 rC 194 418 :M f3_12 sf ( )S 197 418 :M ( )S 200 418 :M ( )S 203 418 :M f4_12 sf (A)S 211 418 :M ( )S 214 418 :M ( )S 217 418 :M ( )S 220 418 :M ( )S 223 418 :M ( )S 226 418 :M ( )S 229 418 :M ( )S 232 418 :M ( )S 235 418 :M ( )S 238 418 :M ( )S 241 418 :M ( )S 244 418 :M ( )S 247 418 :M ( )S 250 418 :M ( )S 253 418 :M ( )S 256 418 :M ( )S 259 418 :M ( )S 262 418 :M ( )S 265 418 :M ( )S 268 418 :M ( )S 271 418 :M ( )S 274 418 :M ( )S 277 418 :M ( )S 280 418 :M (B)S 194 454 :M ( )S 197 454 :M ( )S 200 454 :M ( )S 203 454 :M ( )S 206 454 :M ( )S 209 454 :M ( )S 212 454 :M ( )S 215 454 :M ( )S 218 454 :M ( )S 221 454 :M ( )S 224 454 :M ( )S 227 454 :M ( )S 230 454 :M ( )S 233 454 :M ( )S 236 454 :M ( )S 239 454 :M f3_12 sf <28>S 243 454 :M (i)S 246 454 :M (i)S 249 454 :M <29>S gR gS 77 383 419 187 rC 78.5 383.5 114 79 rS 80 385 111 76 rC 80 418 :M f3_12 sf ( )S 83 418 :M ( )S 86 418 :M ( )S 89 418 :M ( )S 92 418 :M f4_12 sf (A)S 100 418 :M ( )S 103 418 :M ( )S 106 418 :M ( )S 109 418 :M ( )S 112 418 :M ( )S 115 418 :M ( )S 118 418 :M ( )S 121 418 :M ( )S 124 418 :M ( )S 127 418 :M ( )S 130 418 :M ( )S 133 418 :M ( )S 136 418 :M ( )S 139 418 :M ( )S 142 418 :M ( )S 145 418 :M ( )S 148 418 :M ( )S 151 418 :M ( )S 154 418 :M ( )S 157 418 :M ( )S 160 418 :M ( )S 163 418 :M ( )S 166 418 :M ( )S 169 418 :M (B)S 80 454 :M f3_12 sf ( )S 83 454 :M ( )S 86 454 :M ( )S 89 454 :M ( )S 92 454 :M ( )S 95 454 :M ( )S 98 454 :M ( )S 101 454 :M ( )S 104 454 :M ( )S 107 454 :M ( )S 110 454 :M ( )S 113 454 :M ( )S 116 454 :M ( )S 119 454 :M ( )S 122 454 :M ( )S 125 454 :M ( )S 128 454 :M <28>S 132 454 :M (i)S 135 454 :M <29>S gR gS 77 383 419 187 rC 107 416 -1 1 160 415 1 107 415 @a np 157 411 :M 157 419 :L 165 415 :L 157 411 :L eofill -1 -1 158 420 1 1 157 411 @b -1 -1 158 420 1 1 165 415 @b 157 412 -1 1 166 415 1 157 411 @a 222 415 -1 1 276 414 1 222 414 @a np 224 418 :M 224 410 :L 216 414 :L 224 418 :L eofill -1 -1 225 419 1 1 224 410 @b -1 -1 217 415 1 1 224 410 @b 216 415 -1 1 225 418 1 216 414 @a 359.5 395.5 17 14 rS -1 -1 334 431 1 1 367 410 @b np 337 432 :M 333 426 :L 328 433 :L 337 432 :L eofill 333 427 -1 1 338 432 1 333 426 @a -1 -1 329 434 1 1 333 426 @b -1 -1 329 434 1 1 337 432 @b 367 410 -1 1 391 429 1 367 409 @a np 391 425 :M 387 430 :L 395 433 :L 391 425 :L eofill -1 -1 388 431 1 1 391 425 @b 387 431 -1 1 396 433 1 387 430 @a 391 426 -1 1 396 433 1 391 425 @a 13 12 132 494.5 @f 105 495 -1 1 120 494 1 105 494 @a np 117 490 :M 117 498 :L 125 494 :L 117 490 :L eofill -1 -1 118 499 1 1 117 490 @b -1 -1 118 499 1 1 125 494 @b 117 491 -1 1 126 494 1 117 490 @a 146 495 -1 1 163 494 1 146 494 @a np 148 498 :M 148 490 :L 140 494 :L 148 498 :L eofill -1 -1 149 499 1 1 148 490 @b -1 -1 141 495 1 1 148 490 @b 140 495 -1 1 149 498 1 140 494 @a 260.5 464.5 19 12 rS 13 12 246 506.5 @f 220 507 -1 1 235 506 1 220 506 @a np 232 502 :M 232 510 :L 240 506 :L 232 502 :L eofill -1 -1 233 511 1 1 232 502 @b -1 -1 233 511 1 1 240 506 @b 232 503 -1 1 241 506 1 232 502 @a -1 -1 255 495 1 1 268 476 @b np 258 495 :M 252 491 :L 250 499 :L 258 495 :L eofill 252 492 -1 1 259 495 1 252 491 @a -1 -1 251 500 1 1 252 491 @b -1 -1 251 500 1 1 258 495 @b 269 477 -1 1 279 496 1 269 476 @a np 281 492 :M 274 495 :L 281 501 :L 281 492 :L eofill -1 -1 275 496 1 1 281 492 @b 274 496 -1 1 282 501 1 274 495 @a -1 -1 282 502 1 1 281 492 @b 354.5 464.5 17 14 rS 357.5 486.5 17 14 rS -1 -1 334 508 1 1 358 478 @b np 337 507 :M 332 503 :L 329 511 :L 337 507 :L eofill 332 504 -1 1 338 507 1 332 503 @a -1 -1 330 512 1 1 332 503 @b -1 -1 330 512 1 1 337 507 @b 371 479 -1 1 393 506 1 371 478 @a np 393 503 :M 387 507 :L 395 511 :L 393 503 :L eofill -1 -1 388 508 1 1 393 503 @b 387 508 -1 1 396 511 1 387 507 @a 393 504 -1 1 396 511 1 393 503 @a -1 -1 341 513 1 1 365 501 @b np 344 515 :M 341 508 :L 335 515 :L 344 515 :L eofill 341 509 -1 1 345 515 1 341 508 @a -1 -1 336 516 1 1 341 508 @b 335 516 -1 1 345 515 1 335 515 @a 365 501 -1 1 385 510 1 365 500 @a np 384 506 :M 380 512 :L 389 513 :L 384 506 :L eofill -1 -1 381 513 1 1 384 506 @b 380 513 -1 1 390 513 1 380 512 @a 384 507 -1 1 390 513 1 384 506 @a 77.5 542.5 342 26 rS 79 544 339 23 rC 79 553 :M f3_12 sf ( )S 82 553 :M ( )S 85 553 :M ( )S 88 553 :M ( )S 91 553 :M ( )S 94 553 :M ( )S 97 553 :M ( )S 100 553 :M ( )S 103 553 :M ( )S 106 553 :M ( )S 109 553 :M ( )S 112 553 :M ( )S 115 553 :M ( )S 118 553 :M ( )S 121 553 :M ( )S 124 553 :M ( )S 127 553 :M ( )S 130 553 :M ( )S 133 553 :M ( )S 136 553 :M ( )S 139 553 :M ( )S 142 553 :M ( )S 145 553 :M ( )S 148 553 :M ( )S 151 553 :M ( )S 154 553 :M ( )S 157 553 :M ( )S 160 553 :M ( )S 163 553 :M (S)S 170 553 :M (o)S 176 553 :M (m)S 185 553 :M (e)S 190 553 :M ( )S 193 553 :M (M)S 203 553 :M (e)S 208 553 :M (m)S 217 553 :M (b)S 223 553 :M (e)S 228 553 :M (r)S 232 553 :M (s)S 237 553 :M ( )S 240 553 :M (o)S 246 553 :M (f)S 250 553 :M ( )S 253 553 :M f0_12 sf (O)S 262 553 :M (-)S 267 553 :M (E)S 275 553 :M (q)S 282 553 :M (u)S 289 553 :M (i)S 293 553 :M (v)S 300 553 :M f3_12 sf <28>S 304 553 :M f0_12 sf (C)S 313 553 :M (o)S 320 553 :M (n)S 327 553 :M (d)S 334 556 :M f0_7 sf (1)S 338 553 :M f3_12 sf <29>S gR gS 77 383 419 187 rC 420.5 383.5 74 185 rS 422 385 71 182 rC 422 430 :M f3_12 sf ( )S 425 430 :M f4_12 sf ( )S 428 430 :M (A)S 436 430 :M ( )S 439 430 :M f3_12 sf ( )S 442 430 :M (o)S 448 430 :M f4_12 sf ( )S 451 430 :M ( )S 454 430 :M ( )S 457 430 :M ( )S 460 430 :M ( )S 463 430 :M ( )S 466 430 :M ( )S 469 430 :M ( )S 472 430 :M f3_12 sf (o)S 478 430 :M f4_12 sf ( )S 481 430 :M (B)S 422 478 :M ( )S 425 478 :M ( )S 428 478 :M ( )S 431 478 :M ( )S 434 478 :M ( )S 437 478 :M ( )S 440 478 :M ( )S 443 478 :M ( )S 446 478 :M f3_12 sf (P)S 453 478 :M (A)S 461 478 :M (G)S 422 530 :M ( )S 425 530 :M ( )S 428 530 :M f0_12 sf (C)S 437 530 :M (o)S 444 530 :M (n)S 451 530 :M (d)S 458 533 :M f0_7 sf (1)S 462 530 :M f3_12 sf ( )S 465 530 :M (=)S 472 530 :M ( )S 475 530 :M f1_12 sf S gR gS 77 383 419 187 rC 447 428 -1 1 473 427 1 447 427 @a gR gS 0 0 552 730 rC 263 597 :M f0_12 sf (Figure )S 300 597 :M (4)S 77 621 :M f3_12 sf .688 .069(Now suppose that there are latent variables but no selection bias, and that the set of)J 59 639 :M .463 .046(measured variables )J f0_12 sf .177(O)A f3_12 sf .153 .015( = {)J 184 639 :M f4_12 sf .17(A)A f3_12 sf .07(,)A f4_12 sf .17(B)A f3_12 sf .366 .037(}. Then, if we do not limit the number of latent variables in)J 59 657 :M .769 .077(a DAG, there are an infinite number of DAGs that entail )J f0_12 sf .326(Cond)A f0_8 sf 0 3 rm .186(1)A 0 -3 rm f3_12 sf .721 .072( many of which do not)J 59 675 :M .106 .011(contain an edge between )J 181 675 :M f4_12 sf .06(A)A f3_12 sf .076 .008( and )J f4_12 sf .06(B)A f3_12 sf .119 .012(. Two such DAGs are shown in \(iii\) and \(vi\) of )J 449 675 :M .092 .009(Figure 4.)J endp %%Page: 11 11 %%BeginPageSetup initializepage (peter; page: 11 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (11)S gR gS 0 0 552 730 rC 59 56 :M f3_12 sf .826 .083(\(Latent variables in )J f0_12 sf .35(L)A f3_12 sf .778 .078( are represented by variables in boxes.\) The examples in \(iii\) and)J 59 74 :M .254 .025(\(vi\) of )J 93 74 :M .202 .02(Figure 4)J 134 74 :M .231 .023( show that when there are latent variables it is not the case that there is an)J 59 92 :M .174 .017(edge between )J 129 92 :M f4_12 sf .11(A)A f3_12 sf .14 .014( and )J f4_12 sf .11(B)A f3_12 sf .185 .019( if and only if for all subsets )J f0_12 sf (X)S 315 92 :M f3_12 sf .241 .024( of )J 332 92 :M f0_12 sf (O)S f3_12 sf (\\{)S f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf .044 .004(}, )J f4_12 sf (G)S 388 92 :M f3_12 sf <28>S 392 92 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 411 92 :M f3_12 sf (,)S f0_12 sf .083(L)A f3_12 sf .143 .014(\) entails that )J f4_12 sf (A)S 59 110 :M f3_12 sf .577 .058(and )J f4_12 sf .291(B)A f3_12 sf .68 .068( are dependent given )J 195 110 :M f0_12 sf (X)S 204 110 :M f3_12 sf .695 .07(. \(Recall that if there is no selection bias then )J 434 110 :M f4_12 sf .409(A)A f3_12 sf .521 .052( and )J f4_12 sf .409(B)A f3_12 sf .758 .076( are)J 59 128 :M (dependent given )S 141 128 :M f0_12 sf (X)S 150 128 :M f3_12 sf ( )S f1_12 sf S f0_12 sf ( )S f3_12 sf <28>S 169 128 :M f0_12 sf (S =)S 186 128 :M f3_12 sf ( )S f0_12 sf (1)S f3_12 sf (\) if and only if )S f4_12 sf (A)S f3_12 sf ( and )S f4_12 sf (B)S f3_12 sf ( are dependent given )S 407 128 :M f0_12 sf (X.)S 419 128 :M f3_12 sf <29>S 77 152 :M .921 .092(Finally, let us consider the case where there are latent variables and selection bias.)J 59 170 :M .535 .053(Examples of DAGs in)J f0_12 sf .682 .068( O-Equiv)J 216 170 :M f3_12 sf <28>S 220 170 :M f0_12 sf .251(Cond)A f0_8 sf 0 3 rm .144(1)A 0 -3 rm f3_12 sf .557 .056(\) with selection bias are shown in \(iv\) and \(v\) of)J 59 188 :M (Figure 4.)S 77 212 :M 1.679 .168(The DAGs in )J 152 212 :M f0_12 sf (O-Equiv)S f3_12 sf <28>S 200 212 :M f0_12 sf .469(Cond)A f0_8 sf 0 3 rm .268(1)A 0 -3 rm f3_12 sf 1.257 .126(\) seem to have little in common, particularly when)J 59 230 :M .474 .047(there is the possibility of both latent variables and selection bias. While there are a great)J 59 248 :M 1.573 .157(variety of DAGs in )J f0_12 sf .612(O-Equiv)A f3_12 sf <28>S 216 248 :M f0_12 sf 1.126(Cond)A f0_8 sf 0 3 rm .643(1)A 0 -3 rm f3_12 sf 2.031 .203(\), it is not the case that every DAG is in )J f0_12 sf 2.143(O-)A 59 266 :M (Equiv)S 90 266 :M f3_12 sf <28>S 94 266 :M f0_12 sf .881(Cond)A f0_8 sf 0 3 rm .504(1)A 0 -3 rm f3_12 sf 2.014 .201(\). For example, a DAG )J 258 266 :M f4_12 sf (G)S 267 266 :M f3_12 sf <28>S 271 266 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 290 266 :M f3_12 sf .635(,)A f0_12 sf 1.693(L)A f3_12 sf 2.514 .251(\) with no edges at all is not in )J 479 266 :M f0_12 sf (O-)S 59 284 :M (Equiv)S 90 284 :M f3_12 sf <28>S 94 284 :M f0_12 sf (Cond)S f0_8 sf 0 3 rm (1)S 0 -3 rm f3_12 sf (\).)S 77 308 :M .711 .071(If )J f0_12 sf .589(Cond)A f0_8 sf 0 3 rm .336(1)A 0 -3 rm f3_12 sf 1.491 .149( is observed, it is not possible to determine anything about the ancestor)J 59 326 :M .058 .006(relationships between )J 167 326 :M f4_12 sf (A)S f3_12 sf .038 .004( and )J f4_12 sf (B)S f3_12 sf .079 .008( in the causal DAG describing the population. We represent)J 59 344 :M .208 .021(this information in a partial ancestral graph with the edge )J 341 344 :M f4_12 sf .114(A)A f3_12 sf .116 .012( o)J f1_12 sf .186A f3_12 sf .116 .012(o )J f4_12 sf .114(B)A f3_12 sf .231 .023(. The \322o\323 on each end)J 59 362 :M .32 .032(of the edge means that the PAG does not specify whether or not )J 375 362 :M f4_12 sf .189(A)A f3_12 sf .32 .032( is an ancestor of )J 469 362 :M f4_12 sf .149(B)A f3_12 sf .25 .025(, or)J 59 380 :M f4_12 sf .751(B)A f3_12 sf 1.252 .125( is an ancestor of )J f4_12 sf .751(A)A f3_12 sf 1.535 .153(. \(Since there are DAGs in )J f0_12 sf .644(O-Equiv)A f3_12 sf <28>S 363 380 :M f0_12 sf .648(Cond)A f0_8 sf 0 3 rm .37(1)A 0 -3 rm f3_12 sf 1.196 .12(\) in which )J 455 380 :M f4_12 sf 1.16(A)A f3_12 sf 1.602 .16( is an)J 59 398 :M .446 .045(ancestor of )J f4_12 sf .187(B)A f3_12 sf .366 .037(, and others in which )J f4_12 sf .187(B)A f3_12 sf .316 .032( is an ancestor of )J 322 398 :M f4_12 sf .131(A)A f3_12 sf .335 .033(, every PAG which represents )J 479 398 :M f0_12 sf (O-)S 59 416 :M (Equiv)S 90 416 :M f3_12 sf <28>S 94 416 :M f0_12 sf (Cond)S f0_8 sf 0 3 rm (1)S 0 -3 rm f3_12 sf (\) has and \322o\323 on each end of the edge between )S 352 416 :M f4_12 sf (A)S f3_12 sf ( and )S f4_12 sf (B)S f3_12 sf (.\))S 95 440 :M f8_12 sf (B)S 103 440 :M (.)S 106 440 :M ( )S 131 440 :M (Example 2)S 77 464 :M f3_12 sf 1.118 .112(Let )J 98 464 :M f0_12 sf .974(O)A f3_12 sf .841 .084( = {)J 129 464 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf (,)S f4_12 sf (C)S f3_12 sf (,)S f4_12 sf (D)S 169 464 :M f3_12 sf 1.163 .116(} and )J 201 464 :M f0_12 sf .455(Cond)A f0_8 sf 0 3 rm .26(2)A 0 -3 rm f3_12 sf .377 .038( = )J 249 464 :M f3_14 sf ({)S 256 464 :M f4_12 sf (D)S 265 464 :M f3_12 sf 1.454 .145( )J 269 451 16 16 rC -1 -1 276 464 1 1 275 454 @b -1 -1 280 464 1 1 279 454 @b 272 465 -1 1 283 464 1 272 464 @a gR gS 0 0 552 730 rC 285 464 :M f3_12 sf .778 .078( {)J f4_12 sf .781(A)A f3_12 sf .32(,)A f4_12 sf .781(B)A f3_12 sf .629 .063( }| )J f4_12 sf .853(C)A f3_12 sf .581 .058(, )J 345 464 :M f4_12 sf 1.032(A)A f3_12 sf .422 .042( )J 356 451 16 16 rC -1 -1 363 464 1 1 362 454 @b -1 -1 367 464 1 1 366 454 @b 359 465 -1 1 370 464 1 359 464 @a gR gS 0 0 552 730 rC 372 464 :M f3_12 sf 1.454 .145( )J 377 464 :M f4_12 sf (B)S f3_14 sf (})S 391 464 :M f3_12 sf 1.039 .104( and all of the other)J 59 482 :M 1.315 .131(conditional independence relations entailed by these. Once again the simplest case is)J 59 500 :M .158 .016(when it is assumed that there are no latent variables and no selection bias. In that case the)J 59 518 :M (only DAG that entails)S 165 518 :M f0_12 sf ( Cond)S f0_8 sf 0 3 rm (2)S 0 -3 rm f3_12 sf ( is \(i\) in )S 244 518 :M (Figure 5.)S endp %%Page: 12 12 %%BeginPageSetup initializepage (peter; page: 12 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (12)S gR .75 lw gS 123 41 323 262 rC 236.5 41.5 112 81 rS 238 43 109 78 rC 238 52 :M f4_12 sf (A)S 246 52 :M ( )S 249 52 :M ( )S 252 52 :M ( )S 255 52 :M ( )S 258 52 :M ( )S 261 52 :M ( )S 264 52 :M ( )S 267 52 :M ( )S 270 52 :M ( )S 273 52 :M ( )S 276 52 :M ( )S 279 52 :M ( )S 282 52 :M ( )S 285 52 :M ( )S 288 52 :M ( )S 291 52 :M ( )S 294 52 :M ( )S 297 52 :M ( )S 300 52 :M ( )S 303 52 :M ( )S 306 52 :M ( )S 309 52 :M ( )S 312 52 :M (T)S 319 55 :M f3_7 sf (1)S 238 76 :M f4_12 sf ( )S 241 76 :M ( )S 244 76 :M ( )S 247 76 :M ( )S 250 76 :M ( )S 253 76 :M ( )S 256 76 :M ( )S 259 76 :M ( )S 262 76 :M ( )S 265 76 :M ( )S 268 76 :M ( )S 271 76 :M ( )S 274 76 :M ( )S 277 76 :M ( )S 280 76 :M (C)S 288 76 :M ( )S 291 76 :M ( )S 294 76 :M ( )S 297 76 :M ( )S 300 76 :M ( )S 303 76 :M ( )S 306 76 :M ( )S 309 76 :M ( )S 312 76 :M ( )S 315 76 :M ( )S 318 76 :M ( )S 321 76 :M ( )S 324 76 :M (D)S 238 100 :M (B)S 246 100 :M ( )S 249 100 :M ( )S 252 100 :M ( )S 255 100 :M ( )S 258 100 :M ( )S 261 100 :M ( )S 264 100 :M ( )S 267 100 :M ( )S 270 100 :M ( )S 273 100 :M ( )S 276 100 :M ( )S 279 100 :M ( )S 282 100 :M ( )S 285 100 :M ( )S 288 100 :M ( )S 291 100 :M ( )S 294 100 :M ( )S 297 100 :M ( )S 300 100 :M ( )S 303 100 :M ( )S 306 100 :M ( )S 309 100 :M (T)S 316 103 :M f3_7 sf (2)S 238 112 :M f3_12 sf ( )S 241 112 :M ( )S 244 112 :M ( )S 247 112 :M ( )S 250 112 :M ( )S 253 112 :M ( )S 256 112 :M ( )S 259 112 :M ( )S 262 112 :M ( )S 265 112 :M ( )S 268 112 :M ( )S 271 112 :M ( )S 274 112 :M ( )S 277 112 :M ( )S 280 112 :M ( )S 283 112 :M <28>S 287 112 :M (i)S 290 112 :M (i)S 293 112 :M <29>S gR gS 123 41 323 262 rC 123.5 41.5 112 81 rS 125 43 109 78 rC 125 52 :M f4_12 sf (A)S 125 76 :M ( )S 128 76 :M ( )S 131 76 :M ( )S 134 76 :M ( )S 137 76 :M ( )S 140 76 :M ( )S 143 76 :M ( )S 146 76 :M ( )S 149 76 :M ( )S 152 76 :M ( )S 155 76 :M ( )S 158 76 :M ( )S 161 76 :M ( )S 164 76 :M ( )S 167 76 :M (C)S 175 76 :M ( )S 178 76 :M ( )S 181 76 :M ( )S 184 76 :M ( )S 187 76 :M ( )S 190 76 :M ( )S 193 76 :M ( )S 196 76 :M ( )S 199 76 :M ( )S 202 76 :M ( )S 205 76 :M ( )S 208 76 :M ( )S 211 76 :M (D)S 125 100 :M (B)S 125 112 :M f3_12 sf ( )S 128 112 :M ( )S 131 112 :M ( )S 134 112 :M ( )S 137 112 :M ( )S 140 112 :M ( )S 143 112 :M ( )S 146 112 :M ( )S 149 112 :M ( )S 152 112 :M ( )S 155 112 :M ( )S 158 112 :M ( )S 161 112 :M ( )S 164 112 :M ( )S 167 112 :M ( )S 170 112 :M <28>S 174 112 :M (i)S 177 112 :M <29>S gR gS 123 41 323 262 rC 137 52.75 -.75 .75 160.75 65 .75 137 52 @a np 160 61 :M 156 67 :L 165 68 :L 160 61 :L eofill -.75 -.75 156.75 67.75 .75 .75 160 61 @b 156 67.75 -.75 .75 165.75 68 .75 156 67 @a 160 61.75 -.75 .75 165.75 68 .75 160 61 @a -.75 -.75 136.75 95.75 .75 .75 161 79 @b np 157 77 :M 161 83 :L 166 76 :L 157 77 :L eofill 157 77.75 -.75 .75 161.75 83 .75 157 77 @a -.75 -.75 161.75 83.75 .75 .75 166 76 @b -.75 -.75 157.75 77.75 .75 .75 166 76 @b 177 72.75 -.75 .75 202.75 72 .75 177 72 @a np 200 68 :M 200 76 :L 208 72 :L 200 68 :L eofill -.75 -.75 200.75 76.75 .75 .75 200 68 @b -.75 -.75 200.75 76.75 .75 .75 208 72 @b 200 68.75 -.75 .75 208.75 72 .75 200 68 @a 256 96.75 -.75 .75 306.75 96 .75 256 96 @a np 258 100 :M 258 92 :L 250 96 :L 258 100 :L eofill -.75 -.75 258.75 100.75 .75 .75 258 92 @b -.75 -.75 250.75 96.75 .75 .75 258 92 @b 250 96.75 -.75 .75 258.75 100 .75 250 96 @a 295 82.75 -.75 .75 310.75 90 .75 295 82 @a np 295 86 :M 299 80 :L 290 79 :L 295 86 :L eofill -.75 -.75 295.75 86.75 .75 .75 299 80 @b 290 79.75 -.75 .75 299.75 80 .75 290 79 @a 290 79.75 -.75 .75 295.75 86 .75 290 79 @a 293 72.75 -.75 .75 318.75 72 .75 293 72 @a np 316 68 :M 316 76 :L 324 72 :L 316 68 :L eofill -.75 -.75 316.75 76.75 .75 .75 316 68 @b -.75 -.75 316.75 76.75 .75 .75 324 72 @b 316 68.75 -.75 .75 324.75 72 .75 316 68 @a -.75 -.75 297.75 63.75 .75 .75 308 57 @b np 301 65 :M 297 59 :L 292 66 :L 301 65 :L eofill 297 59.75 -.75 .75 301.75 65 .75 297 59 @a -.75 -.75 292.75 66.75 .75 .75 297 59 @b -.75 -.75 292.75 66.75 .75 .75 301 65 @b 263 49.75 -.75 .75 309.75 49 .75 263 49 @a np 265 53 :M 265 45 :L 257 49 :L 265 53 :L eofill -.75 -.75 265.75 53.75 .75 .75 265 45 @b -.75 -.75 257.75 49.75 .75 .75 265 45 @b 257 49.75 -.75 .75 265.75 53 .75 257 49 @a 124.5 120.5 112 81 rS 126 122 109 78 rC 126 131 :M f4_12 sf (A)S 134 131 :M ( )S 137 131 :M ( )S 140 131 :M ( )S 143 131 :M ( )S 146 131 :M ( )S 149 131 :M ( )S 152 131 :M ( )S 155 131 :M ( )S 158 131 :M ( )S 161 131 :M ( )S 164 131 :M ( )S 167 131 :M ( )S 170 131 :M ( )S 173 131 :M ( )S 176 131 :M ( )S 179 131 :M ( )S 182 131 :M ( )S 185 131 :M ( )S 188 131 :M ( )S 191 131 :M ( )S 194 131 :M ( )S 197 131 :M ( )S 200 131 :M (T)S 207 134 :M f3_7 sf (1)S 126 155 :M f4_12 sf ( )S 129 155 :M ( )S 132 155 :M ( )S 135 155 :M ( )S 138 155 :M ( )S 141 155 :M ( )S 144 155 :M ( )S 147 155 :M ( )S 150 155 :M ( )S 153 155 :M ( )S 156 155 :M ( )S 159 155 :M ( )S 162 155 :M ( )S 165 155 :M ( )S 168 155 :M (C)S 176 155 :M ( )S 179 155 :M ( )S 182 155 :M ( )S 185 155 :M ( )S 188 155 :M ( )S 191 155 :M ( )S 194 155 :M ( )S 197 155 :M ( )S 200 155 :M ( )S 203 155 :M ( )S 206 155 :M ( )S 209 155 :M ( )S 212 155 :M (D)S 126 179 :M (B)S 126 191 :M f3_12 sf ( )S 129 191 :M ( )S 132 191 :M ( )S 135 191 :M ( )S 138 191 :M ( )S 141 191 :M ( )S 144 191 :M ( )S 147 191 :M ( )S 150 191 :M ( )S 153 191 :M ( )S 156 191 :M ( )S 159 191 :M ( )S 162 191 :M ( )S 165 191 :M ( )S 168 191 :M ( )S 171 191 :M <28>S 175 191 :M (i)S 178 191 :M (i)S 181 191 :M (i)S 184 191 :M <29>S gR gS 123 41 323 262 rC 180 151.75 -.75 .75 205.75 151 .75 180 151 @a np 203 147 :M 203 155 :L 211 151 :L 203 147 :L eofill -.75 -.75 203.75 155.75 .75 .75 203 147 @b -.75 -.75 203.75 155.75 .75 .75 211 151 @b 203 147.75 -.75 .75 211.75 151 .75 203 147 @a -.75 -.75 185.75 142.75 .75 .75 196 136 @b np 189 144 :M 185 138 :L 180 145 :L 189 144 :L eofill 185 138.75 -.75 .75 189.75 144 .75 185 138 @a -.75 -.75 180.75 145.75 .75 .75 185 138 @b -.75 -.75 180.75 145.75 .75 .75 189 144 @b 148 127.75 -.75 .75 194.75 127 .75 148 127 @a np 150 131 :M 150 123 :L 142 127 :L 150 131 :L eofill -.75 -.75 150.75 131.75 .75 .75 150 123 @b -.75 -.75 142.75 127.75 .75 .75 150 123 @b 142 127.75 -.75 .75 150.75 131 .75 142 127 @a -.75 -.75 140.75 172.75 .75 .75 164 159 @b np 160 157 :M 164 163 :L 169 156 :L 160 157 :L eofill 160 157.75 -.75 .75 164.75 163 .75 160 157 @a -.75 -.75 164.75 163.75 .75 .75 169 156 @b -.75 -.75 160.75 157.75 .75 .75 169 156 @b 236.5 120.5 112 81 rS 238 122 109 78 rC 238 131 :M f4_12 sf (A)S 238 155 :M ( )S 241 155 :M ( )S 244 155 :M ( )S 247 155 :M ( )S 250 155 :M ( )S 253 155 :M ( )S 256 155 :M ( )S 259 155 :M ( )S 262 155 :M ( )S 265 155 :M ( )S 268 155 :M ( )S 271 155 :M ( )S 274 155 :M ( )S 277 155 :M ( )S 280 155 :M (C)S 288 155 :M ( )S 291 155 :M ( )S 294 155 :M ( )S 297 155 :M ( )S 300 155 :M ( )S 303 155 :M ( )S 306 155 :M ( )S 309 155 :M ( )S 312 155 :M ( )S 315 155 :M ( )S 318 155 :M ( )S 321 155 :M ( )S 324 155 :M (D)S 238 179 :M (B)S 246 179 :M ( )S 249 179 :M ( )S 252 179 :M ( )S 255 179 :M ( )S 258 179 :M ( )S 261 179 :M ( )S 264 179 :M ( )S 267 179 :M ( )S 270 179 :M ( )S 273 179 :M ( )S 276 179 :M ( )S 279 179 :M ( )S 282 179 :M ( )S 285 179 :M ( )S 288 179 :M ( )S 291 179 :M ( )S 294 179 :M ( )S 297 179 :M ( )S 300 179 :M ( )S 303 179 :M ( )S 306 179 :M ( )S 309 179 :M (T)S 316 182 :M f3_7 sf (2)S 238 191 :M f3_12 sf ( )S 241 191 :M ( )S 244 191 :M ( )S 247 191 :M ( )S 250 191 :M ( )S 253 191 :M ( )S 256 191 :M ( )S 259 191 :M ( )S 262 191 :M ( )S 265 191 :M ( )S 268 191 :M ( )S 271 191 :M ( )S 274 191 :M ( )S 277 191 :M ( )S 280 191 :M ( )S 283 191 :M <28>S 287 191 :M (i)S 290 191 :M (v)S 296 191 :M <29>S gR gS 123 41 323 262 rC 256 175.75 -.75 .75 306.75 175 .75 256 175 @a np 258 179 :M 258 171 :L 250 175 :L 258 179 :L eofill -.75 -.75 258.75 179.75 .75 .75 258 171 @b -.75 -.75 250.75 175.75 .75 .75 258 171 @b 250 175.75 -.75 .75 258.75 179 .75 250 175 @a 295 161.75 -.75 .75 310.75 169 .75 295 161 @a np 295 165 :M 299 159 :L 290 158 :L 295 165 :L eofill -.75 -.75 295.75 165.75 .75 .75 299 159 @b 290 158.75 -.75 .75 299.75 159 .75 290 158 @a 290 158.75 -.75 .75 295.75 165 .75 290 158 @a 292 151.75 -.75 .75 317.75 151 .75 292 151 @a np 315 147 :M 315 155 :L 323 151 :L 315 147 :L eofill -.75 -.75 315.75 155.75 .75 .75 315 147 @b -.75 -.75 315.75 155.75 .75 .75 323 151 @b 315 147.75 -.75 .75 323.75 151 .75 315 147 @a 250 132.75 -.75 .75 276.75 146 .75 250 132 @a np 276 142 :M 272 148 :L 281 149 :L 276 142 :L eofill -.75 -.75 272.75 148.75 .75 .75 276 142 @b 272 148.75 -.75 .75 281.75 149 .75 272 148 @a 276 142.75 -.75 .75 281.75 149 .75 276 142 @a 124.5 201.5 112 81 rS 126 203 109 78 rC 126 212 :M f4_12 sf (A)S 134 212 :M ( )S 137 212 :M ( )S 140 212 :M ( )S 143 212 :M ( )S 146 212 :M ( )S 149 212 :M ( )S 152 212 :M ( )S 155 212 :M ( )S 158 212 :M ( )S 161 212 :M ( )S 164 212 :M (S)S 171 212 :M ( )S 174 212 :M ( )S 177 212 :M ( )S 180 212 :M ( )S 183 212 :M ( )S 186 212 :M ( )S 189 212 :M ( )S 192 212 :M ( )S 195 212 :M ( )S 198 212 :M ( )S 201 212 :M (T)S 208 215 :M f3_7 sf (1)S 126 236 :M f4_12 sf ( )S 129 236 :M ( )S 132 236 :M ( )S 135 236 :M ( )S 138 236 :M ( )S 141 236 :M ( )S 144 236 :M ( )S 147 236 :M ( )S 150 236 :M ( )S 153 236 :M ( )S 156 236 :M ( )S 159 236 :M ( )S 162 236 :M ( )S 165 236 :M ( )S 168 236 :M (C)S 176 236 :M ( )S 179 236 :M ( )S 182 236 :M ( )S 185 236 :M ( )S 188 236 :M ( )S 191 236 :M ( )S 194 236 :M ( )S 197 236 :M ( )S 200 236 :M ( )S 203 236 :M ( )S 206 236 :M ( )S 209 236 :M ( )S 212 236 :M (D)S 126 260 :M (B)S 126 272 :M f3_12 sf ( )S 129 272 :M ( )S 132 272 :M ( )S 135 272 :M ( )S 138 272 :M ( )S 141 272 :M ( )S 144 272 :M ( )S 147 272 :M ( )S 150 272 :M ( )S 153 272 :M ( )S 156 272 :M ( )S 159 272 :M ( )S 162 272 :M ( )S 165 272 :M ( )S 168 272 :M ( )S 171 272 :M <28>S 175 272 :M (v)S 181 272 :M <29>S gR gS 123 41 323 262 rC 180 232.75 -.75 .75 205.75 232 .75 180 232 @a np 203 228 :M 203 236 :L 211 232 :L 203 228 :L eofill -.75 -.75 203.75 236.75 .75 .75 203 228 @b -.75 -.75 203.75 236.75 .75 .75 211 232 @b 203 228.75 -.75 .75 211.75 232 .75 203 228 @a -.75 -.75 185.75 223.75 .75 .75 196 217 @b np 189 225 :M 185 219 :L 180 226 :L 189 225 :L eofill 185 219.75 -.75 .75 189.75 225 .75 185 219 @a -.75 -.75 180.75 226.75 .75 .75 185 219 @b -.75 -.75 180.75 226.75 .75 .75 189 225 @b -.75 -.75 140.75 253.75 .75 .75 164 240 @b np 160 238 :M 164 244 :L 169 237 :L 160 238 :L eofill 160 238.75 -.75 .75 164.75 244 .75 160 238 @a -.75 -.75 164.75 244.75 .75 .75 169 237 @b -.75 -.75 160.75 238.75 .75 .75 169 237 @b 236.5 201.5 112 81 rS 238 203 109 78 rC 238 212 :M f4_12 sf (A)S 238 236 :M ( )S 241 236 :M ( )S 244 236 :M ( )S 247 236 :M ( )S 250 236 :M ( )S 253 236 :M ( )S 256 236 :M ( )S 259 236 :M ( )S 262 236 :M ( )S 265 236 :M ( )S 268 236 :M ( )S 271 236 :M ( )S 274 236 :M ( )S 277 236 :M ( )S 280 236 :M (C)S 288 236 :M ( )S 291 236 :M ( )S 294 236 :M ( )S 297 236 :M ( )S 300 236 :M ( )S 303 236 :M ( )S 306 236 :M ( )S 309 236 :M ( )S 312 236 :M ( )S 315 236 :M ( )S 318 236 :M ( )S 321 236 :M ( )S 324 236 :M (D)S 238 260 :M (B)S 246 260 :M ( )S 249 260 :M ( )S 252 260 :M ( )S 255 260 :M ( )S 258 260 :M ( )S 261 260 :M ( )S 264 260 :M ( )S 267 260 :M ( )S 270 260 :M ( )S 273 260 :M (S)S 280 260 :M ( )S 283 260 :M ( )S 286 260 :M ( )S 289 260 :M ( )S 292 260 :M ( )S 295 260 :M ( )S 298 260 :M ( )S 301 260 :M ( )S 304 260 :M ( )S 307 260 :M ( )S 310 260 :M (T)S 317 263 :M f3_7 sf (2)S 238 272 :M f3_12 sf ( )S 241 272 :M ( )S 244 272 :M ( )S 247 272 :M ( )S 250 272 :M ( )S 253 272 :M ( )S 256 272 :M ( )S 259 272 :M ( )S 262 272 :M ( )S 265 272 :M ( )S 268 272 :M ( )S 271 272 :M ( )S 274 272 :M ( )S 277 272 :M ( )S 280 272 :M ( )S 283 272 :M <28>S 287 272 :M (v)S 293 272 :M (i)S 296 272 :M <29>S gR gS 123 41 323 262 rC 295 242.75 -.75 .75 310.75 250 .75 295 242 @a np 295 246 :M 299 240 :L 290 239 :L 295 246 :L eofill -.75 -.75 295.75 246.75 .75 .75 299 240 @b 290 239.75 -.75 .75 299.75 240 .75 290 239 @a 290 239.75 -.75 .75 295.75 246 .75 290 239 @a 292 232.75 -.75 .75 317.75 232 .75 292 232 @a np 315 228 :M 315 236 :L 323 232 :L 315 228 :L eofill -.75 -.75 315.75 236.75 .75 .75 315 228 @b -.75 -.75 315.75 236.75 .75 .75 323 232 @b 315 228.75 -.75 .75 323.75 232 .75 315 228 @a 250 213.75 -.75 .75 276.75 227 .75 250 213 @a np 276 223 :M 272 229 :L 281 230 :L 276 223 :L eofill -.75 -.75 272.75 229.75 .75 .75 276 223 @b 272 229.75 -.75 .75 281.75 230 .75 272 229 @a 276 223.75 -.75 .75 281.75 230 .75 276 223 @a 310.5 43.5 17 13 rS 308.5 168.5 17 13 rS 197.5 122.5 17 13 rS 198.5 203.5 17 13 rS 310.5 250.5 17 13 rS 22 13 170.5 209 @f 22 13 279.5 256 @f 141 208.75 -.75 .75 154.75 208 .75 141 208 @a np 152 204 :M 152 212 :L 160 208 :L 152 204 :L eofill -.75 -.75 152.75 212.75 .75 .75 152 204 @b -.75 -.75 152.75 212.75 .75 .75 160 208 @b 152 204.75 -.75 .75 160.75 208 .75 152 204 @a 189 208.75 -.75 .75 198.75 208 .75 189 208 @a np 191 212 :M 191 204 :L 183 208 :L 191 212 :L eofill -.75 -.75 191.75 212.75 .75 .75 191 204 @b -.75 -.75 183.75 208.75 .75 .75 191 204 @b 183 208.75 -.75 .75 191.75 212 .75 183 208 @a 253 256.75 -.75 .75 262.75 256 .75 253 256 @a np 260 252 :M 260 260 :L 268 256 :L 260 252 :L eofill -.75 -.75 260.75 260.75 .75 .75 260 252 @b -.75 -.75 260.75 260.75 .75 .75 268 256 @b 260 252.75 -.75 .75 268.75 256 .75 260 252 @a 300 256.75 -.75 .75 309.75 256 .75 300 256 @a np 302 260 :M 302 252 :L 294 256 :L 302 260 :L eofill -.75 -.75 302.75 260.75 .75 .75 302 252 @b -.75 -.75 294.75 256.75 .75 .75 302 252 @b 294 256.75 -.75 .75 302.75 260 .75 294 256 @a 348.5 41.5 96 260 rS 350 43 93 257 rC 350 100 :M f4_12 sf (A)S 358 100 :M ( )S 361 100 :M (o)S 350 136 :M ( )S 353 136 :M ( )S 356 136 :M ( )S 359 136 :M ( )S 362 136 :M ( )S 365 136 :M ( )S 368 136 :M ( )S 371 136 :M ( )S 374 136 :M ( )S 377 136 :M ( )S 380 136 :M ( )S 383 136 :M ( )S 386 136 :M ( )S 389 136 :M (C)S 397 136 :M ( )S 400 136 :M ( )S 403 136 :M ( )S 406 136 :M ( )S 409 136 :M ( )S 412 136 :M ( )S 415 136 :M ( )S 418 136 :M ( )S 421 136 :M ( )S 424 136 :M ( )S 427 136 :M ( )S 430 136 :M (D)S 350 172 :M (B)S 358 172 :M ( )S 361 172 :M (o)S 350 196 :M ( )S 353 196 :M ( )S 356 196 :M ( )S 359 196 :M ( )S 362 196 :M ( )S 365 196 :M ( )S 368 196 :M ( )S 371 196 :M ( )S 374 196 :M ( )S 377 196 :M ( )S 380 196 :M ( )S 383 196 :M (P)S 390 196 :M (A)S 398 196 :M (G)S 350 232 :M ( )S 353 232 :M ( )S 356 232 :M ( )S 359 232 :M ( )S 362 232 :M ( )S 365 232 :M ( )S 368 232 :M f0_12 sf (C)S 377 232 :M (o)S 384 232 :M (n)S 391 232 :M (d)S 398 235 :M f0_7 sf (2)S 402 232 :M f3_12 sf ( )S 405 232 :M (=)S 350 256 :M ({)S 356 256 :M f4_12 sf ( )S 359 256 :M ( )S 362 256 :M ( )S 365 256 :M (D)S 373 256 :M ( )S 376 256 :M ( )S 379 256 :M ( )S 382 256 :M ( )S 385 256 :M f3_12 sf ({)S 391 256 :M f4_12 sf (A)S 399 256 :M (,)S 403 256 :M (B)S 411 256 :M f3_12 sf (})S 417 256 :M (|)S 419 256 :M ( )S 422 256 :M f4_12 sf (C)S 430 256 :M (,)S 350 268 :M ( )S 353 268 :M ( )S 356 268 :M ( )S 359 268 :M ( )S 362 268 :M ( )S 365 268 :M ( )S 368 268 :M ( )S 371 268 :M ( )S 374 268 :M ( )S 377 268 :M (A)S 385 268 :M ( )S 388 268 :M ( )S 391 268 :M ( )S 394 268 :M ( )S 397 268 :M ( )S 400 268 :M ( )S 403 268 :M (B)S 411 268 :M ( )S 414 268 :M ( )S 417 268 :M f3_12 sf (})S gR gS 123 41 323 262 rC 365 99.75 -.75 .75 385.75 123 .75 365 99 @a np 387 120 :M 381 124 :L 389 128 :L 387 120 :L eofill -.75 -.75 381.75 124.75 .75 .75 387 120 @b 381 124.75 -.75 .75 389.75 128 .75 381 124 @a 387 120.75 -.75 .75 389.75 128 .75 387 120 @a -.75 -.75 364.75 167.75 .75 .75 385 142 @b np 381 141 :M 387 145 :L 389 137 :L 381 141 :L eofill 381 141.75 -.75 .75 387.75 145 .75 381 141 @a -.75 -.75 387.75 145.75 .75 .75 389 137 @b -.75 -.75 381.75 141.75 .75 .75 389 137 @b 399 132.75 -.75 .75 422.75 132 .75 399 132 @a np 420 128 :M 420 136 :L 428 132 :L 420 128 :L eofill -.75 -.75 420.75 136.75 .75 .75 420 128 @b -.75 -.75 420.75 136.75 .75 .75 428 132 @b 420 128.75 -.75 .75 428.75 132 .75 420 128 @a -1 -1 380 258 1 1 379 248 @b -1 -1 382 258 1 1 381 248 @b 376 258 -1 1 385 257 1 376 257 @a -1 -1 396 269 1 1 395 259 @b -1 -1 398 269 1 1 397 259 @b 392 269 -1 1 401 268 1 392 268 @a 1 lw 124.5 282.5 224 19 rS 126 284 221 16 rC 126 293 :M f3_12 sf ( )S 129 293 :M ( )S 132 293 :M ( )S 135 293 :M ( )S 138 293 :M ( )S 141 293 :M ( )S 144 293 :M ( )S 147 293 :M (S)S 154 293 :M (o)S 160 293 :M (m)S 169 293 :M (e)S 174 293 :M ( )S 177 293 :M (M)S 187 293 :M (e)S 192 293 :M (m)S 201 293 :M (b)S 207 293 :M (e)S 212 293 :M (r)S 216 293 :M (s)S 221 293 :M ( )S 224 293 :M (o)S 230 293 :M (f)S 234 293 :M ( )S 237 293 :M f0_12 sf (O)S 246 293 :M (-)S 251 293 :M (E)S 259 293 :M (q)S 266 293 :M (u)S 273 293 :M (i)S 277 293 :M (v)S 284 293 :M f3_12 sf <28>S 288 293 :M f0_12 sf (C)S 297 293 :M (o)S 304 293 :M (n)S 311 293 :M (d)S 318 296 :M f0_7 sf (2)S 322 293 :M f3_12 sf <29>S gR gS 123 41 323 262 rC 309.5 89.5 17 15 rS gR gS 0 0 552 730 rC 263 324 :M f0_12 sf (Figure )S 300 324 :M (5)S 77 348 :M f3_12 sf .225 .023(Now suppose that we consider DAGs with latent variables so )J 378 348 :M f0_12 sf (V)S 387 348 :M f3_12 sf .101 .01( )J f2_12 sf S 397 348 :M f0_12 sf .07A f3_12 sf .197 .02(, but no selection)J 59 366 :M .248 .025(bias. In that case if there is no upper limit to the number of latent variables allowed, then)J 59 384 :M .137 .014(there are an infinite number of DAGs in )J f0_12 sf .053(O-Equiv)A f3_12 sf <28>S 304 384 :M f0_12 sf .058(Cond)A f0_8 sf 0 3 rm (2)S 0 -3 rm f3_12 sf .14 .014(\), several of which are shown in)J 59 402 :M (\(ii\), \(iii\) and \(iv\) of Figure 5.)S 77 426 :M 1.207 .121(Suppose that we now consider DAGs with selection bias. \(v\) and \(vi\) of )J 450 426 :M 1.092 .109(Figure 5)J 59 444 :M .571 .057(show some examples of DAGs that are in )J 269 444 :M f0_12 sf (O-Equiv)S f3_12 sf <28>S 317 444 :M f0_12 sf .172(Cond)A f0_8 sf 0 3 rm .098(2)A 0 -3 rm f3_12 sf .433 .043(\) that have selection bias and)J 59 462 :M (latent variables.)S 77 486 :M 1.077 .108(Is there anything that all of the DAGs in Figure 5)J 329 486 :M 1.134 .113( have in common? In any of the)J 59 504 :M 1.062 .106(DAGs in )J 108 504 :M f0_12 sf (O-Equiv)S f3_12 sf <28>S 156 504 :M f0_12 sf .361(Cond)A f0_8 sf 0 3 rm .206(2)A 0 -3 rm f3_12 sf .694 .069(\), for each of the pairs <)J f4_12 sf .378(A)A f3_12 sf .155(,)A f4_12 sf (D)S 331 504 :M f3_12 sf 1.103 .11(>, <)J 353 504 :M f4_12 sf (B)S f3_12 sf (,)S f4_12 sf (D)S 372 504 :M f3_12 sf 1.062 .106(>, and <)J 415 504 :M f4_12 sf .428(A)A f3_12 sf .175(,)A f4_12 sf .428(B)A f3_12 sf .779 .078(>, there is a)J 59 522 :M 3.518 .352(subset )J f0_12 sf 1.451(Z)A f3_12 sf 1.26 .126( of )J 126 522 :M f0_12 sf 1.194(O)A f3_12 sf 2.144 .214( such that the pair is independent conditional on )J 399 522 :M f0_12 sf 2.505(Z)A f3_12 sf .939 .094( )J 414 522 :M f1_12 sf 1.398A f3_12 sf 1.108 .111( {)J f4_12 sf .911(S)A f3_12 sf 2.307 .231(}. This is)J 59 540 :M 1.06 .106(represented in the PAG by the lack of edges between )J f4_12 sf .459(A)A f3_12 sf .585 .058( and )J f4_12 sf (D)S 373 540 :M f3_12 sf .99 .099(, between )J f4_12 sf .444(B)A f3_12 sf .948 .095( and)J 454 540 :M f4_12 sf .86 .086( D)J f3_12 sf 1.475 .147(, and)J 59 558 :M .915 .091(between )J 104 558 :M f4_12 sf .492(A)A f3_12 sf .626 .063( and )J f4_12 sf .492(B)A f3_12 sf 1.103 .11(. Moreover, in some of the DAGs in Figure 5)J 375 558 :M .43 .043(, )J f4_12 sf .63(A)A f3_12 sf 1.05 .105( is an ancestor of )J f4_12 sf .688(C)A f3_12 sf (,)S 59 576 :M .088 .009(while in others it is not. Note that in none of them is )J 314 576 :M f4_12 sf (C)S f3_12 sf .074 .007( an ancestor of )J f4_12 sf (A)S f3_12 sf .089 .009( or any member of)J 59 594 :M f0_12 sf (S)S 66 594 :M f3_12 sf .229 .023(. It can be shown that in none of the DAGs in )J 291 594 :M f0_12 sf (O-Equiv)S f3_12 sf <28>S 339 594 :M f0_12 sf .101(Cond)A f0_8 sf 0 3 rm .058(2)A 0 -3 rm f3_12 sf .104 .01(\) is )J f4_12 sf .115(C)A f3_12 sf .192 .019( an ancestor of )J 472 594 :M f4_12 sf .114(A)A f3_12 sf .168 .017( or)J 59 612 :M .918 .092(of any member of )J 153 612 :M f0_12 sf (S)S 160 612 :M f3_12 sf .857 .086(. In the PAG representing )J 293 612 :M f0_12 sf (O-Equiv)S f3_12 sf <28>S 341 612 :M f0_12 sf .309(Cond)A f0_8 sf 0 3 rm .177(2)A 0 -3 rm f3_12 sf .643 .064(\) we represent this by )J f4_12 sf (A)S 59 630 :M f3_12 sf (o)S f1_12 sf S 77 630 :M f3_12 sf ( )S f4_12 sf (C)S f3_12 sf (.)S 77 654 :M .09 .009(The \322o\323 on the )J f4_12 sf (A)S f3_12 sf .07 .007( end of the )J 213 654 :M f4_12 sf (A)S f3_12 sf ( o)S f1_12 sf S 241 654 :M f3_12 sf ( )S f4_12 sf (C)S f3_12 sf .081 .008( edge means the PAG does not say whether or not)J 59 672 :M f4_12 sf .831(A)A f3_12 sf 1.385 .139( is an ancestor of )J f4_12 sf .908(C)A f3_12 sf 1.381 .138(; and the \322>\323 on the )J 278 672 :M f4_12 sf .838(C)A f3_12 sf 1.435 .144( end of the edge means that )J f4_12 sf .838(C)A f3_12 sf 1.172 .117( is not an)J endp %%Page: 13 13 %%BeginPageSetup initializepage (peter; page: 13 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (13)S gR gS 0 0 552 730 rC 59 56 :M f3_12 sf .672 .067(ancestor of )J f4_12 sf .282(A)A f3_12 sf .525 .053( or any member of )J 220 56 :M f0_12 sf (S)S 227 56 :M f3_12 sf .739 .074( in )J 243 56 :M f4_12 sf (all)S 256 56 :M f3_12 sf .654 .065( of the DAGs in )J 341 56 :M f0_12 sf (O-Equiv)S f3_12 sf <28>S 389 56 :M f0_12 sf .237(Cond)A f0_8 sf 0 3 rm .136(2)A 0 -3 rm f3_12 sf .414 .041(\). It is also the)J 59 74 :M .408 .041(case that in all of the DAGs in Figure 5)J 253 74 :M .532 .053(, )J 260 74 :M f4_12 sf .254(C)A f3_12 sf .393 .039( is an ancestor of )J 354 74 :M f4_12 sf (D)S 363 74 :M f3_12 sf .418 .042( but not of any member of)J 59 92 :M f0_12 sf (S)S 66 92 :M f3_12 sf .516 .052(, and )J f4_12 sf (D)S 103 92 :M f3_12 sf .652 .065( is not an ancestor of )J 212 92 :M f4_12 sf .336(C)A f3_12 sf .565 .056( or any member of )J f0_12 sf (S)S 323 92 :M f3_12 sf .642 .064(. It can be shown that in all of the)J 59 110 :M .061 .006(DAGs in )J f0_12 sf .023(O-Equiv)A f3_12 sf <28>S 153 110 :M f0_12 sf .053(Cond)A f0_8 sf 0 3 rm (2)S 0 -3 rm f3_12 sf .044 .004(\) )J f4_12 sf .06(C)A f3_12 sf .092 .009( is an ancestor of )J f4_12 sf (D)S 294 110 :M f3_12 sf .122 .012( but not of any member of )J f0_12 sf (S)S 430 110 :M f3_12 sf .146 .015( and )J 454 110 :M f4_12 sf (D)S 463 110 :M f3_12 sf .135 .013( is not)J 59 128 :M .134 .013(an ancestor of )J 130 128 :M f4_12 sf .079(C)A f3_12 sf .135 .013( or any member of )J 231 128 :M f0_12 sf (S)S 238 128 :M f3_12 sf .13 .013(. These facts are represented in the PAG by the edge)J 59 146 :M (between )S 102 146 :M f4_12 sf (C)S f3_12 sf ( and )S f4_12 sf (D)S 142 146 :M f3_12 sf ( having a \322>\323 at the )S 239 146 :M f4_12 sf (D)S 248 146 :M f3_12 sf ( end and a \322)S f1_12 sf (-)S 312 146 :M f3_12 sf (\323 at the )S 350 146 :M f4_12 sf (C)S f3_12 sf ( end.)S 77 170 :M .413 .041(There is an important distinction between the conditional distribution of )J 433 170 :M f4_12 sf (D)S 442 170 :M f3_12 sf .621 .062( on )J 462 170 :M f4_12 sf .393(C)A f3_12 sf .273 .027( = )J f4_12 sf .262(c)A f3_12 sf (,)S 59 188 :M .118 .012(and the distribution that results when )J 241 188 :M f4_12 sf .056(C)A f3_12 sf .119 .012( is forced \(by intervening on the structure\) to have)J 59 206 :M (the value )S 106 206 :M f4_12 sf (c)S f3_12 sf (. The latter quantity depends upon the causal relations between )S 416 206 :M f4_12 sf (C)S f3_12 sf ( and )S f4_12 sf (D)S 456 206 :M f3_12 sf (. If )S 473 206 :M f4_12 sf (C)S f3_12 sf ( is)S 59 224 :M .351 .035(a cause of )J f4_12 sf (D)S 120 224 :M f3_12 sf .379 .038(, then forcing the value )J 237 224 :M f4_12 sf .144(c)A f3_12 sf .203 .02( on )J f4_12 sf .217(C)A f3_12 sf .403 .04( will in general have an effect on the value of)J 59 242 :M f4_12 sf (D)S 68 242 :M f3_12 sf .464 .046(, while if )J f4_12 sf .331(C)A f3_12 sf .442 .044( is an effect of )J f4_12 sf (D)S 208 242 :M f3_12 sf .49 .049(, then forcing a value of )J 330 242 :M f4_12 sf .413 .041(c )J f3_12 sf .744 .074(on )J 355 242 :M f4_12 sf .25(C)A f3_12 sf .455 .046( will not have an effect on)J 59 260 :M 1.005 .101(the value of )J 124 260 :M f4_12 sf (D)S 133 260 :M f3_12 sf .893 .089(. See Spirtes et al. \(1993\) and Pearl \(1995\) for details. In this particular)J 59 278 :M .34 .034(case, it is possible to make both qualitative and quantitative predictions about the effects)J 59 296 :M .159 .016(on the value of )J 135 296 :M f4_12 sf (D)S 144 296 :M f3_12 sf .147 .015( of interventions that set the value of )J 325 296 :M f4_12 sf .063(C)A f3_12 sf .14 .014( from the PAG and the measured)J 59 314 :M .519 .052(conditional distribution of )J f4_12 sf (D)S 199 314 :M f3_12 sf .829 .083( on )J f4_12 sf .884(C)A f3_12 sf .641 .064( = )J 243 314 :M f4_12 sf .193(c)A f3_12 sf .551 .055(. This is because every DAG in )J f0_12 sf .228(O-Equiv)A f3_12 sf <28>S 456 314 :M f0_12 sf (Cond)S f0_7 sf 0 3 rm (2)S 0 -3 rm 488 314 :M f3_12 sf <29>S 59 332 :M .239 .024(makes the same quantitative prediction about the effects of intervening to set the value of)J 59 350 :M f4_12 sf .795(C)A f3_12 sf .634 .063( to )J f4_12 sf .529(c)A f3_12 sf 1.748 .175(. The details of the algorithm for making this prediction are given in Spirtes,)J 59 368 :M 1.41 .141(Glymour and Scheines \(1993\). In this case we can determine that the only source of)J 59 386 :M .622 .062(dependency between )J 165 386 :M f4_12 sf .473(C)A f3_12 sf .552 .055( and )J f4_12 sf (D)S 207 386 :M f3_12 sf .799 .08( are directed paths from )J 330 386 :M f4_12 sf .592(C)A f3_12 sf .472 .047( to )J f4_12 sf (D)S 364 386 :M f3_12 sf .91 .091(, so if P\()J 409 386 :M f4_12 sf (D)S 418 386 :M f3_12 sf .592 .059( | )J f4_12 sf 1.298(C)A f3_12 sf .941 .094( = )J 452 386 :M f4_12 sf .332(c)A f3_12 sf .753 .075(\) is the)J 59 404 :M .314 .031(conditional distribution of )J f4_12 sf (D)S 198 404 :M f3_12 sf .249 .025( on )J f4_12 sf .265(C)A f3_12 sf .502 .05( in the population, and )J 339 404 :M f4_12 sf .258(C)A f3_12 sf .436 .044( is forced to have the value )J 484 404 :M f4_12 sf (c)S f3_12 sf (,)S 59 422 :M (then the new distribution of )S 195 422 :M f4_12 sf (D)S 204 422 :M f3_12 sf ( will be P\()S 254 422 :M f4_12 sf (D)S 263 422 :M f3_12 sf ( | )S f4_12 sf (C)S f3_12 sf ( = )S 292 422 :M f4_12 sf (c)S f3_12 sf (\).)S 95 446 :M f8_12 sf (C)S 103 446 :M (.)S 106 446 :M ( )S 131 446 :M (Example 3)S 77 470 :M f3_12 sf .99 .099(Finally, consider an example in which )J 273 470 :M f0_12 sf .5(Cond)A f0_8 sf 0 3 rm .286(3)A 0 -3 rm f3_12 sf .415 .041( = )J 321 470 :M f3_14 sf ({)S 328 470 :M f4_12 sf (D)S 337 470 :M f3_12 sf 1.6 .16( )J 341 457 16 16 rC -1 -1 348 470 1 1 347 460 @b -1 -1 352 470 1 1 351 460 @b 344 471 -1 1 355 470 1 344 470 @a gR gS 0 0 552 730 rC 357 470 :M f3_12 sf 1.454 .145( {)J 368 470 :M f4_12 sf .768(A)A f3_12 sf .314(,)A f4_12 sf .768(B)A f3_12 sf .618 .062(} | )J f4_12 sf .838(C)A f3_12 sf .571 .057(, )J 418 470 :M f4_12 sf 1.135(A)A f3_12 sf .465 .046( )J 429 457 16 16 rC -1 -1 436 470 1 1 435 460 @b -1 -1 440 470 1 1 439 460 @b 432 471 -1 1 443 470 1 432 470 @a gR gS 0 0 552 730 rC 445 470 :M f3_12 sf 1.454 .145( {)J 456 470 :M f4_12 sf (C)S f3_12 sf (,)S f4_12 sf (D)S 476 470 :M f3_12 sf (})S 482 470 :M f3_14 sf (})S 489 470 :M f3_12 sf (.)S 59 488 :M .123 .012(There is no DAG in )J f0_12 sf .054(O-Equiv)A f3_12 sf <28>S 206 488 :M f0_12 sf .065(Cond)A f0_8 sf 0 3 rm (3)S 0 -3 rm f3_12 sf .129 .013(\) in which both )J 315 488 :M f0_12 sf (V)S 324 488 :M f3_12 sf .067 .007( = )J f0_12 sf .113(O)A f3_12 sf .192 .019( and there is no selection bias.)J 59 506 :M .372 .037(Hence we can conclude that the DAG that entails )J 304 506 :M f0_12 sf .107(Cond)A f0_8 sf 0 3 rm .061(3)A 0 -3 rm f3_12 sf .282 .028( either contains a latent variable)J 59 524 :M .022 .002(or there is selection bias or both. \(i\) and \(ii\) of )J 285 524 :M .021 .002(Figure 6 are examples of DAGs with latent)J 59 542 :M .359 .036(variables that entail )J 158 542 :M f0_12 sf .197(Cond)A f0_8 sf 0 3 rm .112(3)A 0 -3 rm f3_12 sf .381 .038(. Note that in each of them, there is a latent cause of )J f4_12 sf .206(B)A f3_12 sf .262 .026( and )J f4_12 sf .225(C)A f3_12 sf (,)S 59 560 :M f4_12 sf .749(B)A f3_12 sf 1.279 .128( is not an ancestor of )J 180 560 :M f4_12 sf 1.702 .17(C )J 193 560 :M f3_12 sf 1.222 .122(or any member of )J f0_12 sf (S)S 296 560 :M f3_12 sf 1.107 .111(, and )J f4_12 sf .875(C)A f3_12 sf 1.369 .137( is not an ancestor of )J 448 560 :M f4_12 sf .675(B)A f3_12 sf 1.18 .118( or any)J 59 578 :M .253 .025(member of )J f0_12 sf (S)S 122 578 :M f3_12 sf .052(.)A f4_12 sf .048 .005( )J f3_12 sf .291 .029(As long as there is no selection bias, these properties can be shown to hold)J 59 596 :M (of any DAG in )S 134 596 :M f0_12 sf (O-Equiv)S f3_12 sf <28>S 182 596 :M f0_12 sf (Cond)S f0_8 sf 0 3 rm (3)S 0 -3 rm f3_12 sf (\).)S endp %%Page: 14 14 %%BeginPageSetup initializepage (peter; page: 14 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (14)S gR gS 125 41 319 212 rC 125.5 41.5 103 112 rS 127 43 100 109 rC 127 64 :M f3_12 sf ( )S 130 64 :M ( )S 133 64 :M ( )S 136 64 :M ( )S 139 64 :M ( )S 142 64 :M ( )S 145 64 :M ( )S 148 64 :M ( )S 151 64 :M ( )S 154 64 :M ( )S 157 64 :M ( )S 160 64 :M ( )S 163 64 :M ( )S 166 64 :M f4_12 sf (T)S 127 100 :M (A)S 135 100 :M ( )S 138 100 :M ( )S 141 100 :M ( )S 144 100 :M ( )S 147 100 :M ( )S 150 100 :M ( )S 153 100 :M (B)S 161 100 :M ( )S 164 100 :M ( )S 167 100 :M ( )S 170 100 :M ( )S 173 100 :M ( )S 176 100 :M ( )S 179 100 :M ( )S 182 100 :M (C)S 190 100 :M ( )S 193 100 :M ( )S 196 100 :M ( )S 199 100 :M ( )S 202 100 :M ( )S 205 100 :M ( )S 208 100 :M ( )S 211 100 :M (D)S 127 148 :M f3_12 sf ( )S 130 148 :M ( )S 133 148 :M ( )S 136 148 :M ( )S 139 148 :M ( )S 142 148 :M ( )S 145 148 :M ( )S 148 148 :M ( )S 151 148 :M ( )S 154 148 :M ( )S 157 148 :M ( )S 160 148 :M ( )S 163 148 :M <28>S 167 148 :M (i)S 170 148 :M <29>S gR gS 125 41 319 212 rC 229.5 41.5 103 112 rS 231 43 100 109 rC 231 64 :M f3_12 sf ( )S 234 64 :M ( )S 237 64 :M ( )S 240 64 :M ( )S 243 64 :M ( )S 246 64 :M ( )S 249 64 :M ( )S 252 64 :M ( )S 255 64 :M ( )S 258 64 :M ( )S 261 64 :M ( )S 264 64 :M ( )S 267 64 :M ( )S 270 64 :M f4_12 sf (T)S 231 100 :M (A)S 239 100 :M ( )S 242 100 :M ( )S 245 100 :M ( )S 248 100 :M ( )S 251 100 :M ( )S 254 100 :M ( )S 257 100 :M (B)S 265 100 :M ( )S 268 100 :M ( )S 271 100 :M ( )S 274 100 :M ( )S 277 100 :M ( )S 280 100 :M ( )S 283 100 :M ( )S 286 100 :M (C)S 294 100 :M ( )S 297 100 :M ( )S 300 100 :M ( )S 303 100 :M ( )S 306 100 :M ( )S 309 100 :M ( )S 312 100 :M ( )S 315 100 :M (D)S 231 136 :M f3_12 sf ( )S 234 136 :M ( )S 237 136 :M ( )S 240 136 :M ( )S 243 136 :M ( )S 246 136 :M ( )S 249 136 :M ( )S 252 136 :M ( )S 255 136 :M ( )S 258 136 :M ( )S 261 136 :M ( )S 264 136 :M ( )S 267 136 :M ( )S 270 136 :M f4_12 sf (U)S 231 148 :M f3_12 sf ( )S 234 148 :M ( )S 237 148 :M ( )S 240 148 :M ( )S 243 148 :M ( )S 246 148 :M ( )S 249 148 :M ( )S 252 148 :M ( )S 255 148 :M ( )S 258 148 :M ( )S 261 148 :M ( )S 264 148 :M ( )S 267 148 :M <28>S 271 148 :M (i)S 274 148 :M (i)S 277 148 :M <29>S gR gS 125 41 319 212 rC 155.5 52.5 26 15 rS 138 96.75 -.75 .75 145.75 96 .75 138 96 @a np 143 92 :M 143 100 :L 151 96 :L 143 92 :L .75 lw eofill -.75 -.75 143.75 100.75 .75 .75 143 92 @b -.75 -.75 143.75 100.75 .75 .75 151 96 @b 143 92.75 -.75 .75 151.75 96 .75 143 92 @a 202 96.75 -.75 .75 208.75 96 .75 202 96 @a np 204 100 :M 204 92 :L 196 96 :L 204 100 :L eofill -.75 -.75 204.75 100.75 .75 .75 204 92 @b -.75 -.75 196.75 96.75 .75 .75 204 92 @b 196 96.75 -.75 .75 204.75 100 .75 196 96 @a -.75 -.75 160.75 84.75 .75 .75 169 67 @b np 164 84 :M 158 80 :L 157 89 :L 164 84 :L eofill 158 80.75 -.75 .75 164.75 84 .75 158 80 @a -.75 -.75 157.75 89.75 .75 .75 158 80 @b -.75 -.75 157.75 89.75 .75 .75 164 84 @b 170 67.75 -.75 .75 180.75 85 .75 170 67 @a np 182 81 :M 176 85 :L 183 90 :L 182 81 :L eofill -.75 -.75 176.75 85.75 .75 .75 182 81 @b 176 85.75 -.75 .75 183.75 90 .75 176 85 @a 182 81.75 -.75 .75 183.75 90 .75 182 81 @a 1 lw 259.5 52.5 26 15 rS 242 96.75 -.75 .75 249.75 96 .75 242 96 @a np 247 92 :M 247 100 :L 255 96 :L 247 92 :L .75 lw eofill -.75 -.75 247.75 100.75 .75 .75 247 92 @b -.75 -.75 247.75 100.75 .75 .75 255 96 @b 247 92.75 -.75 .75 255.75 96 .75 247 92 @a 306 96.75 -.75 .75 312.75 96 .75 306 96 @a np 308 100 :M 308 92 :L 300 96 :L 308 100 :L eofill -.75 -.75 308.75 100.75 .75 .75 308 92 @b -.75 -.75 300.75 96.75 .75 .75 308 92 @b 300 96.75 -.75 .75 308.75 100 .75 300 96 @a -.75 -.75 264.75 84.75 .75 .75 273 67 @b np 268 84 :M 262 80 :L 261 89 :L 268 84 :L eofill 262 80.75 -.75 .75 268.75 84 .75 262 80 @a -.75 -.75 261.75 89.75 .75 .75 262 80 @b -.75 -.75 261.75 89.75 .75 .75 268 84 @b 274 67.75 -.75 .75 284.75 85 .75 274 67 @a np 286 81 :M 280 85 :L 287 90 :L 286 81 :L eofill -.75 -.75 280.75 85.75 .75 .75 286 81 @b 280 85.75 -.75 .75 287.75 90 .75 280 85 @a 286 81.75 -.75 .75 287.75 90 .75 286 81 @a 1 lw 259.5 123.5 26 15 rS -.75 -.75 272.75 123.75 .75 .75 282 109 @b np 278 108 :M 284 113 :L 286 104 :L 278 108 :L .75 lw eofill 278 108.75 -.75 .75 284.75 113 .75 278 108 @a -.75 -.75 284.75 113.75 .75 .75 286 104 @b -.75 -.75 278.75 108.75 .75 .75 286 104 @b 263 109.75 -.75 .75 272.75 123 .75 263 109 @a np 261 113 :M 267 109 :L 260 104 :L 261 113 :L eofill -.75 -.75 261.75 113.75 .75 .75 267 109 @b 260 104.75 -.75 .75 267.75 109 .75 260 104 @a 260 104.75 -.75 .75 261.75 113 .75 260 104 @a 125.5 153.5 207 75 rS 127 155 204 72 rC 127 176 :M f3_12 sf ( )S 130 176 :M ( )S 133 176 :M ( )S 136 176 :M ( )S 139 176 :M ( )S 142 176 :M ( )S 145 176 :M ( )S 148 176 :M ( )S 151 176 :M ( )S 154 176 :M ( )S 157 176 :M ( )S 160 176 :M ( )S 163 176 :M ( )S 166 176 :M ( )S 169 176 :M ( )S 172 176 :M ( )S 175 176 :M f4_12 sf (T)S 182 179 :M f3_7 sf (1)S 186 176 :M f3_12 sf ( )S 189 176 :M ( )S 192 176 :M ( )S 195 176 :M ( )S 198 176 :M ( )S 201 176 :M ( )S 204 176 :M ( )S 207 176 :M ( )S 210 176 :M ( )S 213 176 :M ( )S 216 176 :M ( )S 219 176 :M f4_12 sf ( )S 222 176 :M (S)S 229 176 :M f3_12 sf ( )S 232 176 :M ( )S 235 176 :M ( )S 238 176 :M ( )S 241 176 :M ( )S 244 176 :M ( )S 247 176 :M ( )S 250 176 :M ( )S 253 176 :M ( )S 256 176 :M ( )S 259 176 :M ( )S 262 176 :M ( )S 265 176 :M ( )S 268 176 :M ( )S 271 176 :M f4_12 sf (T)S 278 179 :M f3_7 sf (2)S 127 212 :M f4_12 sf (A)S 135 212 :M ( )S 138 212 :M ( )S 141 212 :M ( )S 144 212 :M ( )S 147 212 :M ( )S 150 212 :M ( )S 153 212 :M ( )S 156 212 :M ( )S 159 212 :M ( )S 162 212 :M ( )S 165 212 :M ( )S 168 212 :M ( )S 171 212 :M ( )S 174 212 :M ( )S 177 212 :M ( )S 180 212 :M ( )S 183 212 :M ( )S 186 212 :M (B)S 194 212 :M ( )S 197 212 :M ( )S 200 212 :M ( )S 203 212 :M ( )S 206 212 :M ( )S 209 212 :M ( )S 212 212 :M ( )S 215 212 :M ( )S 218 212 :M ( )S 221 212 :M ( )S 224 212 :M ( )S 227 212 :M ( )S 230 212 :M ( )S 233 212 :M ( )S 236 212 :M ( )S 239 212 :M ( )S 242 212 :M ( )S 245 212 :M (C)S 253 212 :M ( )S 256 212 :M ( )S 259 212 :M ( )S 262 212 :M ( )S 265 212 :M ( )S 268 212 :M ( )S 271 212 :M ( )S 274 212 :M ( )S 277 212 :M ( )S 280 212 :M ( )S 283 212 :M ( )S 286 212 :M ( )S 289 212 :M ( )S 292 212 :M ( )S 295 212 :M ( )S 298 212 :M ( )S 301 212 :M ( )S 304 212 :M (D)S 127 224 :M f3_12 sf ( )S 130 224 :M ( )S 133 224 :M ( )S 136 224 :M ( )S 139 224 :M ( )S 142 224 :M ( )S 145 224 :M ( )S 148 224 :M ( )S 151 224 :M ( )S 154 224 :M ( )S 157 224 :M ( )S 160 224 :M ( )S 163 224 :M ( )S 166 224 :M ( )S 169 224 :M ( )S 172 224 :M ( )S 175 224 :M ( )S 178 224 :M ( )S 181 224 :M ( )S 184 224 :M ( )S 187 224 :M ( )S 190 224 :M ( )S 193 224 :M ( )S 196 224 :M ( )S 199 224 :M ( )S 202 224 :M ( )S 205 224 :M ( )S 208 224 :M <28>S 212 224 :M (i)S 215 224 :M (i)S 218 224 :M (i)S 221 224 :M <29>S gR gS 125 41 319 212 rC 169.5 165.5 26 15 rS 265.5 166.5 26 15 rS .75 lw 19 15 228 172 @f 196 172.75 -.75 .75 210.75 172 .75 196 172 @a np 208 168 :M 208 176 :L 216 172 :L 208 168 :L eofill -.75 -.75 208.75 176.75 .75 .75 208 168 @b -.75 -.75 208.75 176.75 .75 .75 216 172 @b 208 168.75 -.75 .75 216.75 172 .75 208 168 @a 249 172.75 -.75 .75 264.75 172 .75 249 172 @a np 251 176 :M 251 168 :L 243 172 :L 251 176 :L eofill -.75 -.75 251.75 176.75 .75 .75 251 168 @b -.75 -.75 243.75 172.75 .75 .75 251 168 @b 243 172.75 -.75 .75 251.75 176 .75 243 172 @a 182 180.75 -.75 .75 188.75 193 .75 182 180 @a np 190 190 :M 184 193 :L 190 199 :L 190 190 :L eofill -.75 -.75 184.75 193.75 .75 .75 190 190 @b 184 193.75 -.75 .75 190.75 199 .75 184 193 @a -.75 -.75 190.75 199.75 .75 .75 190 190 @b -.75 -.75 263.75 197.75 .75 .75 279 181 @b np 267 198 :M 262 193 :L 259 201 :L 267 198 :L eofill 262 193.75 -.75 .75 267.75 198 .75 262 193 @a -.75 -.75 259.75 201.75 .75 .75 262 193 @b -.75 -.75 259.75 201.75 .75 .75 267 198 @b 143 208.75 -.75 .75 177.75 208 .75 143 208 @a np 175 204 :M 175 212 :L 183 208 :L 175 204 :L eofill -.75 -.75 175.75 212.75 .75 .75 175 204 @b -.75 -.75 175.75 212.75 .75 .75 183 208 @b 175 204.75 -.75 .75 183.75 208 .75 175 204 @a 271 208.75 -.75 .75 301.75 208 .75 271 208 @a np 273 212 :M 273 204 :L 265 208 :L 273 212 :L eofill -.75 -.75 273.75 212.75 .75 .75 273 204 @b -.75 -.75 265.75 208.75 .75 .75 273 204 @b 265 208.75 -.75 .75 273.75 212 .75 265 208 @a 125.5 228.5 207 23 rS 127 230 204 20 rC 127 239 :M f3_12 sf ( )S 130 239 :M ( )S 133 239 :M ( )S 136 239 :M ( )S 139 239 :M (S)S 146 239 :M (o)S 152 239 :M (m)S 161 239 :M (e)S 166 239 :M ( )S 169 239 :M (M)S 179 239 :M (e)S 184 239 :M (m)S 193 239 :M (b)S 199 239 :M (e)S 204 239 :M (r)S 208 239 :M (s)S 213 239 :M ( )S 216 239 :M (o)S 222 239 :M (f)S 226 239 :M ( )S 229 239 :M f0_12 sf (O)S 238 239 :M (-)S 243 239 :M (E)S 251 239 :M (q)S 258 239 :M (u)S 265 239 :M (i)S 269 239 :M (v)S 276 239 :M f3_12 sf <28>S 280 239 :M f0_12 sf (C)S 289 239 :M (o)S 296 239 :M (n)S 303 239 :M (d)S 310 242 :M f0_7 sf (3)S 314 239 :M f3_12 sf <29>S gR .75 lw gS 125 41 319 212 rC 332.5 41.5 110 210 rS 334 43 107 207 rC 334 88 :M f4_12 sf ( )S 337 88 :M ( )S 340 88 :M (A)S 348 88 :M f3_12 sf ( )S 351 88 :M (o)S 357 88 :M ( )S 360 88 :M ( )S 363 88 :M ( )S 366 88 :M ( )S 369 88 :M ( )S 372 88 :M ( )S 375 88 :M ( )S 378 88 :M ( )S 381 88 :M ( )S 384 88 :M ( )S 387 88 :M ( )S 390 88 :M ( )S 393 88 :M ( )S 396 88 :M ( )S 399 88 :M ( )S 402 88 :M ( )S 405 88 :M ( )S 408 88 :M ( )S 411 88 :M ( )S 414 88 :M ( )S 417 88 :M (o)S 423 88 :M ( )S 426 88 :M f4_12 sf (D)S 334 136 :M ( )S 337 136 :M ( )S 340 136 :M ( )S 343 136 :M ( )S 346 136 :M ( )S 349 136 :M ( )S 352 136 :M ( )S 355 136 :M (B)S 363 136 :M ( )S 366 136 :M ( )S 369 136 :M ( )S 372 136 :M ( )S 375 136 :M ( )S 378 136 :M ( )S 381 136 :M ( )S 384 136 :M ( )S 387 136 :M ( )S 390 136 :M ( )S 393 136 :M ( )S 396 136 :M (C)S 334 160 :M ( )S 337 160 :M ( )S 340 160 :M ( )S 343 160 :M ( )S 346 160 :M ( )S 349 160 :M ( )S 352 160 :M ( )S 355 160 :M ( )S 358 160 :M ( )S 361 160 :M ( )S 364 160 :M ( )S 367 160 :M ( )S 370 160 :M f3_12 sf (P)S 377 160 :M (A)S 385 160 :M (G)S 334 184 :M ( )S 337 184 :M ( )S 340 184 :M ( )S 343 184 :M ( )S 346 184 :M ( )S 349 184 :M ( )S 352 184 :M ( )S 355 184 :M ( )S 358 184 :M ( )S 361 184 :M ( )S 364 184 :M f0_12 sf (C)S 373 184 :M (o)S 380 184 :M (n)S 387 184 :M (d)S 394 187 :M f0_7 sf (3)S 398 184 :M f3_12 sf ( )S 401 184 :M (=)S 334 208 :M ({)S 340 208 :M ( )S 343 208 :M ({)S 349 208 :M (D)S 357 208 :M (})S 363 208 :M ( )S 366 208 :M ( )S 369 208 :M ( )S 372 208 :M ({)S 378 208 :M (A)S 386 208 :M (,)S 390 208 :M (B)S 398 208 :M (})S 404 208 :M (,)S 334 220 :M ( )S 337 220 :M ( )S 340 220 :M ( )S 343 220 :M ({)S 349 220 :M (A)S 357 220 :M (})S 363 220 :M ( )S 366 220 :M ( )S 369 220 :M ( )S 372 220 :M ({)S 378 220 :M (C)S 386 220 :M (,)S 390 220 :M (D)S 398 220 :M (})S 404 220 :M ( )S 407 220 :M ( )S 410 220 :M (})S gR gS 125 41 319 212 rC 353 88.75 -.75 .75 360.75 118 .75 353 88 @a np 363 115 :M 356 117 :L 361 124 :L 363 115 :L eofill -.75 -.75 356.75 117.75 .75 .75 363 115 @b 356 117.75 -.75 .75 361.75 124 .75 356 117 @a -.75 -.75 361.75 124.75 .75 .75 363 115 @b -.75 -.75 407.75 118.75 .75 .75 419 87 @b np 411 118 :M 405 115 :L 405 124 :L 411 118 :L eofill 405 115.75 -.75 .75 411.75 118 .75 405 115 @a -.75 -.75 405.75 124.75 .75 .75 405 115 @b -.75 -.75 405.75 124.75 .75 .75 411 118 @b 375 133.75 -.75 .75 389.75 133 .75 375 133 @a np 387 129 :M 387 137 :L 395 133 :L 387 129 :L eofill -.75 -.75 387.75 137.75 .75 .75 387 129 @b -.75 -.75 387.75 137.75 .75 .75 395 133 @b 387 129.75 -.75 .75 395.75 133 .75 387 129 @a np 377 137 :M 377 129 :L 369 133 :L 377 137 :L eofill -.75 -.75 377.75 137.75 .75 .75 377 129 @b -.75 -.75 369.75 133.75 .75 .75 377 129 @b 369 133.75 -.75 .75 377.75 137 .75 369 133 @a -1 -1 367 210 1 1 366 200 @b -1 -1 370 210 1 1 369 200 @b 361 210 -1 1 375 209 1 361 209 @a -1 -1 367 221 1 1 366 211 @b -1 -1 370 221 1 1 369 211 @b 361 221 -1 1 375 220 1 361 220 @a gR gS 0 0 552 730 rC 263 274 :M f0_12 sf (Figure )S 300 274 :M (6)S 77 298 :M f3_12 sf .704 .07(Suppose now that we also consider DAGs with selection bias. \(iii\) of Figure 6 is an)J 59 316 :M .283 .028(example of a DAG with selection bias that entails )J 305 316 :M f0_12 sf .129(Cond)A f0_8 sf 0 3 rm .074(3)A 0 -3 rm f3_12 sf .228 .023(. Note that \(iii\) in )J 426 316 :M .257 .026(Figure 6)J 467 316 :M .312 .031( does)J 59 334 :M .119 .012(not contain a latent common cause of )J f4_12 sf .056(C)A f3_12 sf .068 .007( and )J 274 334 :M f4_12 sf .051(B)A f3_12 sf .111 .011(. However, in each of the DAGs in Figure 5)J 59 352 :M f4_12 sf .749(B)A f3_12 sf 1.279 .128( is not an ancestor of )J 180 352 :M f4_12 sf 1.702 .17(C )J 193 352 :M f3_12 sf 1.222 .122(or any member of )J f0_12 sf (S)S 296 352 :M f3_12 sf 1.107 .111(, and )J f4_12 sf .875(C)A f3_12 sf 1.369 .137( is not an ancestor of )J 448 352 :M f4_12 sf .675(B)A f3_12 sf 1.18 .118( or any)J 59 370 :M .807 .081(member of )J f0_12 sf (S)S 124 370 :M f3_12 sf .941 .094(; these properties can be shown to hold of any DAG in )J 405 370 :M f0_12 sf (O-Equiv)S f3_12 sf <28>S 453 370 :M f0_12 sf (Cond)S f0_8 sf 0 3 rm (3)S 0 -3 rm f3_12 sf (\),)S 59 388 :M .95 .095(even when there are latent variables and selection bias. Hence in the PAG we have an)J 59 406 :M .133 .013(edge )J 86 406 :M f4_12 sf .06(B)A f3_12 sf ( )S f1_12 sf S 109 406 :M f3_12 sf ( )S f4_12 sf (C)S f3_12 sf .109 .011(. Thus, if the conditional independence relations in )J f0_12 sf .041(Cond)A f0_8 sf 0 3 rm (3)S 0 -3 rm f3_12 sf .111 .011( are ever observed,)J 59 424 :M .504 .05(it can be reliably concluded that even though there may be latent variables and selection)J 59 442 :M .186 .019(bias, regardless of the causal connections of the latent variables and selection variables to)J 59 460 :M 1.1 .11(other variables, in the causal DAG that generated )J 314 460 :M f0_12 sf .524(Cond)A f0_8 sf 0 3 rm .299(3)A 0 -3 rm f3_12 sf .374 .037(, )J f4_12 sf .549(B)A f3_12 sf 1.038 .104( is not a direct or indirect)J 59 478 :M (cause of )S 102 478 :M f4_12 sf (C)S f3_12 sf ( and )S f4_12 sf (C)S f3_12 sf ( is not a direct or indirect cause of )S f4_12 sf (B)S f3_12 sf (.)S 77 502 :M .685 .068(Suppose P\()J f0_12 sf .235(O)A f3_12 sf .469 .047(\) is a distribution that has just the conditional independence relations in)J 59 520 :M f0_12 sf .141(Cond)A f0_8 sf 0 3 rm .081(3)A 0 -3 rm f3_12 sf .26 .026(. The PAG in )J 160 520 :M .294 .029(Figure 6)J 201 520 :M .31 .031( can be parameterized in such a way that it represents P\()J f0_12 sf .179(O)A f3_12 sf .134(\),)A 59 538 :M .269 .027(and is more parsimonious \(its parameterization is lower dimensional\) than any DAG that)J 59 556 :M .413 .041(contains just the variables in )J 202 556 :M f0_12 sf .195(O)A f3_12 sf .38 .038( and represents P\()J 299 556 :M f0_12 sf .229(O)A f3_12 sf .407 .041(\). \(For example in the linear case, the)J 59 574 :M .317 .032(PAG can be given a complete orientation with all \322o\323 ends removed, and interpreted as a)J 59 592 :M 1.747 .175(linear structural equation models with correlated errors. See Spirtes, et al. 1996 for)J 59 610 :M 1.345 .134(details.\) Hence the PAG can be used to find an unbiased estimator of the population)J 59 628 :M .243 .024(parameters that has lower variance than any unbiased estimator based on a DAG with the)J 59 646 :M 2.871 .287(same set of variables. Even if one is not interested in predicting the effects of)J 59 664 :M .472 .047(interventions, but merely seeks to find a parsimonious representation of a distribution in)J endp %%Page: 15 15 %%BeginPageSetup initializepage (peter; page: 15 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (15)S gR gS 0 0 552 730 rC 59 56 :M f3_12 sf 2.409 .241(order to classify or diagnose members of a population, the PAG in )J 426 56 :M 2.185 .218(Figure 6)J 469 56 :M 2.857 .286( has)J 59 74 :M (advantages over any DAG with the same variables.)S 59 105 :M f0_14 sf (I)S 64 105 :M (V)S 74 105 :M (.)S 77 105 :M ( )S 95 105 :M (Summary of PAG Theo)S 235 105 :M (rems)S 77 128 :M f3_12 sf .712 .071(Note that it follows from the definition of a PAG and the assumed acyclicity of the)J 59 146 :M 1.035 .103(directed graphs, that there are no edges )J 261 146 :M f4_12 sf 1.154(A)A f3_12 sf .472 .047( )J 273 146 :M f1_12 sf .998 .1J f4_12 sf .585(B)A f3_12 sf 1.14 .114( in a PAG, and no directed cycles in a)J 59 164 :M (PAG. \(PAGs can also be used to represent directed cyclic graphs. See Richardson 1996\).)S 77 188 :M .382 .038(Informally, a directed path in a PAG is a path that contains only \322)J 400 188 :M f1_12 sf S 412 188 :M f3_12 sf .329 .033(\323 edges pointing)J 59 206 :M (in the same direction.)S 77 230 :M f0_12 sf .302 .03(Theorem 1:)J 137 230 :M f3_12 sf .325 .033( If )J f5_12 sf (p)S 159 230 :M f3_12 sf .385 .038( is a partial ancestral graph, and there is a directed path )J 433 230 :M f4_12 sf (U)S 442 230 :M f3_12 sf .396 .04( from )J f4_12 sf .258(A)A f3_12 sf .361 .036( to)J 59 248 :M f4_12 sf .501(B)A f3_12 sf .437 .044( in )J f5_12 sf (p)S 90 248 :M f3_12 sf .724 .072(, then in every DAG )J 196 248 :M f4_12 sf (G)S 205 248 :M f3_12 sf <28>S 209 248 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 228 248 :M f3_12 sf .119(,)A f0_12 sf .316(L)A f3_12 sf .562 .056(\) with PAG )J f5_12 sf (p)S 307 248 :M f3_12 sf .693 .069(, there is a directed path from )J 458 248 :M f4_12 sf .444(A)A f3_12 sf .387 .039( to )J f4_12 sf .444(B)A f3_12 sf (,)S 59 266 :M (and )S f4_12 sf (A)S f3_12 sf ( is not an ancestor of )S 189 266 :M f0_12 sf (S)S 196 266 :M f3_12 sf (.)S 77 290 :M (\(This follows directly from the definition of an \322)S f1_12 sf S 322 290 :M f3_12 sf (\323 edge in a PAG.)S 77 314 :M .206 .021(A )J 90 314 :M f0_12 sf .178 .018(semi-directed path from )J f4_12 sf .056(A)A f3_12 sf .051 .005( to )J 240 314 :M f4_12 sf .068(B)A f3_12 sf .137 .014( in a partial ancestral graph )J f5_12 sf (p)S 388 314 :M f3_12 sf .149 .015( is an undirected path)J 59 332 :M f4_12 sf (U)S 68 332 :M f3_12 sf .594 .059( from )J 99 332 :M f4_12 sf .209(A)A f3_12 sf .182 .018( to )J f4_12 sf .209(B)A f3_12 sf .517 .052( in which no edge contains an arrowhead pointing towards )J 420 332 :M f4_12 sf .234(A)A f3_12 sf .443 .044(, \(i.e. there is)J 59 350 :M .412 .041(no arrowhead at )J f4_12 sf .173(A)A f3_12 sf .177 .018( on )J f4_12 sf (U)S 175 350 :M f3_12 sf .426 .043(, and if )J 214 350 :M f4_12 sf .212(X)A f3_12 sf .27 .027( and )J f4_12 sf (Y)S 252 350 :M f3_12 sf .358 .036( are adjacent on the path, and )J f4_12 sf .185(X)A f3_12 sf .342 .034( is between )J f4_12 sf .185(A)A f3_12 sf .394 .039( and)J 59 368 :M f4_12 sf (Y)S 66 368 :M f3_12 sf 1.071 .107( on the path, then there is no arrowhead at the )J f4_12 sf .559(X)A f3_12 sf 1.113 .111( end of the edge between )J 445 368 :M f4_12 sf .62(X)A f3_12 sf .789 .079( and )J f4_12 sf (Y)S 485 368 :M f3_12 sf (\).)S 59 386 :M 1.675 .168(Theorems 4, 5, and 6 give information about what variables appear on causal paths)J 59 404 :M 1.269 .127(between a pair of variables )J f4_12 sf .549(A)A f3_12 sf .698 .07( and )J f4_12 sf .549(B)A f3_12 sf 1.369 .137(, i.e. information about how those paths could be)J 59 422 :M (blocked.)S 77 446 :M f0_12 sf (Theorem )S 127 446 :M (2: )S 140 446 :M f3_12 sf (If )S 151 446 :M f5_12 sf (p)S 158 446 :M f3_12 sf ( is a partial ancestral graph, and there is no semi-directed path from )S 485 446 :M f4_12 sf (A)S 59 464 :M f3_12 sf (to )S f4_12 sf (B)S f3_12 sf .022 .002( in )J f5_12 sf (p)S 100 464 :M f3_12 sf .034 .003( that contains a member of )J 231 464 :M f0_12 sf (C)S 240 464 :M f3_12 sf .032 .003(, then every directed path from )J 392 464 :M f4_12 sf (A)S f3_12 sf ( to )S f4_12 sf (B)S f3_12 sf .04 .004( in every DAG)J 59 482 :M f4_12 sf (G)S 68 482 :M f3_12 sf <28>S 72 482 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 91 482 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (\) with PAG )S f5_12 sf (p)S 167 482 :M f3_12 sf ( that contains a member of )S 298 482 :M f0_12 sf (C)S 307 482 :M f3_12 sf ( also contains a member of )S 442 482 :M f0_12 sf (S)S 449 482 :M f3_12 sf (.)S 77 506 :M f0_12 sf .229 .023(Theorem 3:)J 137 506 :M f3_12 sf .378 .038( If )J 152 506 :M f5_12 sf (p)S 159 506 :M f3_12 sf .3 .03( is a partial ancestral graph of DAG )J 337 506 :M f4_12 sf (G)S 346 506 :M f3_12 sf <28>S 350 506 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 369 506 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (\),)S 387 506 :M f0_12 sf .058 .006( )J f3_12 sf .352 .035(and there is no semi-)J 59 524 :M 1.353 .135(directed path from )J 158 524 :M f4_12 sf 1.223(A)A f3_12 sf 1.066 .107( to )J f4_12 sf 1.223(B)A f3_12 sf 1.066 .107( in )J f5_12 sf (p)S 217 524 :M f3_12 sf 1.427 .143(, then every directed path from )J 381 524 :M f4_12 sf 1.4(A)A f3_12 sf 1.274 .127( to )J 408 524 :M f4_12 sf .608(B)A f3_12 sf 1.402 .14( in every DAG)J 59 542 :M f4_12 sf (G)S 68 542 :M f3_12 sf <28>S 72 542 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 91 542 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (\) with PAG )S f5_12 sf (p)S 167 542 :M f3_12 sf ( contains a member of )S 277 542 :M f0_12 sf (S)S 284 542 :M f3_12 sf (.)S 77 572 :M f0_12 sf .073 .007(Theorem )J 127 572 :M (4:)S 137 572 :M f3_12 sf .071 .007( If )J f5_12 sf (p)S 158 572 :M f3_12 sf .079 .008( is a partial ancestral graph,)J 291 572 :M f0_12 sf ( )S f3_12 sf .089 .009(and every semi-directed path from )J 463 572 :M f4_12 sf .061(A)A f3_12 sf .053 .005( to )J f4_12 sf (B)S 59 590 :M f3_12 sf .895 .09(contains some member of )J 192 590 :M f0_12 sf (C)S 201 590 :M f3_12 sf 1.226 .123( in )J 218 590 :M f5_12 sf (p)S 225 590 :M f3_12 sf .929 .093(, then every directed path from )J 385 590 :M f4_12 sf .512(A)A f3_12 sf .446 .045( to )J f4_12 sf .512(B)A f3_12 sf 1.18 .118( in every DAG)J 59 608 :M f4_12 sf (G)S 68 608 :M f3_12 sf <28>S 72 608 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 91 608 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (\) with PAG )S f5_12 sf (p)S 167 608 :M f3_12 sf ( contains a member of )S 280 608 :M f0_12 sf (S)S 287 608 :M f3_12 sf ( )S f1_12 sf S f3_12 sf ( )S f0_12 sf (C)S 311 608 :M f3_12 sf (.)S 59 645 :M f0_14 sf (V)S 69 645 :M (.)S 72 645 :M ( )S 95 645 :M (An Algo)S 143 645 :M (rithm fo)S 191 645 :M (r Constructing PAGs)S 77 668 :M f3_12 sf .041 .004(We have seen that a PAG contains valuable information about the causal relationships)J 59 686 :M 1.102 .11(between variables; it also represents conditional independence relations in the margin,)J endp %%Page: 16 16 %%BeginPageSetup initializepage (peter; page: 16 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (16)S gR gS 0 0 552 730 rC 59 56 :M f3_12 sf 1.991 .199(and can be used for classification. However, the number of observable conditional)J 59 74 :M 2.061 .206(independence relations grows exponentially with the number of members of )J 463 74 :M f0_12 sf 1.306(O)A f3_12 sf 1.722 .172(. In)J 59 92 :M .942 .094(addition, some of the independence relations are conditional on large sets of variables,)J 59 110 :M 1.188 .119(and often these cannot be reliably tested on reasonable sample sizes. Is it feasible to)J 59 128 :M (construct a PAG from data?)S 77 152 :M 1.373 .137(The Fast Causal Inference \(FCI\) algorithm constructs PAGs that are correct even)J 59 170 :M .822 .082(when selection bias may be present \(under the Causal Markov Assumption, the Causal)J 59 188 :M .531 .053(Faithfulness Assumption, the Population Inference Assumption, and the assumption that)J 59 206 :M .039 .004(conditional independence relations can be reliably tested\). The description in Spirtes et al.)J 59 224 :M .561 .056(1993 did not allow the possibility of selection bias. If the possibility of selection bias is)J 59 242 :M 1.79 .179(allowed, the algorithm described there gives the correct output, \(called a POIPG in)J 59 260 :M .507 .051(Spirtes et al. 1993\) but the conclusions that one can draw from the PAG are the slightly)J 59 278 :M .332 .033(different ones described in section )J 230 278 :M (IV)S 243 278 :M .355 .036(, rather than those described in Spirtes et al. 1993.\))J 59 296 :M 1.149 .115(Since the algorithm decides which conditional independence tests to perform, we will)J 59 314 :M 1.276 .128(assume that for each )J 168 314 :M f0_12 sf (X)S 177 314 :M f3_12 sf 1.74 .174(, )J 185 314 :M f0_12 sf (Y)S 194 314 :M f3_12 sf 1.031 .103(, and )J f0_12 sf .815(Z)A f3_12 sf 1.446 .145( included in )J 298 314 :M f0_12 sf .586(O)A f3_12 sf 1.159 .116(, the algorithm has some method for)J 59 332 :M .222 .022(reliably determining if )J f0_12 sf (X)S 180 332 :M f3_12 sf .297 .03( is independent of )J 271 332 :M f0_12 sf (Y)S 280 332 :M f3_12 sf .332 .033( given )J 314 332 :M f0_12 sf .281(Z)A f3_12 sf .096 .01( )J f1_12 sf .323A f3_12 sf .204 .02( \()J f0_12 sf .308 .031(S )J 352 332 :M f3_12 sf .143 .014(= )J f0_12 sf .106(1)A f3_12 sf .271 .027(\) in the distribution )J f4_12 sf .129(P)A f3_12 sf <28>S 476 332 :M f0_12 sf (V)S 485 332 :M f3_12 sf (\);)S 59 350 :M -.005(we will call this method an )A 192 350 :M f0_12 sf (oracle)S f3_12 sf ( for )S 243 350 :M f4_12 sf (P)S f3_12 sf <28>S 254 350 :M f0_12 sf (V)S 263 350 :M f3_12 sf (\) over )S f0_12 sf (O)S f3_12 sf ( given )S 336 350 :M f0_12 sf (S)S 343 350 :M f3_12 sf ( = )S 356 350 :M f0_12 sf (1)S f3_12 sf -.004(. In practice, the oracle can)A 59 368 :M 1.815 .181(be a statistical test of conditional independence \(which is of course not completely)J 59 386 :M (reliable on finite sample sizes.\))S 77 410 :M f0_12 sf (Theorem )S 127 410 :M (5: )S f3_12 sf (If )S f4_12 sf (P)S f3_12 sf <28>S 162 410 :M f0_12 sf (V)S 171 410 :M f3_12 sf .013 .001(\) is faithful to )J 240 410 :M f4_12 sf (G)S 249 413 :M f3_7 sf (1)S 253 410 :M f3_12 sf <28>S 257 410 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 276 413 :M f0_7 sf (1)S 280 410 :M f3_12 sf (,)S f0_12 sf (L)S f0_7 sf 0 3 rm (1)S 0 -3 rm 295 410 :M f3_12 sf .012 .001(\), and the input to the FCI algorithm is an)J 59 428 :M (oracle for )S f4_12 sf (P)S f3_12 sf (\(V\) over )S 159 428 :M f0_12 sf (O)S f3_12 sf ( given )S 201 428 :M f0_12 sf (S)S 208 428 :M f3_12 sf ( = )S 221 428 :M f0_12 sf (1)S f3_12 sf (, the output is a PAG of )S 344 428 :M f4_12 sf (G)S 353 431 :M f3_7 sf (1)S 357 428 :M f3_12 sf <28>S 361 428 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 380 431 :M f0_7 sf (1)S 384 428 :M f3_12 sf (,)S f0_12 sf (L)S f0_7 sf 0 3 rm (1)S 0 -3 rm 399 428 :M f3_12 sf (\).)S 77 452 :M .53 .053(Even if one drops the assumptions relating causal DAGs to probability distributions,)J 59 470 :M 2.28 .228(then the output PAG is still a parsimonious representation of the marginal of the)J 59 488 :M (distribution.)S 77 512 :M 1.914 .191(In the worst case the number of times the FCI algorithm consults the oracle is)J 59 530 :M 1.198 .12(exponential in the number of variables \(as is any correct algorithm whose output is a)J 59 548 :M .385 .039(function of the answers of a conditional independence oracle\). Even when the maximum)J 59 566 :M 1.077 .108(number of vertices any given vertex is adjacent to is held fixed, in the worse case the)J 59 584 :M .424 .042(algorithm is exponential in the number of variables. In light of this the title \322Fast Causal)J 59 602 :M .098 .01(Inference Algorithm\323 is perhaps somewhat over-optimistic; however, on simulated data it)J 59 620 :M .264 .026(can often be run on up to 100 variables provided the true graph is sparse. This is because)J 59 638 :M 2.776 .278(it is \(usually\) not necessary to examine the entire set of observable conditional)J 59 656 :M 1.024 .102(independence relations; many conditional independence relations are entailed by other)J 59 674 :M 2.171 .217(conditional independence relations. The FCI algorithm relies on this fact to test a)J endp %%Page: 17 17 %%BeginPageSetup initializepage (peter; page: 17 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (17)S gR gS 0 0 552 730 rC 59 56 :M f3_12 sf .166 .017(relatively small set of conditional independence relations, and test independence relations)J 59 74 :M (conditional on as few variables as possible.)S 77 98 :M .151 .015(The FCI algorithm can be divided into two parts. First the adjacencies in the PAG are)J 59 116 :M 1.013 .101(found, and then the edges are oriented. First we will describe how the adjacencies are)J 59 134 :M (found.)S 95 158 :M f8_12 sf (A)S 103 158 :M (.)S 106 158 :M ( )S 131 158 :M (Fast Causal Inference Algorithm - Adjacencies)S 77 182 :M f3_12 sf .717 .072(The details of the adjacency phase of the FCI algorithm are stated at the end of this)J 59 200 :M 1.119 .112(section. Here we will give an informal description and motivation for the steps of the)J 59 218 :M (algorithm.)S 77 242 :M 2.101 .21(There is a very simple, but slow and unreliable way of determining when two)J 59 260 :M 2.492 .249(variables in a PAG are adjacent. Start off with a complete undirected graph. By)J 59 278 :M .442 .044(definition, two variables )J f4_12 sf .151(A)A f3_12 sf .192 .019( and )J f4_12 sf .151(B)A f3_12 sf .221 .022( in a PAG for )J 289 278 :M f4_12 sf (G)S 298 278 :M f3_12 sf <28>S 302 278 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (,)S f0_12 sf (S)S 332 278 :M f3_12 sf .313 .031(\) are adjacent if and only if there)J 59 296 :M .376 .038(is no subset )J 120 296 :M f0_12 sf .177(Z)A f3_12 sf .148 .015( of )J f0_12 sf .207(O)A f3_12 sf .101(\\{)A f4_12 sf .163(A)A f3_12 sf .067(,)A f4_12 sf .163(B)A f3_12 sf .289 .029(} such that )J f4_12 sf .163(A)A f3_12 sf .207 .021( and )J f4_12 sf .163(B)A f3_12 sf .372 .037( are entailed to be independent given )J 458 296 :M f0_12 sf .363(Z)A f3_12 sf .124 .012( )J f1_12 sf .418A f3_12 sf .136 .014( )J 482 296 :M f0_12 sf (S)S 489 296 :M f3_12 sf (.)S 59 314 :M .8 .08(So, for each pair of variables )J 208 314 :M f4_12 sf .531(A)A f3_12 sf .676 .068( and )J f4_12 sf .531(B)A f3_12 sf .916 .092(, and each subset )J f0_12 sf .58(Z)A f3_12 sf .503 .05( of )J 367 314 :M f0_12 sf .18(O)A f3_12 sf .088(\\{)A f4_12 sf .141(A)A f3_12 sf .058(,)A f4_12 sf .141(B)A f3_12 sf .474 .047(}one could simply)J 59 332 :M .126 .013(ask the oracle if )J f4_12 sf .068(A)A f3_12 sf .086 .009( and )J f4_12 sf .068(B)A f3_12 sf .155 .015( are entailed to be independent given )J 358 332 :M f0_12 sf .113(Z)A f3_12 sf ( )S f1_12 sf .13A f3_12 sf ( )S f0_12 sf (S)S 388 332 :M f3_12 sf .133 .013(. The edge between )J f4_12 sf (A)S 59 350 :M f3_12 sf 1.008 .101(and )J f4_12 sf .509(B)A f3_12 sf 1.099 .11( is removed if and only if the oracle answers yes to any of these questions. The)J 59 368 :M .48 .048(problems with this algorithm are that the number of questions asked of the oracle grows)J 59 386 :M -.001(exponentially with the number of variables in )A 281 386 :M f0_12 sf (O)S f3_12 sf -.001(, and in practice, the oracle is unreliable if)A 59 404 :M .166 .017(the number of variables in )J 190 404 :M f0_12 sf .088(Z)A f3_12 sf .165 .016( is large. Clearly it is desirable to ask as few questions of the)J 59 422 :M (oracle as possible, and to ask questions in which )S f0_12 sf (Z)S f3_12 sf ( is as small as possible.)S 77 446 :M .215 .021(Since when the algorithm asks an oracle a question, it is trying to limit the size of the)J 59 464 :M .44 .044(conditioning sets, it makes sense to first ask the oracle about independencies conditional)J 59 482 :M 2.177 .218(on the empty set, then independencies conditional on sets with one variable, then)J 59 500 :M .959 .096(independencies conditional on sets with two variables, etc. However, this still leads to)J 59 518 :M (asking unnecessary questions of the oracle.)S 77 542 :M 1.429 .143(The strategy that the FCI algorithm adopts for avoiding asking such unnecessary)J 59 560 :M 1.184 .118(questions of the oracle is based on the following idea. Suppose the original unknown)J 59 578 :M .375 .038(graph is )J f4_12 sf (G)S 110 578 :M f3_12 sf <28>S 114 578 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (,)S f0_12 sf (S)S 144 578 :M f3_12 sf .549 .055(\) in )J 165 578 :M .405 .04(Figure 7 \(i\), where for purposes of illustration there is no selection)J 59 596 :M .187 .019(bias, so we do not need to condition on )J 253 596 :M f0_12 sf (S)S 260 596 :M f3_12 sf .168 .017(. If we applied the strategy described above, we)J 59 614 :M .803 .08(would first create the complete undirected graph shown in Figure 7\(ii\). After we asked)J 59 632 :M .546 .055(the oracle if )J f4_12 sf (W)S 132 632 :M f3_12 sf .644 .064( is independent of )J 225 632 :M f4_12 sf .513(X)A f3_12 sf .382 .038(, )J 239 632 :M f4_12 sf (Y)S 246 632 :M f3_12 sf .746 .075(, and )J 273 632 :M f4_12 sf (Z)S 280 632 :M f3_12 sf .583 .058( \(in each case receiving the answer \322yes\323\),)J 59 650 :M 1.42 .142(we would obtain the result shown in )J 251 650 :M 1.332 .133(Figure 7 \(iii\). At this point, although the graph)J 59 668 :M .636 .064(created is not a PAG for )J f4_12 sf (G)S 193 668 :M f3_12 sf <28>S 197 668 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (,)S f0_12 sf (S)S 227 668 :M f3_12 sf .586 .059(\) \(because it contains the wrong adjacencies\), and we)J 59 686 :M .298 .03(have very incomplete information about )J f4_12 sf (G)S 266 686 :M f3_12 sf <28>S 270 686 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (,)S f0_12 sf (S)S 300 686 :M f3_12 sf .405 .04(\), it is easy to show that )J 421 686 :M f4_12 sf (W)S 431 686 :M f3_12 sf .396 .04( does not lie)J endp %%Page: 18 18 %%BeginPageSetup initializepage (peter; page: 18 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (18)S gR gS 0 0 552 730 rC 59 56 :M f3_12 sf .379 .038(on any path between )J f4_12 sf .163(X)A f3_12 sf .208 .021( and )J f4_12 sf (Z)S 201 56 :M f3_12 sf .276 .028( in )J f4_12 sf (G)S 226 56 :M f3_12 sf <28>S 230 56 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (,)S f0_12 sf (S)S 260 56 :M f3_12 sf .364 .036(\). Hence if )J f4_12 sf .208(X)A f3_12 sf .264 .026( and )J f4_12 sf (Z)S 353 56 :M f3_12 sf .272 .027( are independent conditional)J 59 74 :M .71 .071(on any subset of )J f0_12 sf .464(O)A f3_12 sf .226(\\{)A f4_12 sf .365(A)A f3_12 sf .149(,)A f4_12 sf .365(B)A f3_12 sf .804 .08(} that contains )J 258 74 :M f4_12 sf (W)S 268 74 :M f3_12 sf .837 .084(, they are also independent given some other)J 59 92 :M 1.01 .101(subset of )J 108 92 :M f0_12 sf .434(O)A f3_12 sf .212(\\{)A f4_12 sf .341(A)A f3_12 sf .14(,)A f4_12 sf .341(B)A f3_12 sf .7 .07(} that does not contain )J f4_12 sf (W)S 271 92 :M f3_12 sf 1.024 .102(. So there is never any need to ever ask the)J 59 110 :M 1.772 .177(oracle if )J 107 110 :M f4_12 sf 1.009(X)A f3_12 sf 1.284 .128( and )J f4_12 sf (Z)S 149 110 :M f3_12 sf 1.578 .158( are independent given any subset containing )J 386 110 :M f4_12 sf (W)S 396 110 :M f3_12 sf 1.64 .164(. For example, the)J 59 128 :M .054 .005(algorithm never asks whether )J 204 128 :M f0_12 sf (X)S 213 128 :M f3_12 sf .036 .004( and )J f0_12 sf (Z)S f3_12 sf .069 .007( are independent given )J 357 128 :M f4_12 sf (W)S 367 128 :M f3_12 sf .057 .006(. This reduces the number)J 59 146 :M (of questions asked of the oracle, and limits the size of the conditioning sets.)S .75 lw 85 155 398 213 rC 217.5 155.5 132 105 rS 219 157 129 102 rC 219 202 :M ( )S 222 202 :M f4_12 sf (X)S 231 202 :M ( )S 234 202 :M ( )S 237 202 :M ( )S 240 202 :M ( )S 243 202 :M ( )S 246 202 :M ( )S 249 202 :M ( )S 252 202 :M ( )S 255 202 :M (Y)S 263 202 :M ( )S 266 202 :M ( )S 269 202 :M ( )S 272 202 :M ( )S 275 202 :M ( )S 278 202 :M ( )S 281 202 :M ( )S 284 202 :M ( )S 287 202 :M ( )S 290 202 :M (Z)S 297 202 :M ( )S 300 202 :M ( )S 303 202 :M ( )S 306 202 :M ( )S 309 202 :M ( )S 312 202 :M ( )S 315 202 :M ( )S 318 202 :M ( )S 321 202 :M ( )S 324 202 :M (W)S 219 250 :M ( )S 222 250 :M ( )S 225 250 :M ( )S 228 250 :M ( )S 231 250 :M ( )S 234 250 :M ( )S 237 250 :M ( )S 240 250 :M ( )S 243 250 :M ( )S 246 250 :M ( )S 249 250 :M ( )S 252 250 :M ( )S 255 250 :M ( )S 258 250 :M ( )S 261 250 :M ( )S 264 250 :M ( )S 267 250 :M ( )S 270 250 :M f3_12 sf ( )S 273 250 :M <28>S 277 250 :M (i)S 280 250 :M (i)S 283 250 :M <29>S gR .75 lw gS 85 155 398 213 rC 85.5 155.5 132 105 rS 87 157 129 102 rC 87 178 :M f3_12 sf ( )S 90 178 :M ( )S 93 178 :M ( )S 96 178 :M ( )S 99 178 :M ( )S 102 178 :M ( )S 105 178 :M ( )S 108 178 :M ( )S 111 178 :M f4_12 sf (T)S 87 202 :M f3_12 sf ( )S 90 202 :M f4_12 sf (X)S 99 202 :M ( )S 102 202 :M ( )S 105 202 :M ( )S 108 202 :M ( )S 111 202 :M ( )S 114 202 :M ( )S 117 202 :M ( )S 120 202 :M ( )S 123 202 :M (Y)S 131 202 :M ( )S 134 202 :M ( )S 137 202 :M ( )S 140 202 :M ( )S 143 202 :M ( )S 146 202 :M ( )S 149 202 :M ( )S 152 202 :M ( )S 155 202 :M ( )S 158 202 :M (Z)S 165 202 :M ( )S 168 202 :M ( )S 171 202 :M ( )S 174 202 :M ( )S 177 202 :M ( )S 180 202 :M ( )S 183 202 :M ( )S 186 202 :M ( )S 189 202 :M ( )S 192 202 :M (W)S 87 226 :M f3_12 sf ( )S 90 226 :M ( )S 93 226 :M ( )S 96 226 :M ( )S 99 226 :M ( )S 102 226 :M ( )S 105 226 :M ( )S 108 226 :M ( )S 111 226 :M ( )S 114 226 :M f0_12 sf (O)S 123 226 :M f3_12 sf ( )S 126 226 :M (=)S 133 226 :M ( )S 136 226 :M ({)S 142 226 :M f4_12 sf (X)S 151 226 :M f3_12 sf (,)S 155 226 :M f4_12 sf (Y)S 163 226 :M f3_12 sf (,)S 167 226 :M f4_12 sf (Z)S 174 226 :M f3_12 sf (})S 87 238 :M ( )S 90 238 :M ( )S 93 238 :M ( )S 96 238 :M ( )S 99 238 :M ( )S 102 238 :M ( )S 105 238 :M ( )S 108 238 :M ( )S 111 238 :M ( )S 114 238 :M ( )S 117 238 :M ( )S 120 238 :M ( )S 123 238 :M f4_12 sf (G)S 132 238 :M f3_12 sf <28>S 136 238 :M f0_12 sf (O)S 145 238 :M f3_12 sf (,)S 149 238 :M f0_12 sf (L)S 157 238 :M f3_12 sf (,)S 161 238 :M f0_12 sf (S)S 169 238 :M f3_12 sf <29>S 87 250 :M ( )S 90 250 :M ( )S 93 250 :M ( )S 96 250 :M ( )S 99 250 :M ( )S 102 250 :M ( )S 105 250 :M ( )S 108 250 :M ( )S 111 250 :M ( )S 114 250 :M ( )S 117 250 :M ( )S 120 250 :M ( )S 123 250 :M ( )S 126 250 :M ( )S 129 250 :M ( )S 132 250 :M ( )S 135 250 :M ( )S 138 250 :M ( )S 141 250 :M ( )S 144 250 :M ( )S 147 250 :M <28>S 151 250 :M (i)S 154 250 :M <29>S gR gS 85 155 398 213 rC 107.5 167.5 22 12 rS -.75 -.75 106.75 187.75 .75 .75 116 180 @b np 110 189 :M 105 183 :L 101 191 :L 110 189 :L eofill 105 183.75 -.75 .75 110.75 189 .75 105 183 @a -.75 -.75 101.75 191.75 .75 .75 105 183 @b -.75 -.75 101.75 191.75 .75 .75 110 189 @b 116 179.75 -.75 .75 124.75 188 .75 116 179 @a np 125 184 :M 120 189 :L 128 192 :L 125 184 :L eofill -.75 -.75 120.75 189.75 .75 .75 125 184 @b 120 189.75 -.75 .75 128.75 192 .75 120 189 @a 125 184.75 -.75 .75 128.75 192 .75 125 184 @a 137 198.75 -.75 .75 151.75 198 .75 137 198 @a np 149 194 :M 149 202 :L 157 198 :L 149 194 :L eofill -.75 -.75 149.75 202.75 .75 .75 149 194 @b -.75 -.75 149.75 202.75 .75 .75 157 198 @b 149 194.75 -.75 .75 157.75 198 .75 149 194 @a 238 199 -1 1 257 198 1 238 198 @a 271 199 -1 1 290 198 1 271 198 @a 304 199 -1 1 323 198 1 304 198 @a 1 lw -180 -90 68 28 264.5 189.5 @n -90 0 74 30 262.5 190.5 @n 90 180 86 32 303.5 205.5 @n 0 90 68 34 297.5 204.5 @n .75 lw 217.5 261.5 132 105 rS 219 263 129 102 rC 219 308 :M f3_12 sf ( )S 222 308 :M f4_12 sf (X)S 231 308 :M ( )S 234 308 :M ( )S 237 308 :M ( )S 240 308 :M ( )S 243 308 :M ( )S 246 308 :M ( )S 249 308 :M ( )S 252 308 :M ( )S 255 308 :M (Y)S 263 308 :M ( )S 266 308 :M ( )S 269 308 :M ( )S 272 308 :M ( )S 275 308 :M ( )S 278 308 :M ( )S 281 308 :M ( )S 284 308 :M ( )S 287 308 :M ( )S 290 308 :M (Z)S 297 308 :M ( )S 300 308 :M ( )S 303 308 :M ( )S 306 308 :M ( )S 309 308 :M ( )S 312 308 :M ( )S 315 308 :M ( )S 318 308 :M ( )S 321 308 :M ( )S 324 308 :M (W)S 219 332 :M f3_12 sf ( )S 222 332 :M ( )S 225 332 :M ( )S 228 332 :M ( )S 231 332 :M ( )S 234 332 :M ({)S 240 332 :M f4_12 sf (Z)S 247 332 :M f3_12 sf (})S 253 332 :M ( )S 256 332 :M ( )S 259 332 :M ( )S 262 332 :M f4_12 sf ( )S 265 332 :M ( )S 268 332 :M f3_12 sf ({)S 274 332 :M f4_12 sf (X)S 283 332 :M f3_12 sf (})S 289 332 :M (|)S 291 332 :M ({)S 297 332 :M f4_12 sf (Y)S 305 332 :M f3_12 sf (})S 219 356 :M ( )S 222 356 :M ( )S 225 356 :M ( )S 228 356 :M ( )S 231 356 :M ( )S 234 356 :M ( )S 237 356 :M ( )S 240 356 :M ( )S 243 356 :M ( )S 246 356 :M ( )S 249 356 :M ( )S 252 356 :M ( )S 255 356 :M ( )S 258 356 :M ( )S 261 356 :M ( )S 264 356 :M ( )S 267 356 :M ( )S 270 356 :M ( )S 273 356 :M <28>S 277 356 :M (i)S 280 356 :M (v)S 286 356 :M <29>S gR gS 85 155 398 213 rC 349.5 155.5 132 105 rS 351 157 129 102 rC 351 202 :M f3_12 sf ( )S 354 202 :M f4_12 sf (X)S 363 202 :M ( )S 366 202 :M ( )S 369 202 :M ( )S 372 202 :M ( )S 375 202 :M ( )S 378 202 :M ( )S 381 202 :M ( )S 384 202 :M ( )S 387 202 :M (Y)S 395 202 :M ( )S 398 202 :M ( )S 401 202 :M ( )S 404 202 :M ( )S 407 202 :M ( )S 410 202 :M ( )S 413 202 :M ( )S 416 202 :M ( )S 419 202 :M ( )S 422 202 :M (Z)S 429 202 :M ( )S 432 202 :M ( )S 435 202 :M ( )S 438 202 :M ( )S 441 202 :M ( )S 444 202 :M ( )S 447 202 :M ( )S 450 202 :M ( )S 453 202 :M ( )S 456 202 :M (W)S 351 226 :M f3_12 sf ({)S 357 226 :M f4_12 sf (W)S 368 226 :M f3_12 sf (})S 374 226 :M f4_12 sf ( )S 377 226 :M ( )S 380 226 :M f3_12 sf ({)S 386 226 :M f4_12 sf (X)S 395 226 :M f3_12 sf (})S 401 226 :M (,)S 405 226 :M ( )S 408 226 :M ({)S 414 226 :M f4_12 sf (W)S 425 226 :M f3_12 sf (})S 431 226 :M f4_12 sf ( )S 434 226 :M ( )S 437 226 :M f3_12 sf ({)S 443 226 :M f4_12 sf (Y)S 451 226 :M f3_12 sf (})S 457 226 :M (,)S 351 238 :M ( )S 354 238 :M ( )S 357 238 :M ( )S 360 238 :M ( )S 363 238 :M ( )S 366 238 :M ( )S 369 238 :M ( )S 372 238 :M ( )S 375 238 :M ( )S 378 238 :M ( )S 381 238 :M ({)S 387 238 :M f4_12 sf (W)S 398 238 :M f3_12 sf (})S 404 238 :M f4_12 sf ( )S 407 238 :M ( )S 410 238 :M f3_12 sf ({)S 416 238 :M f4_12 sf (Z)S 423 238 :M f3_12 sf (})S 351 250 :M ( )S 354 250 :M ( )S 357 250 :M ( )S 360 250 :M ( )S 363 250 :M ( )S 366 250 :M ( )S 369 250 :M ( )S 372 250 :M ( )S 375 250 :M ( )S 378 250 :M ( )S 381 250 :M ( )S 384 250 :M ( )S 387 250 :M ( )S 390 250 :M ( )S 393 250 :M ( )S 396 250 :M ( )S 399 250 :M ( )S 402 250 :M ( )S 405 250 :M <28>S 409 250 :M (i)S 412 250 :M (i)S 415 250 :M (i)S 418 250 :M <29>S gR gS 85 155 398 213 rC 369 199 -1 1 388 198 1 369 198 @a 402 199 -1 1 421 198 1 402 198 @a 1 lw 90 180 114 42 287.5 208.5 @n 0 90 110 48 282.5 205.5 @n -180 -90 68 28 396.5 189.5 @n -90 0 74 30 394.5 190.5 @n -1 -1 376 228 1 1 375 217 @b -1 -1 379 228 1 1 378 217 @b 372 228 -1 1 383 227 1 372 227 @a -1 -1 433 228 1 1 432 217 @b -1 -1 436 228 1 1 435 217 @b 429 228 -1 1 440 227 1 429 227 @a -1 -1 406 240 1 1 405 229 @b -1 -1 409 240 1 1 408 229 @b 402 240 -1 1 413 239 1 402 239 @a 237 304 -1 1 256 303 1 237 303 @a 270 304 -1 1 289 303 1 270 303 @a -1 -1 260 334 1 1 259 323 @b -1 -1 263 334 1 1 262 323 @b 256 334 -1 1 267 333 1 256 333 @a gR gS 0 0 552 730 rC 262 389 :M f0_12 sf (Figure )S 299 389 :M (7)S 77 413 :M f3_12 sf .564 .056(The adjacency phase of the FCI algorithm, which is stated at the end of this section,)J 59 431 :M (contains four steps, A\), B\), C\) and D\).)S 77 455 :M (Step A\) of the algorithm simply creates a complete undirected graph.)S 77 479 :M .538 .054(In step B\), in searching for a subset )J f0_12 sf .294(Z)A f3_12 sf .255 .026( of )J 281 479 :M f0_12 sf .223(O)A f3_12 sf .109(\\{)A f4_12 sf .175(A)A f3_12 sf .072(,)A f4_12 sf .175(B)A f3_12 sf .319 .032(} such that )J 373 479 :M f4_12 sf .346(A)A f3_12 sf .458 .046( and )J 405 479 :M f4_12 sf .122(B)A f3_12 sf .378 .038( are independent)J 59 497 :M .766 .077(conditional on )J 134 497 :M f0_12 sf .897(Z)A f3_12 sf .306 .031( )J f1_12 sf 1.033A f3_12 sf .336 .034( )J 160 497 :M f0_12 sf (S)S 167 497 :M f3_12 sf .804 .08(, the algorithm restricts the search to subsets of variables that are)J 59 515 :M .756 .076(adjacent to )J f4_12 sf .321(A)A f3_12 sf .75 .075( in the undirected graph it has constructed thus far, or subsets of variables)J 59 533 :M .523 .052(adjacent to )J 116 533 :M f4_12 sf .25(B)A f3_12 sf .523 .052(. If there were no latent variables or selection bias in )J 387 533 :M f4_12 sf (G)S 396 533 :M f3_12 sf .507 .051(, no more questions)J 59 551 :M .488 .049(would need to be asked of the oracle in order to determine the correct set of adjacencies)J 59 569 :M 1.113 .111(in the PAG. Unfortunately, if there are latent variables or selection bias, some further)J 59 587 :M (questions are needed. This is done in step D\) of the FCI algorithm.)S 77 611 :M .305 .031(Consider the DAG )J f4_12 sf (G)S 180 611 :M f3_12 sf <28>S 184 611 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (,)S f0_12 sf (S)S 214 611 :M f3_12 sf .392 .039(\) shown in )J 270 611 :M .369 .037(Figure 8 \(i\). The PAG for )J 399 611 :M f4_12 sf (G)S 408 611 :M f3_12 sf <28>S 412 611 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (,)S f0_12 sf (S)S 442 611 :M f3_12 sf .361 .036(\) is shown)J 59 629 :M .215 .021(in Figure 10)J 118 629 :M .214 .021( \(ii\). In the PAG, )J f4_12 sf .128(X)A f3_7 sf 0 3 rm (3)S 0 -3 rm 214 629 :M f3_12 sf .232 .023( is not adjacent to either )J 334 629 :M f4_12 sf (X)S f3_7 sf 0 3 rm (1)S 0 -3 rm 345 629 :M f3_12 sf .289 .029( or )J 362 629 :M f4_12 sf (X)S f3_7 sf 0 3 rm (5)S 0 -3 rm 373 629 :M f3_12 sf .201 .02(. However, in )J f4_12 sf (G)S 451 629 :M f3_12 sf <28>S 455 629 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (,)S f0_12 sf (S)S 485 629 :M f3_12 sf (\),)S 59 647 :M .042 .004(the only subset )J f0_12 sf (Z)S f3_12 sf ( of )S f0_12 sf (O)S f3_12 sf .031 .003( such that )J f4_12 sf (X)S f3_7 sf 0 3 rm (1)S 0 -3 rm 227 647 :M f3_12 sf .046 .005( and )J 251 647 :M f4_12 sf (X)S f3_7 sf 0 3 rm (5)S 0 -3 rm 262 647 :M f3_12 sf .039 .004( are independent conditional on )J f0_12 sf (Z)S f3_12 sf ( )S f1_12 sf S f3_12 sf ( )S f0_12 sf (S)S 446 647 :M f3_12 sf .029 .003(, contains)J 59 665 :M f4_12 sf (X)S f3_7 sf 0 3 rm (3)S 0 -3 rm 70 665 :M f3_12 sf 1.45 .145(. Hence we need to consider asking independence questions conditional on sets of)J 59 683 :M .224 .022(variables that contain variables not adjacent to either )J 318 683 :M f4_12 sf (X)S f3_7 sf 0 3 rm (1)S 0 -3 rm 329 683 :M f3_12 sf .188 .019( or )J f4_12 sf .206(X)A f3_7 sf 0 3 rm (5)S 0 -3 rm 356 683 :M f3_12 sf .225 .023(. The algorithm constructs a)J endp %%Page: 19 19 %%BeginPageSetup initializepage (peter; page: 19 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (19)S gR gS 0 0 552 730 rC 59 56 :M f3_12 sf 1.093 .109(set of variables called )J f0_12 sf .385(Possible-D-Sep)A 255 56 :M f3_12 sf <28>S 259 56 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf (,)S f5_12 sf (p)S 286 56 :M f3_12 sf 2.117 .212(\), which is a function of )J 422 56 :M f4_12 sf 1.649(A)A f3_12 sf 1.227 .123(, )J 438 56 :M f4_12 sf .91(B)A f3_12 sf 1.817 .182(, and the)J 59 74 :M -.004(graphical object )A 139 74 :M f5_12 sf (p)S 146 74 :M f3_12 sf -.005( constructed by the algorithm thus far which has the following property:)A 59 92 :M .69 .069(if )J f4_12 sf .637(A)A f3_12 sf .81 .081( and )J f4_12 sf .637(B)A f3_12 sf 1.532 .153( are independent conditional on any subset of )J 350 92 :M f0_12 sf .442(O)A f3_12 sf .215(\\{)A f4_12 sf .347(A)A f3_12 sf .142(,)A f4_12 sf .347(B)A f3_12 sf .377 .038(} )J 397 92 :M f1_12 sf 1.036A f3_12 sf .307 .031( )J f0_12 sf (S)S 418 92 :M f3_12 sf 1.447 .145(, then they are)J 59 110 :M (independent given some subset of )S 225 110 :M f0_12 sf (Possible-D-Sep)S 302 110 :M f3_12 sf <28>S 306 110 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf (,)S f5_12 sf (p)S 333 110 :M f3_12 sf (\) or )S 353 110 :M f0_12 sf (Possible-D-Sep)S 430 110 :M f3_12 sf <28>S 434 110 :M f4_12 sf (B)S f3_12 sf (,)S f4_12 sf (A)S f3_12 sf (,)S f5_12 sf (p)S 461 110 :M f3_12 sf (\).)S 77 134 :M f4_12 sf .085(A)A f3_12 sf .058 .006(, )J f4_12 sf .085(B)A f3_12 sf .122 .012(, and )J 124 134 :M f4_12 sf .051(C)A f3_12 sf .063 .006( form a )J f0_12 sf .036(triangle)A 210 134 :M f3_12 sf .138 .014( in a graph or a PAG if and only if )J 380 134 :M f4_12 sf .064(A)A f3_12 sf .081 .008( and )J f4_12 sf .064(B)A f3_12 sf .132 .013( are adjacent, )J 485 134 :M f4_12 sf (B)S 59 152 :M f3_12 sf 1.022 .102(and )J f4_12 sf .704 .07(C )J 93 152 :M f3_12 sf .502 .05(are adjacent,)J 155 152 :M f4_12 sf .144 .014( )J f3_12 sf .765 .077(and )J f4_12 sf .386(A)A f3_12 sf .492 .049( and )J f4_12 sf .422(C)A f3_12 sf 1.016 .102( are adjacent.)J 286 152 :M f4_12 sf .721(V)A f3_12 sf .719 .072( is in )J 323 152 :M f0_12 sf (Possible-D-Sep)S 400 152 :M f3_12 sf <28>S 404 152 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B,)S f5_12 sf (p)S 431 152 :M f3_12 sf .658 .066(\) in )J f5_12 sf (p)S 459 152 :M f3_12 sf .849 .085( if and)J 59 170 :M .396 .04(only if there is an undirected path )J f4_12 sf (U)S 236 170 :M f3_12 sf .451 .045( between )J 284 170 :M f4_12 sf .309(A)A f3_12 sf .393 .039( and )J f4_12 sf .309(B)A f3_12 sf .269 .027( in )J f5_12 sf (p)S 345 170 :M f3_12 sf .431 .043( such that for every subpath )J 485 170 :M f4_12 sf (X)S 59 188 :M f3_12 sf .057(*)A f1_12 sf .114A f3_12 sf .078 .008(* )J 87 188 :M f4_12 sf (Y)S 94 188 :M f3_12 sf .105 .01( *)J f1_12 sf .167A f3_12 sf .105 .01(* )J f4_12 sf (Z)S 131 188 :M f3_12 sf .222 .022( of )J 148 188 :M f4_12 sf (U)S 157 188 :M f3_12 sf .152 .015( either )J f4_12 sf (Y)S 197 188 :M f3_12 sf .178 .018( is a collider on the subpath, or )J f4_12 sf .099(X)A f3_12 sf .067 .007(, )J f4_12 sf (Y)S 370 188 :M f3_12 sf .205 .02(, and )J 397 188 :M f4_12 sf (Z)S 404 188 :M f3_12 sf .176 .018( form a triangle in)J 59 206 :M f5_12 sf (p)S 66 206 :M f3_12 sf 1.421 .142(.. In )J 92 206 :M .551 .055(Figure 8 \(vi\), )J f0_12 sf .219(Possible-D-Sep)A 240 206 :M f3_12 sf <28>S 244 206 :M f4_12 sf (X)S f3_7 sf 0 3 rm (1)S 0 -3 rm 255 206 :M f3_12 sf (,)S f4_12 sf (X)S f3_7 sf 0 3 rm (5)S 0 -3 rm 269 206 :M f5_12 sf (p)S 276 206 :M f3_12 sf 1.544 .154(\) = )J 297 206 :M f0_12 sf (Possible-D-Sep)S 374 206 :M f3_12 sf <28>S 378 206 :M f4_12 sf (X)S f3_7 sf 0 3 rm (5)S 0 -3 rm 389 206 :M f3_12 sf (,)S f4_12 sf (X)S f3_7 sf 0 3 rm (1)S 0 -3 rm 403 206 :M f3_12 sf .455(,)A f5_12 sf 1.212 .121( p)J f3_12 sf 1.424 .142(\) = {)J 444 206 :M f4_12 sf (X)S f3_7 sf 0 3 rm (2)S 0 -3 rm 455 206 :M f3_12 sf (,)S f4_12 sf (X)S f3_7 sf 0 3 rm (3)S 0 -3 rm 469 206 :M f3_12 sf (,)S f4_12 sf (X)S f3_7 sf 0 3 rm (4)S 0 -3 rm 483 206 :M f3_12 sf (}.)S 59 224 :M .658 .066(Thus in step D\) of the algorithm, the only independence questions that are asked of the)J 59 242 :M 1.188 .119(oracle for a given pair of variables )J 240 242 :M f4_12 sf .441(A)A f3_12 sf .561 .056( and )J f4_12 sf .441(B)A f3_12 sf .947 .095( are conditional on subsets of )J f0_12 sf .349(Possible-D-)A 59 260 :M (Sep)S 78 260 :M f3_12 sf <28>S 82 260 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf (,)S f5_12 sf (p)S 109 260 :M f3_12 sf (\) or )S 129 260 :M f0_12 sf (Possible-D-Sep)S 206 260 :M f3_12 sf <28>S 210 260 :M f4_12 sf (B)S f3_12 sf (,)S f4_12 sf (A)S f3_12 sf (,)S f5_12 sf (p)S 237 260 :M f3_12 sf (\).)S 77 284 :M 3.843 .384(The construction of )J 193 284 :M f0_12 sf (Possible-D-Sep)S 270 284 :M f3_12 sf <28>S 274 284 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf (,)S f5_12 sf (p)S 301 284 :M f3_12 sf 3.393 .339(\) requires some limited orientation)J 59 302 :M .396 .04(information about the edges in the PAG. Step C\) performs some orientation of the PAG,)J 59 320 :M 1.754 .175(so that the membership of )J 201 320 :M f0_12 sf (Possible-D-Sep)S 278 320 :M f3_12 sf <28>S 282 320 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf (,)S f5_12 sf (p)S 309 320 :M f3_12 sf 1.771 .177(\) can be calculated in step D\). The)J 59 338 :M .487 .049(orientation principles used in step C\) are essentially the same as those used in step F\) of)J 59 356 :M .061 .006(the orientation phase of the algorithm. Step F\) will be discussed in the next section, so we)J 59 374 :M (will not discuss step C\) here.)S 77 398 :M 1.447 .145(When the algorithm removes an edge between )J f4_12 sf .531(A)A f3_12 sf .704 .07( and )J 352 398 :M f4_12 sf .708(B)A f3_12 sf 1.358 .136(, it does so because it has)J 59 416 :M .113 .011(found some subset )J 152 416 :M f0_12 sf .068(Z)A f3_12 sf .056 .006( of )J f0_12 sf .079(O)A f3_12 sf .039(\\{)A f4_12 sf .062(A)A f3_12 sf (,)S f4_12 sf .062(B)A f3_12 sf .11 .011(} such that )J f4_12 sf .062(A)A f3_12 sf .082 .008( and )J 297 416 :M f4_12 sf (B)S f3_12 sf .121 .012( are independent conditional on )J f0_12 sf .052(Z)A f3_12 sf ( )S f1_12 sf .059A f3_12 sf ( )S f0_12 sf (S)S 489 416 :M f3_12 sf (.)S 59 434 :M .681 .068(The subset )J 116 434 :M f0_12 sf .292(Z)A f3_12 sf .465 .046( is recorded in )J f0_12 sf .238(Sepset)A 232 434 :M f3_12 sf <28>S 236 434 :M f4_12 sf .193(A)A f3_12 sf .079(,)A f4_12 sf .193(B)A f3_12 sf .276 .028(\) and )J f0_12 sf .172(Sepset)A 315 434 :M f3_12 sf <28>S 319 434 :M f4_12 sf .222(B)A f3_12 sf .091(,)A f4_12 sf .222(A)A f3_12 sf .558 .056(\). This information is used later)J 59 452 :M .93 .093(in the orientation phase of the algorithm. Because each edge is removed at most once,)J 59 470 :M f0_12 sf (Sepset)S 92 470 :M f3_12 sf <28>S 96 470 :M f4_12 sf .169(A)A f3_12 sf .069(,)A f4_12 sf .169(B)A f3_12 sf .345 .035(\) contains at most one subset of )J f0_12 sf .215(O)A f3_12 sf .105(\\{)A f4_12 sf .169(A)A f3_12 sf .069(,)A f4_12 sf .169(B)A f3_12 sf .365 .037(}. In the algorithm, )J 404 470 :M f0_12 sf (Adjacencies)S f3_12 sf <28>S 469 470 :M f4_12 sf (Q)S 478 470 :M f3_12 sf (,)S f4_12 sf (X)S f3_12 sf <29>S 59 488 :M 2.084 .208(is the set of vertices that are adjacent to )J f4_12 sf 1.065(X)A f3_12 sf .928 .093( in )J f4_12 sf (Q)S 314 488 :M f3_12 sf 2.684 .268(. )J 323 488 :M f0_12 sf (Adjacencies)S f3_12 sf <28>S 388 488 :M f4_12 sf (Q)S 397 488 :M f3_12 sf .307(,)A f4_12 sf .75(X)A f3_12 sf 1.734 .173(\) changes as the)J 59 506 :M .235 .024(algorithm progresses, because the algorithm removes edges from )J f4_12 sf (Q)S 386 506 :M f3_12 sf .279 .028(. \(However, )J 447 506 :M f0_12 sf (Possible-)S 59 524 :M (D-Sep)S f3_12 sf ( is calculated only once, and remains fixed, even as the graph changes.)S 164 578 :M f0_12 sf (Fast Causal Inference Algorithm - Adjacencies)S 95 605 :M f3_12 sf (A\). Form the complete undirected graph )S f4_12 sf (Q)S 300 605 :M f3_12 sf ( on the vertex set )S 385 605 :M f0_12 sf (V)S 394 605 :M f3_12 sf (.)S 95 620 :M (B\). )S 113 620 :M f4_12 sf (n)S f3_12 sf ( = 0.)S 95 635 :M (repeat)S 104 650 :M (repeat)S 113 665 :M 1.132 .113(select an ordered pair of variables )J f4_12 sf .482(X)A f3_12 sf .614 .061( and )J f4_12 sf (Y)S 328 665 :M f3_12 sf 1.173 .117( that are adjacent in )J 434 665 :M f4_12 sf (Q)S 443 665 :M f3_12 sf 1.149 .115( such that)J 113 680 :M f0_12 sf (Adjacencies)S f3_12 sf <28>S 178 680 :M f4_12 sf (Q)S 187 680 :M f3_12 sf (,)S f4_12 sf (X)S f3_12 sf (\)\\{)S f4_12 sf (Y)S 217 680 :M f3_12 sf (} that contains at least )S f4_12 sf (n)S f3_12 sf ( members,)S endp %%Page: 20 20 %%BeginPageSetup initializepage (peter; page: 20 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (20)S gR gS 0 0 552 730 rC 113 53 :M f3_12 sf (repeat)S 131 68 :M .051 .005(select a subset )J 203 68 :M f0_12 sf (T)S f3_12 sf ( of )S f0_12 sf .01(Adjacencies)A f3_12 sf <28>S 292 68 :M f4_12 sf (Q)S 301 68 :M f3_12 sf (,)S f4_12 sf (X)S f3_12 sf (\)\\{)S f4_12 sf (Y)S 331 68 :M f3_12 sf .059 .006(} with )J 365 68 :M f4_12 sf (n)S f3_12 sf .051 .005( members, and if )J 455 68 :M f4_12 sf (X)S f3_12 sf .038 .004( and )J f4_12 sf (Y)S 131 83 :M f3_12 sf .155 .016(are independent given )J f0_12 sf .059(T)A f3_12 sf ( )S 252 83 :M f1_12 sf .108A f3_12 sf ( )S f0_12 sf (S)S 271 83 :M f3_12 sf .158 .016( delete the edge between )J f4_12 sf .072(X)A f3_12 sf .091 .009( and )J f4_12 sf (Y)S 430 83 :M f3_12 sf .177 .018( from )J 460 83 :M f4_12 sf (Q)S 469 83 :M f3_12 sf .158 .016(, and)J 131 98 :M (record )S 165 98 :M f0_12 sf (T)S f3_12 sf ( in )S f0_12 sf (Sepset)S 221 98 :M f3_12 sf <28>S 225 98 :M f4_12 sf (X)S f3_12 sf (,)S f4_12 sf (Y)S 242 98 :M f3_12 sf (\) and )S f0_12 sf (Sepset)S 302 98 :M f3_12 sf <28>S 306 98 :M f4_12 sf (Y)S 313 98 :M f3_12 sf (,)S f4_12 sf (X)S f3_12 sf <29>S 113 113 :M 1.43 .143(until all subsets of )J 213 113 :M f0_12 sf (Adjacencies)S f3_12 sf <28>S 278 113 :M f4_12 sf (Q)S 287 113 :M f3_12 sf (,)S f4_12 sf (X)S f3_12 sf (\)\\{)S f4_12 sf (Y)S 317 113 :M f3_12 sf 1.018 .102(} of size )J f4_12 sf .563(n)A f3_12 sf 1.701 .17( have been checked for)J 113 128 :M (independence given )S 212 128 :M f0_12 sf (T)S f3_12 sf ( )S f1_12 sf S f3_12 sf ( )S f0_12 sf (S)S 242 128 :M f3_12 sf ( or there is no edge between )S 380 128 :M f4_12 sf (X)S f3_12 sf ( and )S f4_12 sf (Y)S 417 128 :M f3_12 sf (;)S 104 143 :M 5.367 .537(until all ordered pairs of adjacent variables )J 374 143 :M f4_12 sf 3.463(X)A f3_12 sf 4.408 .441( and )J f4_12 sf (Y)S 428 143 :M f3_12 sf 6.081 .608( such that)J 104 158 :M f0_12 sf (Adjacencies)S f3_12 sf <28>S 169 158 :M f4_12 sf (Q)S 178 158 :M f3_12 sf (,)S f4_12 sf (X)S f3_12 sf (\)\\{)S f4_12 sf (Y)S 208 158 :M f3_12 sf (} has at least )S f4_12 sf (n)S f3_12 sf ( members have been selected;)S 104 173 :M f4_12 sf (n)S f3_12 sf ( = )S 123 173 :M f4_12 sf (n)S f3_12 sf ( + 1;)S 95 188 :M .952 .095(until for each ordered pair of adjacent vertices )J 332 188 :M f4_12 sf .838(X)A f3_12 sf .623 .062(, )J 347 188 :M f4_12 sf (Y)S 354 188 :M f3_12 sf .107 .011(, )J f0_12 sf .119(Adjacencies)A f3_12 sf <28>S 426 188 :M f4_12 sf (Q)S 435 188 :M f3_12 sf (,)S f4_12 sf (X)S f3_12 sf (\)\\{)S f4_12 sf (Y)S 465 188 :M f3_12 sf 1.088 .109(} has)J 95 203 :M (fewer than )S 149 203 :M f4_12 sf (n )S f3_12 sf ( members.)S 95 218 :M .054 .005(C\). Let )J f5_12 sf (p)S 139 221 :M f3_7 sf (0)S 143 218 :M f4_12 sf ( )S f3_12 sf .059 .006(be the undirected graph resulting from step B\). Orient each edge as \322)J f1_12 sf (-)S 484 218 :M f3_12 sf S 95 233 :M .053 .005(For each triple of vertices )J 222 233 :M f4_12 sf (A)S f3_12 sf (, )S f4_12 sf (B)S f3_12 sf (, )S f4_12 sf (C)S f3_12 sf .06 .006( such that the pair )J f4_12 sf (A)S f3_12 sf (, )S f4_12 sf (B)S f3_12 sf .056 .006( and the pair )J 428 233 :M f4_12 sf (B)S f3_12 sf (, )S f4_12 sf (C)S f3_12 sf .058 .006( are each)J 95 248 :M .046 .005(adjacent in )J f5_12 sf (p)S 157 251 :M f3_7 sf (0)S 161 248 :M f3_12 sf .063 .006( but the pair )J 223 248 :M f4_12 sf (A)S f3_12 sf (, )S f4_12 sf (C)S f3_12 sf .054 .005( are not adjacent in )J f5_12 sf (p)S 345 251 :M f3_7 sf (0)S 349 248 :M f3_12 sf .094 .009(, orient )J f4_12 sf .054(A)A f3_12 sf ( )S f1_12 sf .089A f3_12 sf ( )S f4_12 sf .054(B)A f3_12 sf ( )S 428 248 :M f1_12 sf .078A f3_12 sf ( )S f4_12 sf .052(C)A f3_12 sf .043 .004( as )J f4_12 sf (A)S f3_12 sf ( )S f1_12 sf S 95 263 :M f4_12 sf (B)S f3_12 sf ( )S f1_12 sf S 117 263 :M f3_12 sf ( )S f4_12 sf (C)S f3_12 sf ( if and only if )S f4_12 sf (B)S f3_12 sf ( is not in )S 248 263 :M f0_12 sf (Sepset)S 281 263 :M f3_12 sf <28>S 285 263 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (C)S f3_12 sf (\).)S 95 278 :M 2.162 .216(D\). Let )J 139 278 :M f5_12 sf (p)S 146 281 :M f4_7 sf (1)S 150 278 :M f3_12 sf 1.904 .19( be the undirected graph resulting from step C.\) For each pair of)J 95 293 :M 1.461 .146(variables )J f4_12 sf .463(A)A f3_12 sf .589 .059( and )J f4_12 sf .463(B)A f3_12 sf .876 .088( adjacent in )J 245 293 :M f5_12 sf (p)S 252 296 :M f3_7 sf (1)S 256 293 :M f3_12 sf 1.098 .11(, if there is a subset )J 361 293 :M f0_12 sf 1.001(T)A f3_12 sf .869 .087( of )J 388 293 :M f0_12 sf (Possible-D-SEP)S 468 293 :M f3_12 sf <28>S 472 293 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf (,)S 95 308 :M f5_12 sf (p)S 102 311 :M f3_7 sf (1)S 106 308 :M f3_12 sf .019(\)\\{)A f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf .039 .004(} or of )J f0_12 sf .026(Possible-D-SEP)A 251 308 :M f3_12 sf <28>S 255 308 :M f4_12 sf .05(B)A f3_12 sf (,)S f4_12 sf .05(A)A f3_12 sf (,)S f5_12 sf .06 .006( p)J 286 311 :M f3_7 sf (1)S 290 308 :M f3_12 sf .039(\)\\{)A f4_12 sf .066(A)A f3_12 sf (,)S f4_12 sf .066(B)A f3_12 sf .117 .012(} such that )J f4_12 sf .066(A)A f3_12 sf .087 .009( and )J 406 308 :M f4_12 sf (B)S f3_12 sf .109 .011( are independent)J 95 323 :M .254 .025(conditional on )J 168 323 :M f0_12 sf .231(T)A f3_12 sf .079 .008( )J f1_12 sf .266A f3_12 sf .079 .008( )J f0_12 sf (S)S 198 323 :M f3_12 sf .288 .029(, remove the edge between )J 332 323 :M f4_12 sf .206(A)A f3_12 sf .273 .027( and )J 362 323 :M f4_12 sf .146(B)A f3_12 sf .225 .023( from )J f5_12 sf (p)S 406 326 :M f3_7 sf (1)S 410 323 :M f3_12 sf .308 .031(, and record )J 472 323 :M f0_12 sf .17(T)A f3_12 sf .218 .022( in)J 95 338 :M f0_12 sf (Sepset)S 128 338 :M f3_12 sf <28>S 132 338 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf (\) and )S f0_12 sf (Sepset)S 209 338 :M f3_12 sf <28>S 213 338 :M f4_12 sf (B)S f3_12 sf (,)S f4_12 sf (A)S f3_12 sf (\).)S 95 353 :M (E.\) Orient each edge as \322o)S 221 353 :M f1_12 sf S f3_12 sf (o\323. Call this graph )S 324 353 :M f5_12 sf (p)S 331 356 :M f3_7 sf (2)S 335 353 :M f3_12 sf (.)S 77 377 :M 1.173 .117(Figure 8 illustrates the application of the adjacency phase of the FCI algorithm to)J 59 395 :M 2.204 .22(DAG )J 91 395 :M f4_12 sf (G)S 100 395 :M f3_12 sf <28>S 104 395 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (,)S f0_12 sf (S)S 134 395 :M f3_12 sf 1.889 .189(\). We show only those steps which make changes to the PAG being)J 59 413 :M (created.)S endp %%Page: 21 21 %%BeginPageSetup initializepage (peter; page: 21 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (21)S gR .75 lw gS 95 41 433 388 rC 95.5 41.5 215 96 rS 97 43 212 93 rC 97 52 :M f3_12 sf ( )S 100 52 :M ( )S 103 52 :M ( )S 106 52 :M ( )S 109 52 :M ( )S 112 52 :M ( )S 115 52 :M ( )S 118 52 :M f4_12 sf ( )S 121 52 :M (T)S 128 55 :M f3_7 sf (1)S 132 52 :M f3_12 sf ( )S 135 52 :M ( )S 138 52 :M ( )S 141 52 :M ( )S 144 52 :M ( )S 147 52 :M ( )S 150 52 :M ( )S 153 52 :M ( )S 156 52 :M ( )S 159 52 :M ( )S 162 52 :M ( )S 165 52 :M ( )S 168 52 :M ( )S 171 52 :M ( )S 174 52 :M ( )S 177 52 :M ( )S 180 52 :M ( )S 183 52 :M ( )S 186 52 :M ( )S 189 52 :M ( )S 192 52 :M ( )S 195 52 :M ( )S 198 52 :M ( )S 201 52 :M ( )S 204 52 :M ( )S 207 52 :M ( )S 210 52 :M ( )S 213 52 :M ( )S 216 52 :M ( )S 219 52 :M ( )S 222 52 :M ( )S 225 52 :M ( )S 228 52 :M ( )S 231 52 :M ( )S 234 52 :M ( )S 237 52 :M f4_12 sf (T)S 244 55 :M f3_7 sf (2)S 97 76 :M f4_12 sf (X)S 106 79 :M f3_7 sf (1)S 110 76 :M f3_12 sf ( )S 113 76 :M ( )S 116 76 :M ( )S 119 76 :M ( )S 122 76 :M ( )S 125 76 :M ( )S 128 76 :M ( )S 131 76 :M ( )S 134 76 :M f4_12 sf (X)S 143 79 :M f3_7 sf (2)S 147 76 :M f3_12 sf ( )S 150 76 :M ( )S 153 76 :M ( )S 156 76 :M ( )S 159 76 :M ( )S 162 76 :M ( )S 165 76 :M ( )S 168 76 :M ( )S 171 76 :M f4_12 sf (X)S 180 79 :M f3_7 sf (3)S 184 76 :M f3_12 sf ( )S 187 76 :M ( )S 190 76 :M ( )S 193 76 :M ( )S 196 76 :M ( )S 199 76 :M ( )S 202 76 :M ( )S 205 76 :M ( )S 208 76 :M f4_12 sf (X)S 217 79 :M f3_7 sf (4)S 221 76 :M f3_12 sf ( )S 224 76 :M ( )S 227 76 :M ( )S 230 76 :M ( )S 233 76 :M ( )S 236 76 :M ( )S 239 76 :M ( )S 242 76 :M ( )S 245 76 :M f4_12 sf (X)S 254 79 :M f3_7 sf (5)S 97 124 :M f3_12 sf ( )S 100 124 :M ( )S 103 124 :M ( )S 106 124 :M ( )S 109 124 :M ( )S 112 124 :M ( )S 115 124 :M ( )S 118 124 :M ( )S 121 124 :M ( )S 124 124 :M ( )S 127 124 :M ( )S 130 124 :M ( )S 133 124 :M ( )S 136 124 :M ( )S 139 124 :M ( )S 142 124 :M ( )S 145 124 :M ( )S 148 124 :M ( )S 151 124 :M ( )S 154 124 :M ( )S 157 124 :M ( )S 160 124 :M ( )S 163 124 :M ( )S 166 124 :M ( )S 169 124 :M <28>S 173 124 :M (i)S 176 124 :M <29>S 180 124 :M ( )S 183 124 :M f4_12 sf (G)S 192 124 :M f3_12 sf <28>S 196 124 :M f0_12 sf (O)S 205 124 :M f3_12 sf (,)S 209 124 :M f0_12 sf (L)S 217 124 :M f3_12 sf (,)S 221 124 :M f0_12 sf (S)S 229 124 :M f3_12 sf <29>S gR gS 95 41 433 388 rC -.75 -.75 112.75 64.75 .75 .75 122 56 @b np 115 66 :M 111 60 :L 107 68 :L 115 66 :L eofill 111 60.75 -.75 .75 115.75 66 .75 111 60 @a -.75 -.75 107.75 68.75 .75 .75 111 60 @b -.75 -.75 107.75 68.75 .75 .75 115 66 @b 123 56.75 -.75 .75 131.75 65 .75 123 56 @a np 132 61 :M 127 66 :L 135 69 :L 132 61 :L eofill -.75 -.75 127.75 66.75 .75 .75 132 61 @b 127 66.75 -.75 .75 135.75 69 .75 127 66 @a 132 61.75 -.75 .75 135.75 69 .75 132 61 @a -.75 -.75 226.75 64.75 .75 .75 238 56 @b np 230 66 :M 226 60 :L 221 67 :L 230 66 :L eofill 226 60.75 -.75 .75 230.75 66 .75 226 60 @a -.75 -.75 221.75 67.75 .75 .75 226 60 @b -.75 -.75 221.75 67.75 .75 .75 230 66 @b 240 55.75 -.75 .75 247.75 62 .75 240 55 @a np 248 58 :M 243 63 :L 251 66 :L 248 58 :L eofill -.75 -.75 243.75 63.75 .75 .75 248 58 @b 243 63.75 -.75 .75 251.75 66 .75 243 63 @a 248 58.75 -.75 .75 251.75 66 .75 248 58 @a 155 73.75 -.75 .75 168.75 73 .75 155 73 @a np 157 77 :M 157 69 :L 149 73 :L 157 77 :L eofill -.75 -.75 157.75 77.75 .75 .75 157 69 @b -.75 -.75 149.75 73.75 .75 .75 157 69 @b 149 73.75 -.75 .75 157.75 77 .75 149 73 @a 188 73.75 -.75 .75 201.75 73 .75 188 73 @a np 199 69 :M 199 77 :L 207 73 :L 199 69 :L eofill -.75 -.75 199.75 77.75 .75 .75 199 69 @b -.75 -.75 199.75 77.75 .75 .75 207 73 @b 199 69.75 -.75 .75 207.75 73 .75 199 69 @a -.75 -.75 103.75 87.75 .75 .75 103 85 @b np 99 87 :M 107 87 :L 103 79 :L 99 87 :L eofill 99 87.75 -.75 .75 107.75 87 .75 99 87 @a 103 79.75 -.75 .75 107.75 87 .75 103 79 @a -.75 -.75 99.75 87.75 .75 .75 103 79 @b 90 180 112 40 160.5 87.5 @n 0 90 120 52 153.5 81.5 @n 0 90 130 40 188.5 88.5 @n 90 180 102 50 197.5 83.5 @n 1 lw 310.5 41.5 215 96 rS 312 43 212 93 rC 312 76 :M f4_12 sf (X)S 321 79 :M f3_7 sf (1)S 325 76 :M f3_12 sf ( )S 328 76 :M ( )S 331 76 :M ( )S 334 76 :M ( )S 337 76 :M ( )S 340 76 :M ( )S 343 76 :M ( )S 346 76 :M ( )S 349 76 :M f4_12 sf (X)S 358 79 :M f3_7 sf (2)S 362 76 :M f3_12 sf ( )S 365 76 :M ( )S 368 76 :M ( )S 371 76 :M ( )S 374 76 :M ( )S 377 76 :M ( )S 380 76 :M ( )S 383 76 :M ( )S 386 76 :M f4_12 sf (X)S 395 79 :M f3_7 sf (3)S 399 76 :M f3_12 sf ( )S 402 76 :M ( )S 405 76 :M ( )S 408 76 :M ( )S 411 76 :M ( )S 414 76 :M ( )S 417 76 :M ( )S 420 76 :M ( )S 423 76 :M f4_12 sf (X)S 432 79 :M f3_7 sf (4)S 436 76 :M f3_12 sf ( )S 439 76 :M ( )S 442 76 :M ( )S 445 76 :M ( )S 448 76 :M ( )S 451 76 :M ( )S 454 76 :M ( )S 457 76 :M ( )S 460 76 :M f4_12 sf (X)S 469 79 :M f3_7 sf (5)S 312 124 :M f3_12 sf ( )S 315 124 :M ( )S 318 124 :M ( )S 321 124 :M ( )S 324 124 :M ( )S 327 124 :M ( )S 330 124 :M ( )S 333 124 :M ( )S 336 124 :M ( )S 339 124 :M ( )S 342 124 :M ( )S 345 124 :M ( )S 348 124 :M ( )S 351 124 :M ( )S 354 124 :M ( )S 357 124 :M ( )S 360 124 :M ( )S 363 124 :M ( )S 366 124 :M ( )S 369 124 :M ( )S 372 124 :M ( )S 375 124 :M ( )S 378 124 :M ( )S 381 124 :M ( )S 384 124 :M ( )S 387 124 :M <28>S 391 124 :M (i)S 394 124 :M (i)S 397 124 :M <29>S 401 124 :M ( )S 404 124 :M (A)S 412 124 :M (.)S gR gS 95 41 433 388 rC 328 74 -1 1 348 73 1 328 73 @a 365 74 -1 1 385 73 1 365 73 @a 402 74 -1 1 422 73 1 402 73 @a 440 74 -1 1 460 73 1 440 73 @a 1 lw -180 -90 82 28 360.5 63.5 @n -90 0 70 32 361.5 65.5 @n -90 0 80 32 391.5 65.5 @n -180 -90 80 32 399.5 65.5 @n -90 0 76 26 432.5 62.5 @n -180 -90 70 34 432.5 66.5 @n 90 180 106 26 372.5 81.5 @n 0 90 110 28 373.5 80.5 @n 0 90 112 26 412.5 81.5 @n 90 180 112 26 416.5 81.5 @n 90 180 158 52 397.5 81.5 @n 0 90 154 52 393.5 81.5 @n 96.5 138.5 215 96 rS 98 140 212 93 rC 98 173 :M f4_12 sf (X)S 107 176 :M f3_7 sf (1)S 111 173 :M f3_12 sf ( )S 114 173 :M ( )S 117 173 :M ( )S 120 173 :M ( )S 123 173 :M ( )S 126 173 :M ( )S 129 173 :M ( )S 132 173 :M ( )S 135 173 :M f4_12 sf (X)S 144 176 :M f3_7 sf (2)S 148 173 :M f3_12 sf ( )S 151 173 :M ( )S 154 173 :M ( )S 157 173 :M ( )S 160 173 :M ( )S 163 173 :M ( )S 166 173 :M ( )S 169 173 :M ( )S 172 173 :M f4_12 sf (X)S 181 176 :M f3_7 sf (3)S 185 173 :M f3_12 sf ( )S 188 173 :M ( )S 191 173 :M ( )S 194 173 :M ( )S 197 173 :M ( )S 200 173 :M ( )S 203 173 :M ( )S 206 173 :M ( )S 209 173 :M f4_12 sf (X)S 218 176 :M f3_7 sf (4)S 222 173 :M f3_12 sf ( )S 225 173 :M ( )S 228 173 :M ( )S 231 173 :M ( )S 234 173 :M ( )S 237 173 :M ( )S 240 173 :M ( )S 243 173 :M ( )S 246 173 :M f4_12 sf (X)S 255 176 :M f3_7 sf (5)S 98 209 :M f4_12 sf ( )S 101 209 :M ( )S 104 209 :M ( )S 107 209 :M ( )S 110 209 :M ( )S 113 209 :M ( )S 116 209 :M ( )S 119 209 :M ( )S 122 209 :M ( )S 125 209 :M ( )S 128 209 :M ( )S 131 209 :M ( )S 134 209 :M ( )S 137 209 :M ( )S 140 209 :M ( )S 143 209 :M ( )S 146 209 :M ( )S 149 209 :M ( )S 152 209 :M ( )S 155 209 :M ( )S 158 209 :M ( )S 161 209 :M ( )S 164 209 :M ( )S 167 209 :M ( )S 170 209 :M ( )S 173 209 :M ( )S 176 209 :M ( )S 179 209 :M ( )S 182 209 :M ( )S 185 209 :M ( )S 188 209 :M ( )S 191 209 :M ( )S 194 209 :M ( )S 197 209 :M ( )S 200 209 :M ( )S 203 209 :M ( )S 206 209 :M ( )S 209 209 :M ( )S 212 209 :M ( )S 215 209 :M ( )S 218 209 :M ( )S 221 209 :M ( )S 224 209 :M ( )S 227 209 :M ( )S 230 209 :M ( )S 233 209 :M ( )S 236 209 :M ( )S 239 209 :M ( )S 242 209 :M ( )S 245 209 :M ( )S 248 209 :M ( )S 251 209 :M ( )S 254 209 :M ( )S 257 209 :M ( )S 260 209 :M (X)S 269 212 :M f3_7 sf (5)S 273 209 :M f3_12 sf ( )S 276 209 :M ( )S 279 209 :M ( )S 282 209 :M ( )S 285 209 :M ( )S 288 209 :M f4_12 sf (X)S 297 212 :M f3_7 sf (3)S 98 221 :M f3_12 sf ( )S 101 221 :M ( )S 104 221 :M ( )S 107 221 :M ( )S 110 221 :M ( )S 113 221 :M ( )S 116 221 :M ( )S 119 221 :M ( )S 122 221 :M ( )S 125 221 :M ( )S 128 221 :M ( )S 131 221 :M ( )S 134 221 :M ( )S 137 221 :M ( )S 140 221 :M ( )S 143 221 :M ( )S 146 221 :M ( )S 149 221 :M ( )S 152 221 :M ( )S 155 221 :M ( )S 158 221 :M ( )S 161 221 :M ( )S 164 221 :M ( )S 167 221 :M <28>S 171 221 :M (i)S 174 221 :M (i)S 177 221 :M (i)S 180 221 :M <29>S 184 221 :M ( )S 187 221 :M ( )S 190 221 :M (B)S 198 221 :M (.)S 202 221 :M (0)S 208 221 :M (.)S 212 221 :M ( )S 215 221 :M ( )S 218 221 :M ( )S 221 221 :M ( )S 224 221 :M ( )S 227 221 :M ( )S 230 221 :M ( )S 233 221 :M ( )S 236 221 :M ( )S 239 221 :M ( )S 242 221 :M ( )S 245 221 :M ( )S 248 221 :M ( )S 251 221 :M ( )S 254 221 :M ( )S 257 221 :M ( )S 260 221 :M f4_12 sf (X)S 269 224 :M f3_7 sf (1)S 273 221 :M f3_12 sf ( )S 276 221 :M ( )S 279 221 :M ( )S 282 221 :M ( )S 285 221 :M ( )S 288 221 :M f4_12 sf (X)S 297 224 :M f3_7 sf (3)S gR gS 95 41 433 388 rC 114 171 -1 1 134 170 1 114 170 @a 151 171 -1 1 171 170 1 151 170 @a 188 171 -1 1 208 170 1 188 170 @a 226 171 -1 1 246 170 1 226 170 @a 1 lw -90 0 80 32 177.5 162.5 @n 0 90 112 26 198.5 178.5 @n 90 180 112 26 202.5 178.5 @n 90 180 158 52 183.5 178.5 @n 0 90 154 52 179.5 178.5 @n -180 -90 78 30 183.5 161.5 @n 311.5 138.5 215 96 rS 313 140 212 93 rC 313 173 :M f4_12 sf (X)S 322 176 :M f3_7 sf (1)S 326 173 :M f3_12 sf ( )S 329 173 :M ( )S 332 173 :M ( )S 335 173 :M ( )S 338 173 :M ( )S 341 173 :M ( )S 344 173 :M ( )S 347 173 :M ( )S 350 173 :M f4_12 sf (X)S 359 176 :M f3_7 sf (2)S 363 173 :M f3_12 sf ( )S 366 173 :M ( )S 369 173 :M ( )S 372 173 :M ( )S 375 173 :M ( )S 378 173 :M ( )S 381 173 :M ( )S 384 173 :M ( )S 387 173 :M f4_12 sf (X)S 396 176 :M f3_7 sf (3)S 400 173 :M f3_12 sf ( )S 403 173 :M ( )S 406 173 :M ( )S 409 173 :M ( )S 412 173 :M ( )S 415 173 :M ( )S 418 173 :M ( )S 421 173 :M ( )S 424 173 :M f4_12 sf (X)S 433 176 :M f3_7 sf (4)S 437 173 :M f3_12 sf ( )S 440 173 :M ( )S 443 173 :M ( )S 446 173 :M ( )S 449 173 :M ( )S 452 173 :M ( )S 455 173 :M ( )S 458 173 :M ( )S 461 173 :M f4_12 sf (X)S 470 176 :M f3_7 sf (5)S 313 221 :M f3_12 sf ( )S 316 221 :M ( )S 319 221 :M ( )S 322 221 :M ( )S 325 221 :M ( )S 328 221 :M ( )S 331 221 :M ( )S 334 221 :M ( )S 337 221 :M ( )S 340 221 :M ( )S 343 221 :M ( )S 346 221 :M ( )S 349 221 :M ( )S 352 221 :M ( )S 355 221 :M ( )S 358 221 :M ( )S 361 221 :M ( )S 364 221 :M ( )S 367 221 :M ( )S 370 221 :M ( )S 373 221 :M ( )S 376 221 :M <28>S 380 221 :M (i)S 383 221 :M (v)S 389 221 :M <29>S 393 221 :M ( )S 396 221 :M ( )S 399 221 :M (B)S 407 221 :M (.)S 411 221 :M (1)S 417 221 :M ( )S 420 221 :M ( )S 423 221 :M ( )S 426 221 :M ( )S 429 221 :M ( )S 432 221 :M ( )S 435 221 :M ( )S 438 221 :M ( )S 441 221 :M ( )S 444 221 :M ( )S 447 221 :M ( )S 450 221 :M ( )S 453 221 :M f4_12 sf (X)S 462 224 :M f3_7 sf (2)S 466 221 :M f3_12 sf ( )S 469 221 :M ( )S 472 221 :M ( )S 475 221 :M ( )S 478 221 :M ( )S 481 221 :M f4_12 sf (X)S 490 224 :M f3_7 sf (4)S 494 221 :M f3_12 sf (|)S 496 221 :M f4_12 sf (X)S 505 224 :M f3_7 sf (3)S gR gS 95 41 433 388 rC 329 171 -1 1 349 170 1 329 170 @a 366 171 -1 1 386 170 1 366 170 @a 403 171 -1 1 423 170 1 403 170 @a 441 171 -1 1 461 170 1 441 170 @a 1 lw 0 90 112 26 413.5 178.5 @n 90 180 112 26 417.5 178.5 @n 90 180 158 52 398.5 178.5 @n 0 90 154 52 394.5 178.5 @n 96.5 235.5 215 96 rS 98 237 212 93 rC 98 270 :M f4_12 sf (X)S 107 273 :M f3_7 sf (1)S 111 270 :M f3_12 sf ( )S 114 270 :M ( )S 117 270 :M ( )S 120 270 :M ( )S 123 270 :M ( )S 126 270 :M ( )S 129 270 :M ( )S 132 270 :M ( )S 135 270 :M f4_12 sf (X)S 144 273 :M f3_7 sf (2)S 148 270 :M f3_12 sf ( )S 151 270 :M ( )S 154 270 :M ( )S 157 270 :M ( )S 160 270 :M ( )S 163 270 :M ( )S 166 270 :M ( )S 169 270 :M ( )S 172 270 :M f4_12 sf (X)S 181 273 :M f3_7 sf (3)S 185 270 :M f3_12 sf ( )S 188 270 :M ( )S 191 270 :M ( )S 194 270 :M ( )S 197 270 :M ( )S 200 270 :M ( )S 203 270 :M ( )S 206 270 :M ( )S 209 270 :M f4_12 sf (X)S 218 273 :M f3_7 sf (4)S 222 270 :M f3_12 sf ( )S 225 270 :M ( )S 228 270 :M ( )S 231 270 :M ( )S 234 270 :M ( )S 237 270 :M ( )S 240 270 :M ( )S 243 270 :M ( )S 246 270 :M f4_12 sf (X)S 255 273 :M f3_7 sf (5)S 98 318 :M f3_12 sf ( )S 101 318 :M ( )S 104 318 :M ( )S 107 318 :M ( )S 110 318 :M ( )S 113 318 :M ( )S 116 318 :M ( )S 119 318 :M ( )S 122 318 :M ( )S 125 318 :M ( )S 128 318 :M ( )S 131 318 :M ( )S 134 318 :M ( )S 137 318 :M ( )S 140 318 :M ( )S 143 318 :M ( )S 146 318 :M ( )S 149 318 :M ( )S 152 318 :M ( )S 155 318 :M ( )S 158 318 :M ( )S 161 318 :M ( )S 164 318 :M ( )S 167 318 :M ( )S 170 318 :M ( )S 173 318 :M ( )S 176 318 :M ( )S 179 318 :M ( )S 182 318 :M <28>S 186 318 :M (v)S 192 318 :M <29>S 196 318 :M ( )S 199 318 :M ( )S 202 318 :M (C)S 210 318 :M (.)S gR gS 95 41 433 388 rC 120 267.75 -.75 .75 127.75 267 .75 120 267 @a np 125 263 :M 125 271 :L 133 267 :L 125 263 :L eofill -.75 -.75 125.75 271.75 .75 .75 125 263 @b -.75 -.75 125.75 271.75 .75 .75 133 267 @b 125 263.75 -.75 .75 133.75 267 .75 125 263 @a np 122 271 :M 122 263 :L 114 267 :L 122 271 :L eofill -.75 -.75 122.75 271.75 .75 .75 122 263 @b -.75 -.75 114.75 267.75 .75 .75 122 263 @b 114 267.75 -.75 .75 122.75 271 .75 114 267 @a 157 267.75 -.75 .75 170.75 267 .75 157 267 @a np 159 271 :M 159 263 :L 151 267 :L 159 271 :L eofill -.75 -.75 159.75 271.75 .75 .75 159 263 @b -.75 -.75 151.75 267.75 .75 .75 159 263 @b 151 267.75 -.75 .75 159.75 271 .75 151 267 @a 188 267.75 -.75 .75 201.75 267 .75 188 267 @a np 199 263 :M 199 271 :L 207 267 :L 199 263 :L eofill -.75 -.75 199.75 271.75 .75 .75 199 263 @b -.75 -.75 199.75 271.75 .75 .75 207 267 @b 199 263.75 -.75 .75 207.75 267 .75 199 263 @a 232 267.75 -.75 .75 239.75 267 .75 232 267 @a np 237 263 :M 237 271 :L 245 267 :L 237 263 :L eofill -.75 -.75 237.75 271.75 .75 .75 237 263 @b -.75 -.75 237.75 271.75 .75 .75 245 267 @b 237 263.75 -.75 .75 245.75 267 .75 237 263 @a np 234 271 :M 234 263 :L 226 267 :L 234 271 :L eofill -.75 -.75 234.75 271.75 .75 .75 234 263 @b -.75 -.75 226.75 267.75 .75 .75 234 263 @b 226 267.75 -.75 .75 234.75 271 .75 226 267 @a 1 lw 0 90 112 26 194.5 278.5 @n 90 180 112 26 198.5 278.5 @n 90 180 158 52 183.5 275.5 @n 0 90 154 52 183.5 275.5 @n -.75 -.75 254.75 87.75 .75 .75 254 85 @b np 250 87 :M 258 87 :L 254 79 :L 250 87 :L .75 lw eofill 250 87.75 -.75 .75 258.75 87 .75 250 87 @a 254 79.75 -.75 .75 258.75 87 .75 254 79 @a -.75 -.75 250.75 87.75 .75 .75 254 79 @b -1 -1 280 210 1 1 279 201 @b -1 -1 283 210 1 1 282 201 @b 275 210 -1 1 287 209 1 275 209 @a -1 -1 280 222 1 1 279 213 @b -1 -1 283 222 1 1 282 213 @b 275 222 -1 1 287 221 1 275 221 @a -1 -1 475 222 1 1 474 213 @b -1 -1 478 222 1 1 477 213 @b 470 222 -1 1 482 221 1 470 221 @a 1 lw 311.5 235.5 215 96 rS 313 237 212 93 rC 313 270 :M f4_12 sf (X)S 322 273 :M f3_7 sf (1)S 326 270 :M f3_12 sf ( )S 329 270 :M ( )S 332 270 :M ( )S 335 270 :M ( )S 338 270 :M ( )S 341 270 :M ( )S 344 270 :M ( )S 347 270 :M ( )S 350 270 :M f4_12 sf (X)S 359 273 :M f3_7 sf (2)S 363 270 :M f3_12 sf ( )S 366 270 :M ( )S 369 270 :M ( )S 372 270 :M ( )S 375 270 :M ( )S 378 270 :M ( )S 381 270 :M ( )S 384 270 :M ( )S 387 270 :M f4_12 sf (X)S 396 273 :M f3_7 sf (3)S 400 270 :M f3_12 sf ( )S 403 270 :M ( )S 406 270 :M ( )S 409 270 :M ( )S 412 270 :M ( )S 415 270 :M ( )S 418 270 :M ( )S 421 270 :M ( )S 424 270 :M f4_12 sf (X)S 433 273 :M f3_7 sf (4)S 437 270 :M f3_12 sf ( )S 440 270 :M ( )S 443 270 :M ( )S 446 270 :M ( )S 449 270 :M ( )S 452 270 :M ( )S 455 270 :M ( )S 458 270 :M ( )S 461 270 :M f4_12 sf (X)S 470 273 :M f3_7 sf (5)S 313 318 :M f3_12 sf ( )S 316 318 :M ( )S 319 318 :M ( )S 322 318 :M ( )S 325 318 :M ( )S 328 318 :M ( )S 331 318 :M ( )S 334 318 :M ( )S 337 318 :M ( )S 340 318 :M ( )S 343 318 :M ( )S 346 318 :M ( )S 349 318 :M ( )S 352 318 :M ( )S 355 318 :M ( )S 358 318 :M ( )S 361 318 :M ( )S 364 318 :M ( )S 367 318 :M ( )S 370 318 :M ( )S 373 318 :M ( )S 376 318 :M ( )S 379 318 :M <28>S 383 318 :M (v)S 389 318 :M (i)S 392 318 :M <29>S 396 318 :M ( )S 399 318 :M ( )S 402 318 :M (D)S 410 318 :M ( )S 413 318 :M ( )S 416 318 :M f4_12 sf (X)S 425 321 :M f3_7 sf (1)S 429 318 :M f3_12 sf ( )S 432 318 :M ( )S 435 318 :M ( )S 438 318 :M ( )S 441 318 :M ( )S 444 318 :M f4_12 sf (X)S 453 321 :M f3_7 sf (5)S 457 318 :M f3_12 sf (|)S 459 318 :M ({)S 465 318 :M f4_12 sf (X)S 474 321 :M f3_7 sf (2)S 478 318 :M f3_12 sf (,)S 482 318 :M f4_12 sf (X)S 491 321 :M f3_7 sf (3)S 495 318 :M f3_12 sf (,)S 499 318 :M f4_12 sf (X)S 508 321 :M f3_7 sf (4)S 512 318 :M f3_12 sf (})S gR gS 95 41 433 388 rC 335 267.75 -.75 .75 342.75 267 .75 335 267 @a np 340 263 :M 340 271 :L 348 267 :L 340 263 :L eofill -.75 -.75 340.75 271.75 .75 .75 340 263 @b -.75 -.75 340.75 271.75 .75 .75 348 267 @b 340 263.75 -.75 .75 348.75 267 .75 340 263 @a np 337 271 :M 337 263 :L 329 267 :L 337 271 :L eofill -.75 -.75 337.75 271.75 .75 .75 337 263 @b -.75 -.75 329.75 267.75 .75 .75 337 263 @b 329 267.75 -.75 .75 337.75 271 .75 329 267 @a 372 267.75 -.75 .75 385.75 267 .75 372 267 @a np 374 271 :M 374 263 :L 366 267 :L 374 271 :L eofill -.75 -.75 374.75 271.75 .75 .75 374 263 @b -.75 -.75 366.75 267.75 .75 .75 374 263 @b 366 267.75 -.75 .75 374.75 271 .75 366 267 @a 403 267.75 -.75 .75 416.75 267 .75 403 267 @a np 414 263 :M 414 271 :L 422 267 :L 414 263 :L eofill -.75 -.75 414.75 271.75 .75 .75 414 263 @b -.75 -.75 414.75 271.75 .75 .75 422 267 @b 414 263.75 -.75 .75 422.75 267 .75 414 263 @a 447 267.75 -.75 .75 454.75 267 .75 447 267 @a np 452 263 :M 452 271 :L 460 267 :L 452 263 :L eofill -.75 -.75 452.75 271.75 .75 .75 452 263 @b -.75 -.75 452.75 271.75 .75 .75 460 267 @b 452 263.75 -.75 .75 460.75 267 .75 452 263 @a np 449 271 :M 449 263 :L 441 267 :L 449 271 :L eofill -.75 -.75 449.75 271.75 .75 .75 449 263 @b -.75 -.75 441.75 267.75 .75 .75 449 263 @b 441 267.75 -.75 .75 449.75 271 .75 441 267 @a 1 lw 0 90 112 26 413.5 280.5 @n 90 180 112 26 417.5 280.5 @n -1 -1 436 320 1 1 435 309 @b -1 -1 439 320 1 1 438 309 @b 430 320 -1 1 444 319 1 430 319 @a -180 -90 150 36 181.5 161.5 @n -90 0 90 42 177.5 164.5 @n -180 -90 150 36 181.5 255.5 @n -90 0 90 42 177.5 258.5 @n -180 -90 150 36 396.5 160.5 @n -90 0 90 42 392.5 163.5 @n -180 -90 150 36 395.5 255.5 @n -90 0 90 42 391.5 258.5 @n -.75 -.75 250.75 280.75 .75 .75 250 278 @b np 246 280 :M 254 280 :L 250 272 :L 246 280 :L .75 lw eofill 246 280.75 -.75 .75 254.75 280 .75 246 280 @a 250 272.75 -.75 .75 254.75 280 .75 250 272 @a -.75 -.75 246.75 280.75 .75 .75 250 272 @b -.75 -.75 106.75 255.75 .75 .75 106 254 @b np 110 253 :M 102 253 :L 106 261 :L 110 253 :L eofill 102 253.75 -.75 .75 110.75 253 .75 102 253 @a 102 253.75 -.75 .75 106.75 261 .75 102 253 @a -.75 -.75 106.75 261.75 .75 .75 110 253 @b -.75 -.75 320.75 254.75 .75 .75 320 253 @b np 324 252 :M 316 252 :L 320 260 :L 324 252 :L eofill 316 252.75 -.75 .75 324.75 252 .75 316 252 @a 316 252.75 -.75 .75 320.75 260 .75 316 252 @a -.75 -.75 320.75 260.75 .75 .75 324 252 @b -.75 -.75 469.75 280.75 .75 .75 469 278 @b np 465 280 :M 473 280 :L 469 272 :L 465 280 :L eofill 465 280.75 -.75 .75 473.75 280 .75 465 280 @a 469 272.75 -.75 .75 473.75 280 .75 469 272 @a -.75 -.75 465.75 280.75 .75 .75 469 272 @b 1 lw 113.5 42.5 26 14 rS 229.5 42.5 26 14 rS 207.5 331.5 215 96 rS 209 333 212 93 rC 209 354 :M f3_12 sf ( )S 212 354 :M ( )S 215 354 :M (o)S 221 354 :M ( )S 224 354 :M ( )S 227 354 :M ( )S 230 354 :M ( )S 233 354 :M ( )S 236 354 :M ( )S 239 354 :M ( )S 242 354 :M ( )S 245 354 :M ( )S 248 354 :M ( )S 251 354 :M ( )S 254 354 :M ( )S 257 354 :M ( )S 260 354 :M ( )S 263 354 :M ( )S 266 354 :M ( )S 269 354 :M ( )S 272 354 :M ( )S 275 354 :M ( )S 278 354 :M ( )S 281 354 :M ( )S 284 354 :M ( )S 287 354 :M ( )S 290 354 :M ( )S 293 354 :M ( )S 296 354 :M ( )S 299 354 :M ( )S 302 354 :M ( )S 305 354 :M ( )S 308 354 :M ( )S 311 354 :M ( )S 314 354 :M ( )S 317 354 :M ( )S 320 354 :M ( )S 323 354 :M ( )S 326 354 :M ( )S 329 354 :M ( )S 332 354 :M ( )S 335 354 :M (o)S 209 366 :M f4_12 sf (X)S 218 369 :M f3_7 sf (1)S 222 366 :M f3_12 sf ( )S 225 366 :M (o)S 231 366 :M ( )S 234 366 :M ( )S 237 366 :M ( )S 240 366 :M ( )S 243 366 :M (o)S 249 366 :M f4_12 sf (X)S 258 369 :M f3_7 sf (2)S 262 366 :M f3_12 sf (o)S 268 366 :M ( )S 271 366 :M ( )S 274 366 :M ( )S 277 366 :M ( )S 280 366 :M ( )S 283 366 :M (o)S 289 366 :M ( )S 292 366 :M f4_12 sf (X)S 301 369 :M f3_7 sf (3)S 305 366 :M f3_12 sf (o)S 311 366 :M ( )S 314 366 :M ( )S 317 366 :M ( )S 320 366 :M ( )S 323 366 :M (o)S 329 366 :M f4_12 sf (X)S 338 369 :M f3_7 sf (4)S 342 366 :M f3_12 sf (o)S 348 366 :M ( )S 351 366 :M ( )S 354 366 :M ( )S 357 366 :M ( )S 360 366 :M ( )S 363 366 :M (o)S 369 366 :M f4_12 sf (X)S 378 369 :M f3_7 sf (5)S 209 378 :M f3_12 sf ( )S 212 378 :M ( )S 215 378 :M ( )S 218 378 :M ( )S 221 378 :M ( )S 224 378 :M ( )S 227 378 :M ( )S 230 378 :M ( )S 233 378 :M ( )S 236 378 :M ( )S 239 378 :M ( )S 242 378 :M ( )S 245 378 :M ( )S 248 378 :M ( )S 251 378 :M ( )S 254 378 :M (o)S 260 378 :M ( )S 263 378 :M ( )S 266 378 :M ( )S 269 378 :M ( )S 272 378 :M ( )S 275 378 :M ( )S 278 378 :M ( )S 281 378 :M ( )S 284 378 :M ( )S 287 378 :M ( )S 290 378 :M ( )S 293 378 :M ( )S 296 378 :M ( )S 299 378 :M ( )S 302 378 :M ( )S 305 378 :M ( )S 308 378 :M ( )S 311 378 :M ( )S 314 378 :M ( )S 317 378 :M ( )S 320 378 :M ( )S 323 378 :M ( )S 326 378 :M ( )S 329 378 :M ( )S 332 378 :M ( )S 335 378 :M ( )S 338 378 :M ( )S 341 378 :M ( )S 344 378 :M ( )S 347 378 :M ( )S 350 378 :M ( )S 353 378 :M ( )S 356 378 :M ( )S 359 378 :M ( )S 362 378 :M ( )S 365 378 :M (o)S 209 414 :M ( )S 212 414 :M ( )S 215 414 :M ( )S 218 414 :M ( )S 221 414 :M ( )S 224 414 :M ( )S 227 414 :M ( )S 230 414 :M ( )S 233 414 :M ( )S 236 414 :M ( )S 239 414 :M ( )S 242 414 :M ( )S 245 414 :M ( )S 248 414 :M ( )S 251 414 :M ( )S 254 414 :M ( )S 257 414 :M ( )S 260 414 :M ( )S 263 414 :M ( )S 266 414 :M ( )S 269 414 :M ( )S 272 414 :M ( )S 275 414 :M ( )S 278 414 :M ( )S 281 414 :M ( )S 284 414 :M ( )S 287 414 :M ( )S 290 414 :M ( )S 293 414 :M <28>S 297 414 :M (v)S 303 414 :M (i)S 306 414 :M <29>S 310 414 :M ( )S 313 414 :M ( )S 316 414 :M (E)S 323 414 :M (.)S gR 1 lw gS 95 41 433 388 rC 0 90 112 26 309.5 377.5 @n 90 180 112 26 313.5 377.5 @n 230 364 -1 1 244 363 1 230 363 @a 268 364 -1 1 282 363 1 268 363 @a 309 364 -1 1 323 363 1 309 363 @a 347 363 -1 1 361 362 1 347 362 @a .75 lw -180 -90 140 20 288.5 349.5 @n -90 0 96 20 288.5 349.5 @n gR gS 0 0 552 730 rC 262 456 :M f0_12 sf (Figure )S 299 456 :M (8)S 95 480 :M f8_12 sf (B)S 103 480 :M (.)S 106 480 :M ( )S 131 480 :M (Fast Causal Inference Algorithm - Orientations)S 77 504 :M f3_12 sf .789 .079(Step F\) states that for each triple of vertices )J f4_12 sf .376(A)A f3_12 sf .28 .028(, )J 314 504 :M f4_12 sf .454(B)A f3_12 sf .309 .031(, )J f4_12 sf .495(C)A f3_12 sf .781 .078( such that the pair )J f4_12 sf .454(A)A f3_12 sf .309 .031(, )J f4_12 sf .454(B)A f3_12 sf .871 .087( and the)J 59 522 :M .195 .019(pair )J 81 522 :M f4_12 sf .15(B)A f3_12 sf .112 .011(, )J 95 522 :M f4_12 sf .095(C)A f3_12 sf .166 .017( are each adjacent in )J f5_12 sf (p)S 212 525 :M f3_7 sf (2)S 216 522 :M f3_12 sf .206 .021( but the pair )J 278 522 :M f4_12 sf .104(A)A f3_12 sf .071 .007(, )J f4_12 sf .113(C)A f3_12 sf .191 .019( are not adjacent in )J 395 522 :M f5_12 sf (p)S 402 525 :M f1_7 sf (2)S 406 522 :M f1_12 sf .058(,)A f3_12 sf .235 .023( orient )J f4_12 sf .142(A)A f3_12 sf .106 .011( *)J 463 522 :M f1_12 sf .116A f3_12 sf .072 .007(* )J f4_12 sf (B)S 59 540 :M f3_12 sf .382(*)A f1_12 sf .764A f3_12 sf .478 .048(* )J f4_12 sf .51(C)A f3_12 sf .424 .042( as )J f4_12 sf .467(A)A f3_12 sf .332 .033( *)J f1_12 sf S 146 540 :M f3_12 sf .304 .03( )J f4_12 sf .817(B)A f3_12 sf .304 .03( )J f1_12 sf S 177 540 :M f3_12 sf .552 .055(* )J f4_12 sf .589(C)A f3_12 sf .773 .077( if and only if )J 269 540 :M f4_12 sf .59(B)A f3_12 sf .719 .072( is not in )J 325 540 :M f0_12 sf (Sepset)S 358 540 :M f3_12 sf <28>S 362 540 :M f4_12 sf .182(A)A f3_12 sf .075(,)A f4_12 sf .199(C)A f3_12 sf .516 .052(\). The intuition behind)J 59 558 :M .306 .031(rule F\) is the following. It is easy to see if a DAG )J 306 558 :M f4_12 sf (G)S 315 558 :M f3_12 sf .281 .028( with a set of variables )J f0_12 sf (V)S 437 558 :M f3_12 sf .295 .03( contains )J 484 558 :M f4_12 sf (A)S 59 576 :M f1_12 sf S 71 576 :M f3_12 sf .29 .029( )J f4_12 sf .781(B)A f3_12 sf .29 .029( )J f1_12 sf S 102 576 :M f3_12 sf .132 .013( )J f4_12 sf .387(C)A f3_12 sf .489 .049(, and )J f4_12 sf .354(A)A f3_12 sf .451 .045( and )J f4_12 sf .387(C)A f3_12 sf .777 .078( are not adjacent, then the path between )J 383 576 :M f4_12 sf .351(A)A f3_12 sf .447 .045( and )J f4_12 sf .384(C)A f3_12 sf .635 .064( entails that )J f4_12 sf (A)S 59 594 :M f3_12 sf .602 .06(and )J f4_12 sf .332(C)A f3_12 sf .698 .07( are dependent given every subset of )J 274 594 :M f0_12 sf (V)S 283 594 :M f3_12 sf .146(\\{)A f4_12 sf .235(A)A f3_12 sf .096(,)A f4_12 sf .257(C)A f3_12 sf .519 .052(} that contains )J 386 594 :M f4_12 sf .253(B)A f3_12 sf .604 .06(. Alternatively, if )J 482 594 :M f4_12 sf (G)S 59 612 :M f3_12 sf .151 .015(contains )J 103 612 :M f4_12 sf .158(A)A f3_12 sf .059 .006( )J f1_12 sf S 128 612 :M f3_12 sf .136 .014( )J f4_12 sf .366(B)A f3_12 sf .143 .014( )J 145 612 :M f1_12 sf S 157 612 :M f3_12 sf .087 .009( )J f4_12 sf .256(C)A f3_12 sf .174 .017(, )J 175 612 :M f4_12 sf .158(A)A f3_12 sf .059 .006( )J f1_12 sf S 200 612 :M f3_12 sf .111 .011( )J f4_12 sf .299(B)A f3_12 sf .122 .012( )J 214 612 :M f1_12 sf S 226 612 :M f3_12 sf .085 .009( )J f4_12 sf .251(C)A f3_12 sf .238 .024(, or )J f4_12 sf .23(A)A f3_12 sf .09 .009( )J 270 612 :M f1_12 sf S 282 612 :M f4_12 sf .299(B)A f3_12 sf .116 .012( )J 296 612 :M f1_12 sf S 308 612 :M f3_12 sf .054 .005( )J f4_12 sf .16(C)A f3_12 sf .202 .02(, and )J f4_12 sf .147(A)A f3_12 sf .194 .019( and )J 376 612 :M f4_12 sf .077(C)A f3_12 sf .163 .016( are not adjacent, then)J 59 630 :M f4_12 sf .401(A)A f3_12 sf .531 .053( and )J 90 630 :M f4_12 sf .274(C)A f3_12 sf .544 .054( are entailed to be dependent given any subset of )J 344 630 :M f0_12 sf (V)S 353 630 :M f3_12 sf .112(\\{)A f4_12 sf .181(A)A f3_12 sf .074(,)A f4_12 sf .197(C)A f3_12 sf .431 .043(} that does not contain)J 59 648 :M f4_12 sf .122(B)A f3_12 sf .284 .028(. This property generalizes to PAGs as well. Hence, if a PAG contains )J 412 648 :M f4_12 sf .337(A)A f3_12 sf .251 .025( *)J 432 648 :M f1_12 sf .397A f3_12 sf .248 .025(* )J f4_12 sf .242(B)A f3_12 sf .18 .018( *)J 473 648 :M f1_12 sf S f3_12 sf (*)S 59 666 :M f4_12 sf .124(C)A f3_12 sf .114 .011( and )J f4_12 sf .114(A)A f3_12 sf .145 .014( and )J f4_12 sf .124(C)A f3_12 sf .225 .022( are not adjacent in the PAG, a PAG can be oriented as )J 403 666 :M f4_12 sf .222(A)A f3_12 sf .165 .017( *)J 423 666 :M f1_12 sf S 435 666 :M f3_12 sf .085 .008( )J f4_12 sf .229(B)A f3_12 sf .085 .008( )J f1_12 sf S 463 666 :M f3_12 sf .266 .027(* )J 473 666 :M f4_12 sf .128(C)A f3_12 sf .137 .014( if)J 59 684 :M .964 .096(and only if )J f0_12 sf .481(Sepset)A 153 684 :M f3_12 sf <28>S 157 684 :M f4_12 sf .589(A)A f3_12 sf .241(,)A f4_12 sf .644(C)A f3_12 sf 1.229 .123(\) does not contain )J 273 684 :M f4_12 sf .626(B)A f3_12 sf 1.37 .137(. \(In the version of the algorithm that we)J endp %%Page: 22 22 %%BeginPageSetup initializepage (peter; page: 22 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (22)S gR gS 0 0 552 730 rC 59 56 :M f3_12 sf .037 .004(implemented for the simulation studies in this paper, we have actually replaced step F\) by)J 59 74 :M (a more complicated step which is more reliable on small samples.\))S 77 98 :M .744 .074(The proofs of correctness of the orientation rules in step G\) of the algorithm are all)J 59 116 :M .528 .053(inductive arguments that show that the n+1)J 272 111 :M f3_7 sf (st)S 277 116 :M f3_12 sf .542 .054( application of an orientation rule is correct)J 59 134 :M (if the first n application of the orientation rules are correct.)S 77 158 :M .302 .03(G\(i\) states that if there is a directed path from )J 304 158 :M f4_12 sf .291(A)A f3_12 sf .265 .026( to )J 326 158 :M f4_12 sf .183(B)A f3_12 sf .307 .031(, and an edge )J f4_12 sf .183(A)A f3_12 sf .205 .02( *)J 418 158 :M f1_12 sf .248A f3_12 sf .155 .016(* )J f4_12 sf .152(B)A f3_12 sf .273 .027(, orient )J 484 158 :M f4_12 sf (A)S 59 176 :M f3_12 sf .201(*)A f1_12 sf .402A f3_12 sf .274 .027(* )J 87 176 :M f4_12 sf .569(B)A f3_12 sf .539 .054( as )J 112 176 :M f4_12 sf .406(A)A f3_12 sf .453 .045( *)J 129 176 :M f1_12 sf S 141 176 :M f3_12 sf .905 .09( )J 145 176 :M f4_12 sf .325(B)A f3_12 sf .662 .066(. This is correct because if there is a directed path from )J f4_12 sf .325(A)A f3_12 sf .284 .028( to )J f4_12 sf .325(B)A f3_12 sf .284 .028( in )J f5_12 sf (p)S 484 179 :M f1_7 sf (2)S 488 176 :M f3_12 sf (,)S 59 194 :M .128 .013(and )J f5_12 sf (p)S 86 197 :M f1_7 sf (2)S 90 194 :M f3_12 sf .155 .015( has been oriented correctly thus far, then )J 294 194 :M f4_12 sf .08(A)A f3_12 sf .133 .013( is an ancestor of )J f4_12 sf .08(B)A f3_12 sf .183 .018( in every member of)J 59 212 :M .672 .067(the )J 78 212 :M f0_12 sf .217(O)A f3_12 sf .471 .047(-equivalence class of )J 194 212 :M f4_12 sf (G)S 203 212 :M f3_12 sf <28>S 207 212 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 226 212 :M f3_12 sf .071(,)A f0_12 sf .189(L)A f3_12 sf .396 .04(\). Because each member of the )J f0_12 sf .221(O)A f3_12 sf .767 .077(-equivalence class)J 59 230 :M .218 .022(of )J f4_12 sf (G)S 81 230 :M f3_12 sf <28>S 85 230 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 104 230 :M f3_12 sf .061(,)A f0_12 sf .164(L)A f3_12 sf .294 .029(\) is acyclic, it follows that )J 246 230 :M f4_12 sf .189(B)A f3_12 sf .323 .032( is not an ancestor of )J 359 230 :M f4_12 sf .18(A)A f3_12 sf .328 .033( in any member of the )J 478 230 :M f0_12 sf (O)S f3_12 sf (-)S 59 248 :M 1.466 .147(equivalence class of )J 167 248 :M f4_12 sf (G)S 176 248 :M f3_12 sf <28>S 180 248 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 199 248 :M f3_12 sf .324(,)A f0_12 sf .865(L)A f3_12 sf 1.627 .163(\). Hence the edge can be oriented as )J f4_12 sf .792(A)A f3_12 sf .884 .088( *)J 424 248 :M f1_12 sf S 436 248 :M f3_12 sf .417 .042( )J f4_12 sf 1.12(B)A f3_12 sf 2.054 .205(. \(In the)J 59 266 :M .272 .027(version of the algorithm that we implemented for the simulation studies in this paper, we)J 59 284 :M .446 .045(have actually deleted step G \(i\) because, although theoretically correct it is expensive to)J 59 302 :M (calculate, and leads to errors on small samples.\))S 77 326 :M 1.112 .111(G\(ii\) states that if )J f4_12 sf .569(P)A f3_12 sf .636 .064( *)J 188 326 :M f1_12 sf S 200 0 5 730 rC 200 326 :M f3_12 sf 12 f6_1 :p 4.61 :m 1.61 .161( )J 200 326 :M 9.22 :m 1.533 .153( )J gR gS 0 0 552 730 rC 200 326 :M f3_12 sf 12 f6_1 :p 4.61 :m 1.61 .161( )J 200 0 5 730 rC 200 326 :M 4.61 :m 1.61 .161( )J 200 326 :M 9.22 :m 1.533 .153( )J gR gS 205 0 10 730 rC 205 326 :M f4_12 sf 12 f7_1 :p 9.22 :m 1.533 .153( )J 210 326 :M 9.22 :m 1.533 .153( )J gR gS 0 0 552 730 rC 205 326 :M f4_12 sf 12 f7_1 :p 9.993 :m (M)S 205 0 10 730 rC 205 326 :M 9.22 :m 1.533 .153( )J 210 326 :M 9.22 :m 1.533 .153( )J gR gS 215 0 4 730 rC 214 326 :M f3_12 sf 12 f6_1 :p 9.22 :m 1.533 .153( )J gR gS 0 0 552 730 rC 215 326 :M f3_12 sf 12 f6_1 :p 10.61 :m 1.463 .146( *)J 219 0 6 730 rC 219 326 :M 4.61 :m 1.61 .161( )J 220 326 :M 9.22 :m 1.533 .153( )J gR gS 0 0 552 730 rC 225 326 :M f1_12 sf .92A f3_12 sf .627 .063(* )J 248 326 :M f4_12 sf .626(R)A f3_12 sf 1.077 .108( then orient as )J f4_12 sf .626(P)A f3_12 sf .698 .07( *)J 350 326 :M f1_12 sf S 362 326 :M f3_12 sf 1.61 .161( )J 367 326 :M f4_12 sf (M)S 377 326 :M f3_12 sf .296 .03( )J f1_12 sf S 393 326 :M f3_12 sf 1.61 .161( )J 398 326 :M f4_12 sf .261(R)A f3_12 sf .822 .082(. The underlining)J 59 344 :M .259 .026(means that )J 115 344 :M f4_12 sf (M)S 125 344 :M f3_12 sf .274 .027( is an ancestor of either )J 242 344 :M f4_12 sf .245(P)A f3_12 sf .232 .023( or )J 266 344 :M f4_12 sf .137(R)A f3_12 sf .265 .026( in every member of the )J 394 344 :M f0_12 sf (O)S f3_12 sf .13 .013(-equivalence class)J 59 362 :M .631 .063(of )J f4_12 sf (G)S 82 362 :M f3_12 sf <28>S 86 362 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 105 362 :M f3_12 sf .195(,)A f0_12 sf .521(L)A f3_12 sf .542 .054(\). )J 128 362 :M f4_12 sf .356(P)A f3_12 sf .364 .036( *)J f1_12 sf S 157 362 :M f3_12 sf .287 .029( )J f4_12 sf (M)S 171 362 :M f3_12 sf 1.001 .1( means that )J 234 362 :M f4_12 sf (M)S 244 362 :M f3_12 sf .932 .093( is not an ancestor of P in any member of the )J f0_12 sf .663(O)A f3_12 sf (-)S 59 380 :M .423 .042(equivalence class of )J 161 380 :M f4_12 sf (G)S 170 380 :M f3_12 sf <28>S 174 380 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 193 380 :M f3_12 sf .084(,)A f0_12 sf .223(L)A f3_12 sf .379 .038(\). It follows that )J f4_12 sf (M)S 297 380 :M f3_12 sf .41 .041( is an ancestor of )J f4_12 sf .246(R)A f3_12 sf .564 .056( in every member of)J 59 398 :M (the )S 77 398 :M f0_12 sf (O)S f3_12 sf (-equivalence class of )S 190 398 :M f4_12 sf (G)S 199 398 :M f3_12 sf <28>S 203 398 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 222 398 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (\). Hence the edge can be oriented as )S f4_12 sf (M)S 419 398 :M f3_12 sf ( )S f1_12 sf S 434 398 :M f3_12 sf ( )S f4_12 sf (R)S f3_12 sf (.)S 77 422 :M 1.208 .121(G\(iii\) states that if )J 175 422 :M f4_12 sf .639(B)A f3_12 sf 1.207 .121( is a collider along <)J 290 422 :M f4_12 sf .45(A)A f3_12 sf .184(,)A f4_12 sf .45(B)A f3_12 sf .184(,)A f4_12 sf .491(C)A f3_12 sf .714 .071(> \(i.e. )J 352 422 :M f4_12 sf .827(A)A f3_12 sf .923 .092( *)J 370 422 :M f1_12 sf S 382 422 :M f4_12 sf 2.379 .238( B)J f3_12 sf .829 .083( )J 399 422 :M f1_12 sf S 411 422 :M f3_12 sf 1.675 .167(* )J 422 422 :M f4_12 sf (C)S f3_12 sf <29>S 434 422 :M f4_12 sf .458 .046( )J f3_12 sf 1.595 .159(in )J f5_12 sf (p)S 459 425 :M f1_7 sf (2)S 463 422 :M f3_12 sf 1.675 .167(, )J 471 422 :M f4_12 sf .737(B)A f3_12 sf .921 .092( is)J 59 440 :M .17 .017(adjacent to )J 115 440 :M f4_12 sf (D)S 124 440 :M f3_12 sf .152 .015(, and )J f4_12 sf (D)S 159 440 :M f3_12 sf .095 .01( is in )J f0_12 sf .087(Sepset)A 219 440 :M f3_12 sf <28>S 223 440 :M f4_12 sf .07(A)A f3_12 sf (,)S f4_12 sf .076(C)A f3_12 sf .14 .014(\), then orient )J 307 440 :M f4_12 sf .084(B)A f3_12 sf .086 .009( *)J f1_12 sf .137A f3_12 sf .086 .009(* )J f4_12 sf (D)S 353 440 :M f3_12 sf .229 .023( as )J 370 440 :M f4_12 sf .087(B)A f3_12 sf ( )S f1_12 sf S 392 440 :M f3_12 sf .112 .011(* )J f4_12 sf (D)S 410 440 :M f3_12 sf .179 .018(. Suppose that )J 482 440 :M f4_12 sf (D)S 59 458 :M f3_12 sf 3.324 .332(is in )J 91 458 :M f0_12 sf (Sepset)S 124 458 :M f3_12 sf <28>S 128 458 :M f4_12 sf .739(A)A f3_12 sf .302(,)A f4_12 sf .806(C)A f3_12 sf 2.211 .221(\). Because the PAG contains orientation information about every)J 59 476 :M 1.128 .113(member of the )J 137 476 :M f0_12 sf .401(O)A f3_12 sf .872 .087(-equivalence class of )J 255 476 :M f4_12 sf (G)S 264 476 :M f3_12 sf <28>S 268 476 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 287 476 :M f3_12 sf .203(,)A f0_12 sf .543(L)A f3_12 sf 1.032 .103(\), in some cases it is possible to show)J 59 494 :M .239 .024(from even a partially oriented PAG that )J 256 494 :M f4_12 sf .209 .021(X )J f3_12 sf .38 .038(and )J 287 494 :M f4_12 sf (Y)S 294 494 :M f3_12 sf .224 .022( are entailed to be dependent conditional)J 59 512 :M .592 .059(on )J 75 512 :M f0_12 sf .214(Z)A f3_12 sf .376 .038( in every member of the )J f0_12 sf .25(O)A f3_12 sf .543 .054(-equivalence class of )J 320 512 :M f4_12 sf (G)S 329 512 :M f3_12 sf <28>S 333 512 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 352 512 :M f3_12 sf .063(,)A f0_12 sf .167(L)A f3_12 sf .401 .04(\). Suppose that there were)J 59 530 :M .843 .084(some member )J 133 530 :M f4_12 sf <47D5>S 146 530 :M f3_12 sf <28>S 150 530 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf <53D5>S 173 530 :M f3_12 sf (,)S f0_12 sf <4CD5>S 188 530 :M f3_12 sf 1.031 .103(\) of the )J 230 530 :M f0_12 sf .301(O)A f3_12 sf .642 .064(-equivalence class of )J f4_12 sf (G)S 355 530 :M f3_12 sf <28>S 359 530 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 378 530 :M f3_12 sf .188(,)A f0_12 sf .501(L)A f3_12 sf .807 .081(\) in which )J 445 530 :M f4_12 sf .442(B)A f3_12 sf .835 .083( was an)J 59 548 :M .706 .071(ancestor of )J 118 548 :M f4_12 sf (D)S 127 548 :M f3_12 sf .814 .081(. We have shown that in that DAG, )J f4_12 sf .396(A)A f3_12 sf .525 .052( and )J 340 548 :M f4_12 sf .301(C)A f3_12 sf .663 .066( are entailed to be dependent)J 59 566 :M 2.153 .215(conditional on any set containing )J f4_12 sf (D)S 249 566 :M f3_12 sf 2.332 .233(. But this is a contradiction, because )J f4_12 sf (D)S 460 566 :M f3_12 sf 3.048 .305( is in)J 59 584 :M f0_12 sf (Sepset)S 92 584 :M f3_12 sf <28>S 96 584 :M f4_12 sf .144(A)A f3_12 sf .059(,)A f4_12 sf .157(C)A f3_12 sf .151 .015(\), )J f4_12 sf .144(A)A f3_12 sf .183 .018( and )J f4_12 sf .157(C)A f3_12 sf .329 .033( are entailed to be independent given )J 347 584 :M f0_12 sf (Sepset)S 380 584 :M f3_12 sf <28>S 384 584 :M f4_12 sf .1(A)A f3_12 sf (,)S f4_12 sf .109(C)A f3_12 sf .261 .026(\) in every member)J 59 602 :M .145 .015(of the )J 91 602 :M f0_12 sf (O)S f3_12 sf .093 .009(-equivalence class of )J f4_12 sf (G)S 213 602 :M f3_12 sf <28>S 217 602 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 236 602 :M f3_12 sf (,)S f0_12 sf .073(L)A f3_12 sf .107 .011(\), and )J 278 602 :M f0_12 sf (Sepset)S 311 602 :M f3_12 sf <28>S 315 602 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf .052(C)A f3_12 sf .098 .01(\) does not contain )J f4_12 sf (D)S 431 602 :M f3_12 sf .138 .014(. Hence B is)J 59 620 :M (not an ancestor of )S 148 620 :M f4_12 sf (D)S 157 620 :M f3_12 sf ( in any member of the )S 266 620 :M f0_12 sf (O)S f3_12 sf (-equivalence class of )S 379 620 :M f4_12 sf (G)S 388 620 :M f3_12 sf <28>S 392 620 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 411 620 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (\).)S 77 644 :M .145 .015(Rule G\(iv\) states that if )J 194 644 :M f4_12 sf .293(B)A f3_12 sf .112 .011( )J 211 644 :M f1_12 sf S 223 644 :M f3_12 sf .062(*)A f4_12 sf .083(C)A f3_12 sf .052 .005(, )J f4_12 sf .076(B)A f3_12 sf ( )S f1_12 sf S 265 644 :M f3_12 sf .051 .005( )J f4_12 sf (D)S 277 644 :M f3_12 sf .174 .017(, and )J 304 644 :M f4_12 sf (D)S 313 644 :M f3_12 sf .1 .01( o)J f1_12 sf .159A f3_12 sf .1 .01(* )J f4_12 sf .106(C)A f3_12 sf .152 .015(, then orient as )J 429 644 :M f4_12 sf (D)S 438 644 :M f3_12 sf .217 .022( )J 442 644 :M f1_12 sf S 454 644 :M f3_12 sf .088 .009(* )J f4_12 sf .094(C)A f3_12 sf .181 .018(. By)J 59 662 :M .141 .014(hypothesis, )J 117 662 :M f4_12 sf .117(B)A f3_12 sf .197 .02( is not an ancestor of )J f4_12 sf .127(C)A f3_12 sf .202 .02(, but is an ancestor of )J 344 662 :M f4_12 sf (D)S 353 662 :M f3_12 sf .219 .022(. Hence )J 394 662 :M f4_12 sf (D)S 403 662 :M f3_12 sf .204 .02( is not an ancestor)J 59 680 :M (of )S 72 680 :M f4_12 sf (C)S f3_12 sf (.)S endp %%Page: 23 23 %%BeginPageSetup initializepage (peter; page: 23 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (23)S gR gS 0 0 552 730 rC 77 56 :M f3_12 sf .234 .023(Rule G\(v\) is somewhat complicated, and not often applied, so the reader may wish to)J 59 74 :M .145 .015(skip the following discussion of it. G\(v\) uses definite discriminating paths to orient edges)J 59 92 :M .686 .069(in the PAG. The concept of a definite discriminating path is illustrated in Figure 9, and)J 59 110 :M (defined more formally below.)S 77 134 :M .645 .065(Consider the graph shown in )J 223 134 :M .626 .063(Figure 9. All of the orientations shown can be derived)J 59 152 :M 1.16 .116(without using step G\(v\) of the FCI algorithm. We have proved that in every member)J 59 170 :M f4_12 sf <47D5>S 72 170 :M f3_12 sf <28>S 76 170 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf <53D5>S 99 170 :M f3_12 sf (,)S f0_12 sf <4CD5>S 114 170 :M f3_12 sf .282 .028(\) of the )J 153 170 :M f0_12 sf .089(O)A f3_12 sf .193 .019(-equivalence class of )J 267 170 :M f4_12 sf (G)S 276 170 :M f3_12 sf <28>S 280 170 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 299 170 :M f3_12 sf (,)S f0_12 sf .13(L)A f3_12 sf .127 .013(\), if )J f4_12 sf .205<47D5>A 344 170 :M f3_12 sf <28>S 348 170 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf <53D5>S 371 170 :M f3_12 sf (,)S f0_12 sf <4CD5>S 386 170 :M f3_12 sf .285 .029(\) entails that )J f4_12 sf .177 .018(A )J f3_12 sf .192 .019( and )J f4_12 sf (C)S 59 188 :M f3_12 sf .161 .016(are entailed to be independent conditional on )J 280 188 :M f0_12 sf .142(Z)A f3_12 sf .048 .005( )J f1_12 sf .163A f3_12 sf .048 .005( )J f0_12 sf (S)S 310 188 :M f3_12 sf .211 .021(, and )J 337 188 :M f4_12 sf <47D5>S 350 188 :M f3_12 sf <28>S 354 188 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf <53D5>S 377 188 :M f3_12 sf (,)S f0_12 sf <4CD5>S 392 188 :M f3_12 sf .173 .017(\) does not entail that)J 59 206 :M f4_12 sf .283 .028(A )J f3_12 sf .306 .031( and )J f4_12 sf .263(C)A f3_12 sf .59 .059( are independent conditional on any proper subset of )J 365 206 :M f0_12 sf .591(Z)A f3_12 sf .221 .022( )J 377 206 :M f1_12 sf .327A f3_12 sf .097 .01( )J f0_12 sf .378<53D5>A 400 206 :M f3_12 sf .625 .062(, then )J 432 206 :M f0_12 sf (Sepset)S 465 206 :M f3_12 sf <28>S 469 206 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (C)S f3_12 sf <29>S 59 224 :M 1.263 .126(contains )J f4_12 sf .409(B)A f3_12 sf .787 .079( if and only if the edges between )J 280 224 :M f4_12 sf .587(B)A f3_12 sf .746 .075( and )J f4_12 sf .64(C)A f3_12 sf .843 .084(, and )J 349 224 :M f4_12 sf .622(B)A f3_12 sf .824 .082( and )J 382 224 :M f4_12 sf (D)S 391 224 :M f3_12 sf .838 .084( do not collide at )J f4_12 sf .507(B)A f3_12 sf (.)S 59 242 :M .665 .066(This orientation rule, when applied repeatedly, can lead to \322long distance\323 orientations,)J 59 260 :M .661 .066(i.e. a conditional independence relation between )J 301 260 :M f4_12 sf .569(A)A f3_12 sf .754 .075( and )J 334 260 :M f4_12 sf .419(C)A f3_12 sf .768 .077( can orient edges into )J 454 260 :M f4_12 sf .242(B)A f3_12 sf .631 .063(, even)J 59 278 :M .816 .082(though the shortest path from )J 210 278 :M f4_12 sf .425(A)A f3_12 sf .37 .037( to )J f4_12 sf .425(B)A f3_12 sf .903 .09( is arbitrarily long. The sorts of paths that have to)J 59 296 :M .054 .005(exist in order to perform this \322long distance\323 orientation are called definite discriminating)J 59 314 :M .768 .077(paths. \(In a partial ancestral graph )J 233 314 :M f5_12 sf (p)S 240 314 :M f3_12 sf .419 .042(, )J f4_12 sf (U)S 256 314 :M f3_12 sf .286 .029( is a )J f0_12 sf 1.372 .137(definite discriminating path)J f3_12 sf .126 .013( )J f0_12 sf .236(for)A f3_12 sf .138 .014( )J 451 314 :M f4_12 sf .474(B)A f3_12 sf .762 .076( if and)J 59 332 :M .408 .041(only if )J f4_12 sf (U)S 104 332 :M f3_12 sf .476 .048( is an undirected path between )J 258 332 :M f4_12 sf .281(X)A f3_12 sf .357 .036( and )J f4_12 sf (Y)S 296 332 :M f3_12 sf .649 .065(, )J 303 332 :M f4_12 sf .221(B)A f3_12 sf .431 .043( is the predecessor of )J f4_12 sf (Y)S 424 332 :M f3_12 sf .621 .062( on )J 444 332 :M f4_12 sf (U)S 453 332 :M f3_12 sf .649 .065(, )J 460 332 :M f4_12 sf .309(B)A f3_12 sf .115 .012( )J f1_12 sf S 477 332 :M f3_12 sf .714 .071( )J 481 332 :M f4_12 sf (X)S f3_12 sf (,)S 59 350 :M .652 .065(every vertex on )J 140 350 :M f4_12 sf (U)S 149 350 :M f3_12 sf .634 .063( except for the endpoints and possibly )J 341 350 :M f4_12 sf .425(B)A f3_12 sf .667 .067( is a collider on )J 430 350 :M f4_12 sf (U)S 439 350 :M f3_12 sf .652 .065(, for every)J 59 368 :M 1.062 .106(vertex )J 94 368 :M f4_12 sf .733(V)A f3_12 sf .75 .075( on )J f4_12 sf (U)S 131 368 :M f3_12 sf 1.158 .116( except for the endpoints there is an edge )J 346 368 :M f4_12 sf 1.206(V)A f3_12 sf .493 .049( )J 358 368 :M f1_12 sf S 370 368 :M f3_12 sf 1.699 .17( )J 375 368 :M f4_12 sf (Y)S 382 368 :M f3_12 sf 1.208 .121(, and )J f4_12 sf .874(X)A f3_12 sf 1.159 .116( and )J 445 368 :M f4_12 sf (Y)S 452 368 :M f3_12 sf 1.258 .126( are not)J 59 386 :M (adjacent.\). An example of a definite discriminating path is given in Figure 9.)S 77 497 :M ( )S 108 396 228 100 rC 126 429 :M f4_12 sf ( )S 129 429 :M ( )S 132 429 :M ( )S 135 429 :M ( )S 138 429 :M ( )S 141 429 :M ( )S 144 429 :M ( )S 147 429 :M ( )S 150 429 :M ( )S 153 429 :M ( )S 156 429 :M ( )S 159 429 :M ( )S 162 429 :M ( )S 165 429 :M ( )S 168 429 :M ( )S 171 429 :M ( )S 174 429 :M ( )S 177 429 :M ( )S 180 429 :M ( )S 183 429 :M ( )S 186 429 :M ( )S 189 429 :M ( )S 192 429 :M ( )S 195 429 :M ( )S 198 429 :M ( )S 201 429 :M ( )S 204 429 :M ( )S 207 429 :M ( )S 210 429 :M ( )S 213 429 :M ( )S 216 429 :M ( )S 219 429 :M ( )S 222 429 :M ( )S 225 429 :M ( )S 228 429 :M ( )S 231 429 :M ( )S 234 429 :M ( )S 237 429 :M ( )S 240 429 :M ( )S 243 429 :M ( )S 246 429 :M ( )S 249 429 :M ( )S 252 429 :M ( )S 255 429 :M ( )S 258 429 :M ( )S 261 429 :M ( )S 264 429 :M ( )S 267 429 :M ( )S 270 429 :M ( )S 273 429 :M ( )S 276 429 :M ( )S 279 429 :M ( )S 282 429 :M ( )S 285 429 :M ( )S 288 429 :M ( )S 291 429 :M ( )S 294 429 :M ( )S 297 429 :M ( )S 300 429 :M ( )S 303 429 :M ( )S 306 429 :M ( )S 309 429 :M ( )S 312 429 :M ( )S 315 429 :M (C)S 126 465 :M ( )S 129 465 :M ( )S 132 465 :M ( )S 135 465 :M ( )S 138 465 :M ( )S 141 465 :M ( )S 144 465 :M ( )S 147 465 :M ( )S 150 465 :M ( )S 153 465 :M ( )S 156 465 :M f3_12 sf ( )S 159 465 :M ( )S 162 465 :M ( )S 165 465 :M ( )S 168 465 :M ( )S 171 465 :M ( )S 174 465 :M ( )S 177 465 :M ( )S 180 465 :M ( )S 183 465 :M ( )S 186 465 :M ( )S 189 465 :M ( )S 192 465 :M ( )S 195 465 :M ( )S 198 465 :M ( )S 201 465 :M ( )S 204 465 :M ( )S 207 465 :M ( )S 210 465 :M ( )S 213 465 :M ( )S 216 465 :M ( )S 219 465 :M ( )S 222 465 :M ( )S 225 465 :M ( )S 228 465 :M ( )S 231 465 :M ( )S 234 465 :M ( )S 237 465 :M ( )S 240 465 :M ( )S 243 465 :M ( )S 246 465 :M ( )S 249 465 :M ( )S 252 465 :M ( )S 255 465 :M ( )S 258 465 :M ( )S 261 465 :M ( )S 264 465 :M ( )S 267 465 :M ( )S 270 465 :M ( )S 273 465 :M ( )S 276 465 :M ( )S 279 465 :M ( )S 282 465 :M ( )S 285 465 :M ( )S 288 465 :M ( )S 291 465 :M ( )S 294 465 :M ( )S 297 465 :M ( )S 300 465 :M ( )S 303 465 :M ( )S 306 465 :M ( )S 309 465 :M ( )S 312 465 :M ( )S 315 465 :M ( )S 318 465 :M ( )S 321 465 :M ( )S 324 465 :M (o)S 126 477 :M f4_12 sf ( )S 129 477 :M ( )S 132 477 :M ( )S 135 477 :M ( )S 138 477 :M ( )S 141 477 :M ( )S 144 477 :M ( )S 147 477 :M ( )S 150 477 :M ( )S 153 477 :M ( )S 156 477 :M ( )S 159 477 :M ( )S 162 477 :M ( )S 165 477 :M ( )S 168 477 :M ( )S 171 477 :M ( )S 174 477 :M ( )S 177 477 :M ( )S 180 477 :M ( )S 183 477 :M ( )S 186 477 :M ( )S 189 477 :M ( )S 192 477 :M ( )S 195 477 :M ( )S 198 477 :M ( )S 201 477 :M ( )S 204 477 :M ( )S 207 477 :M ( )S 210 477 :M ( )S 213 477 :M ( )S 216 477 :M ( )S 219 477 :M ( )S 222 477 :M ( )S 225 477 :M ( )S 228 477 :M (A)S 236 477 :M ( )S 239 477 :M f3_12 sf (o)S 245 477 :M f4_12 sf ( )S 248 477 :M ( )S 251 477 :M ( )S 254 477 :M ( )S 257 477 :M ( )S 260 477 :M ( )S 263 477 :M ( )S 266 477 :M ( )S 269 477 :M ( )S 272 477 :M ( )S 275 477 :M (D)S 283 477 :M ( )S 286 477 :M ( )S 289 477 :M ( )S 292 477 :M ( )S 295 477 :M ( )S 298 477 :M ( )S 301 477 :M ( )S 304 477 :M ( )S 307 477 :M ( )S 310 477 :M ( )S 313 477 :M ( )S 316 477 :M f3_12 sf (o)S 322 477 :M f4_12 sf ( )S 325 477 :M (B)S gR gS 107 395 230 102 rC 293 472.75 -.75 .75 312.75 472 .75 293 472 @a np 295 476 :M 295 468 :L 287 472 :L 295 476 :L .75 lw eofill -.75 -.75 295.75 476.75 .75 .75 295 468 @b -.75 -.75 287.75 472.75 .75 .75 295 468 @b 287 472.75 -.75 .75 295.75 476 .75 287 472 @a 243 473.75 -.75 .75 263.75 473 .75 243 473 @a np 261 469 :M 261 477 :L 269 473 :L 261 469 :L eofill -.75 -.75 261.75 477.75 .75 .75 261 469 @b -.75 -.75 261.75 477.75 .75 .75 269 473 @b 261 469.75 -.75 .75 269.75 473 .75 261 469 @a -.75 -.75 280.75 467.75 .75 .75 311 433 @b np 307 433 :M 312 437 :L 315 429 :L 307 433 :L eofill 307 433.75 -.75 .75 312.75 437 .75 307 433 @a -.75 -.75 312.75 437.75 .75 .75 315 429 @b -.75 -.75 307.75 433.75 .75 .75 315 429 @b 321 437.75 -.75 .75 326.75 458 .75 321 437 @a np 318 440 :M 325 438 :L 320 431 :L 318 440 :L eofill -.75 -.75 318.75 440.75 .75 .75 325 438 @b 320 431.75 -.75 .75 325.75 438 .75 320 431 @a -.75 -.75 318.75 440.75 .75 .75 320 431 @b gR gS 0 0 552 730 rC 138 506 :M f0_12 sf (Figure )S 175 506 :M (9: )S 188 506 :M f8_12 sf ( )S f3_12 sf (<)S 198 506 :M f4_12 sf (A, D,B,C)S f3_12 sf (> is a definite discriminating path for )S 423 506 :M f4_12 sf (B)S 163 542 :M f0_12 sf (Fast Causal Inference Algorithm - Orientations)S 95 557 :M f3_12 sf .479 .048(F. For each triple of vertices )J f4_12 sf .23(A)A f3_12 sf .171 .017(, )J 252 557 :M f4_12 sf .375(B)A f3_12 sf .279 .028(, )J 266 557 :M f4_12 sf .299(C)A f3_12 sf .471 .047( such that the pair )J f4_12 sf .274(A)A f3_12 sf .204 .02(, )J 379 557 :M f4_12 sf .286(B)A f3_12 sf .461 .046( and the pair )J 452 557 :M f4_12 sf .257(B)A f3_12 sf .175 .018(, )J f4_12 sf .28(C)A f3_12 sf .475 .048( are)J 95 572 :M .265 .027(each adjacent in )J f5_12 sf (p)S 183 575 :M f3_7 sf .409 .041(2 )J 189 572 :M f3_12 sf .321 .032(but the pair )J 248 572 :M f4_12 sf .247(A)A f3_12 sf .184 .018(, )J 262 572 :M f4_12 sf .193(C)A f3_12 sf .32 .032( are not adjacent in )J f4_12 sf .177(F)A f3_12 sf .314 .031(', orient )J f4_12 sf .177(A)A f3_12 sf .131 .013( *)J 433 572 :M f1_12 sf .402A f3_12 sf .251 .025(* )J f4_12 sf .245(B)A f3_12 sf .182 .018( *)J 474 572 :M f1_12 sf S f3_12 sf (*)S 95 587 :M f4_12 sf (C)S f3_12 sf ( as )S 119 587 :M f4_12 sf (A)S f3_12 sf ( *)S f1_12 sf S 150 587 :M f3_12 sf ( )S f4_12 sf (B)S f3_12 sf ( )S f1_12 sf S 178 587 :M f3_12 sf (* )S f4_12 sf (C)S f3_12 sf ( if and only if )S f4_12 sf (B)S f3_12 sf ( is not in )S 315 587 :M f0_12 sf (Sepset)S 348 587 :M f3_12 sf <28>S 352 587 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (C)S f3_12 sf (\).)S 95 602 :M (G. repeat)S 131 617 :M .575 .057(\(i\) If there is a directed path from )J 301 617 :M f4_12 sf .539(A)A f3_12 sf .49 .049( to )J 324 617 :M f4_12 sf .332(B)A f3_12 sf .566 .057(, and an edge )J 401 617 :M f4_12 sf .374(A)A f3_12 sf .417 .042( *)J 418 617 :M f1_12 sf .459A f3_12 sf .287 .029(* )J f4_12 sf .281(B)A f3_12 sf .506 .051(, orient )J 485 617 :M f4_12 sf (A)S 131 632 :M f3_12 sf (*)S f1_12 sf S f3_12 sf (* )S f4_12 sf (B)S f3_12 sf ( as )S 181 632 :M f4_12 sf (A)S f3_12 sf ( *)S f1_12 sf S 209 632 :M f3_12 sf ( )S f4_12 sf (B)S f3_12 sf (,)S 131 647 :M (\(ii\) else if )S 181 647 :M f4_12 sf (P)S f3_12 sf ( *)S f1_12 sf S 209 0 3 730 rC 209 647 :M f3_12 sf 12 f6_1 :p 3 :m ( )S 209 647 :M 6 :m ( )S gR gS 0 0 552 730 rC 209 647 :M f3_12 sf 12 f6_1 :p 3 :m ( )S 209 0 3 730 rC 209 647 :M 3 :m ( )S 209 647 :M 6 :m ( )S gR gS 212 0 10 730 rC 212 647 :M f4_12 sf 12 f7_1 :p 9 :m ( )S 219 647 :M 6 :m ( )S gR gS 0 0 552 730 rC 212 647 :M f4_12 sf 12 f7_1 :p 9.993 :m (M)S 212 0 10 730 rC 212 647 :M 9 :m ( )S 219 647 :M 6 :m ( )S gR gS 222 0 3 730 rC 222 647 :M f3_12 sf 12 f6_1 :p 3 :m ( )S 222 647 :M 6 :m ( )S gR gS 0 0 552 730 rC 222 647 :M f3_12 sf 12 f6_1 :p 9 :m ( *)S 225 0 6 730 rC 225 647 :M 6 :m ( )S 228 647 :M 6 :m ( )S gR gS 0 0 552 730 rC 231 647 :M f1_12 sf S f3_12 sf (* )S f4_12 sf (R)S f3_12 sf ( then orient as )S 330 647 :M f4_12 sf (P)S f3_12 sf ( *)S f1_12 sf S 358 647 :M f3_12 sf ( )S f4_12 sf (M)S 371 647 :M f3_12 sf ( )S f1_12 sf S 386 647 :M f3_12 sf ( )S f4_12 sf (R)S f3_12 sf (.,)S endp %%Page: 24 24 %%BeginPageSetup initializepage (peter; page: 24 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (24)S gR gS 0 0 552 730 rC 131 53 :M f3_12 sf .303 .03(\(iii\) else if )J f4_12 sf .185(B)A f3_12 sf .349 .035( is a collider along <)J 293 53 :M f4_12 sf .128(A)A f3_12 sf .052(,)A f4_12 sf .128(B)A f3_12 sf .052(,)A f4_12 sf .14(C)A f3_12 sf .154 .015(> in )J f5_12 sf (p)S 351 56 :M f1_7 sf (2)S 355 53 :M f3_12 sf .148 .015(, )J f4_12 sf .218(A)A f3_12 sf .377 .038( is not adjacent to )J 459 53 :M f4_12 sf .248(C)A f3_12 sf .155 .015(, )J f4_12 sf .227(B)A f3_12 sf .284 .028( is)J 131 68 :M .309 .031(adjacent to )J f4_12 sf (D)S 196 68 :M f3_12 sf .457 .046(, and )J f4_12 sf .478 .048(D )J 236 68 :M f3_12 sf .325 .032( is a non-collider along <)J f4_12 sf .15(A)A f3_12 sf .061(,)A f4_12 sf (D)S 378 68 :M f3_12 sf .066(,)A f4_12 sf .175(C)A f3_12 sf .335 .033(>, then orient )J 457 68 :M f4_12 sf .519 .052(B )J 468 68 :M f3_12 sf (*)S f1_12 sf S f3_12 sf (*)S 131 83 :M f4_12 sf (D)S 140 83 :M f3_12 sf ( as )S 156 83 :M f4_12 sf (B )S f3_12 sf ( )S f1_12 sf S 181 83 :M f3_12 sf ( * )S f4_12 sf (D)S 202 83 :M f3_12 sf (,)S 131 98 :M (\(iv\) else if )S f4_12 sf (B)S f3_12 sf ( )S f1_12 sf S 211 98 :M f3_12 sf ( *)S f4_12 sf (C)S f3_12 sf (, )S f4_12 sf (B)S f3_12 sf ( )S f1_12 sf S 256 98 :M f3_12 sf ( )S f4_12 sf (D)S 268 98 :M f3_12 sf (, and )S f4_12 sf (D)S 303 98 :M f3_12 sf ( o)S f1_12 sf S f3_12 sf (* )S f4_12 sf (C)S f3_12 sf (, orient as )S 391 98 :M f4_12 sf (D )S 403 98 :M f3_12 sf ( )S f1_12 sf S 418 98 :M f3_12 sf ( * )S f4_12 sf (C)S f3_12 sf (;)S 131 113 :M .226 .023(\(v\) If )J 160 113 :M f4_12 sf (U)S 169 113 :M f3_12 sf .184 .018( is a definite discriminating path between )J 372 113 :M f4_12 sf .147(A)A f3_12 sf .187 .019( and )J f4_12 sf .161(C)A f3_12 sf .167 .017( for )J 431 113 :M f4_12 sf .147(B)A f3_12 sf .128 .013( in )J f5_12 sf (p)S 460 115 :M f3_10 sf .134(2)A f3_12 sf 0 -2 rm .146 .015(, )J 0 2 rm 472 113 :M f4_12 sf (D)S 481 113 :M f3_12 sf .245 .025( is)J 131 128 :M 2.456 .246(adjacent to )J f4_12 sf 1.137(C)A f3_12 sf 1.066 .107( on )J f4_12 sf (U)S 233 128 :M f3_12 sf 2.557 .256(, and )J 266 128 :M f4_12 sf (D)S 275 128 :M f3_12 sf .942 .094(, )J f4_12 sf 1.382(B)A f3_12 sf 1.908 .191(, and )J f4_12 sf 1.509(C)A f3_12 sf 2.355 .235( for a triangle, then if )J 456 128 :M f4_12 sf 1.528(B)A f3_12 sf 1.946 .195( is in)J 131 143 :M f0_12 sf (Sepset)S 164 143 :M f3_12 sf <28>S 168 143 :M f4_12 sf .287(A)A f3_12 sf .118(,)A f4_12 sf .314(C)A f3_12 sf .544 .054(\) then mark )J f4_12 sf .287(B)A f3_12 sf .56 .056( as a non-collider on subpath )J f4_12 sf (D)S 413 143 :M f3_12 sf .844 .084( *)J 423 143 :M f1_12 sf S 435 0 6 730 rC 435 143 :M f3_12 sf 12 f6_1 :p 3.929 :m .928 .093( )J 437 143 :M 7.857 :m .884 .088( )J gR gS 0 0 552 730 rC 435 143 :M f3_12 sf 12 f6_1 :p 9.929 :m .844 .084(* )J 441 0 4 730 rC 441 143 :M 3.929 :m .928 .093( )J 441 143 :M 7.857 :m .884 .088( )J gR gS 445 0 7 730 rC 445 143 :M f4_12 sf 12 f7_1 :p 3.929 :m .928 .093( )J 448 143 :M 7.857 :m .884 .088( )J gR gS 0 0 552 730 rC 445 143 :M f4_12 sf 12 f7_1 :p 7.33 :m (B)S 445 0 7 730 rC 445 143 :M 3.929 :m .928 .093( )J 448 143 :M 7.857 :m .884 .088( )J gR gS 452 0 4 730 rC 452 143 :M f3_12 sf 12 f6_1 :p 3.929 :m .928 .093( )J 452 143 :M 7.857 :m .884 .088( )J gR gS 0 0 552 730 rC 452 143 :M f3_12 sf 12 f6_1 :p 9.929 :m .844 .084( *)J 456 0 6 730 rC 456 143 :M 3.929 :m .928 .093( )J 458 143 :M 7.857 :m .884 .088( )J gR gS 0 0 552 730 rC 462 143 :M f1_12 sf .531A f3_12 sf .362 .036(* )J 484 143 :M f4_12 sf (C)S 131 158 :M f3_12 sf (else orient )S 184 158 :M f4_12 sf (D)S 193 158 :M f3_12 sf ( *)S f1_12 sf S f3_12 sf (* )S f4_12 sf (B)S f3_12 sf ( *)S f1_12 sf S f3_12 sf (* )S f4_12 sf (C)S f3_12 sf ( as )S 284 158 :M f4_12 sf (D)S 293 158 :M f3_12 sf ( *)S f1_12 sf S 314 158 :M f3_12 sf ( )S f4_12 sf (B)S f3_12 sf ( )S f1_12 sf S 342 158 :M f3_12 sf ( * )S f4_12 sf (C)S f3_12 sf (.)S 77 173 :M ( )S 104 173 :M (until no more edges can be oriented.)S 77 200 :M 1.046 .105(Figure 10 shows the application of the orientation part of the FCI algorithm to the)J 59 218 :M (example begun in Figure 8.)S .75 lw 77 227 433 99 rC 293.5 228.5 215 96 rS 295 230 212 93 rC 295 263 :M f4_12 sf (X)S 304 266 :M f3_7 sf (1)S 308 263 :M f3_12 sf ( )S 311 263 :M ( )S 314 263 :M ( )S 317 263 :M ( )S 320 263 :M ( )S 323 263 :M ( )S 326 263 :M ( )S 329 263 :M ( )S 332 263 :M f4_12 sf (X)S 341 266 :M f3_7 sf (2)S 345 263 :M f3_12 sf ( )S 348 263 :M ( )S 351 263 :M ( )S 354 263 :M ( )S 357 263 :M ( )S 360 263 :M ( )S 363 263 :M (o)S 369 263 :M f4_12 sf (X)S 378 266 :M f3_7 sf (3)S 382 263 :M f3_12 sf ( )S 385 263 :M (o)S 391 263 :M ( )S 394 263 :M ( )S 397 263 :M ( )S 400 263 :M ( )S 403 263 :M ( )S 406 263 :M ( )S 409 263 :M f4_12 sf (X)S 418 266 :M f3_7 sf (4)S 422 263 :M f3_12 sf ( )S 425 263 :M ( )S 428 263 :M ( )S 431 263 :M ( )S 434 263 :M ( )S 437 263 :M ( )S 440 263 :M ( )S 443 263 :M ( )S 446 263 :M f4_12 sf (X)S 455 266 :M f3_7 sf (5)S 295 311 :M f3_12 sf ( )S 298 311 :M ( )S 301 311 :M ( )S 304 311 :M ( )S 307 311 :M ( )S 310 311 :M ( )S 313 311 :M ( )S 316 311 :M ( )S 319 311 :M ( )S 322 311 :M ( )S 325 311 :M ( )S 328 311 :M ( )S 331 311 :M ( )S 334 311 :M ( )S 337 311 :M ( )S 340 311 :M ( )S 343 311 :M ( )S 346 311 :M ( )S 349 311 :M ( )S 352 311 :M ( )S 355 311 :M ( )S 358 311 :M ( )S 361 311 :M <28>S 365 311 :M (i)S 368 311 :M (i)S 371 311 :M <29>S 375 311 :M ( )S 378 311 :M ( )S 381 311 :M (G)S 390 311 :M <28>S 394 311 :M (i)S 397 311 :M (i)S 400 311 :M <29>S 404 311 :M (.)S gR gS 77 227 433 99 rC 317 260.75 -.75 .75 324.75 260 .75 317 260 @a np 322 256 :M 322 264 :L 330 260 :L 322 256 :L .75 lw eofill -.75 -.75 322.75 264.75 .75 .75 322 256 @b -.75 -.75 322.75 264.75 .75 .75 330 260 @b 322 256.75 -.75 .75 330.75 260 .75 322 256 @a np 319 264 :M 319 256 :L 311 260 :L 319 264 :L eofill -.75 -.75 319.75 264.75 .75 .75 319 256 @b -.75 -.75 311.75 260.75 .75 .75 319 256 @b 311 260.75 -.75 .75 319.75 264 .75 311 260 @a 354 259.75 -.75 .75 361.75 259 .75 354 259 @a np 356 263 :M 356 255 :L 348 259 :L 356 263 :L eofill -.75 -.75 356.75 263.75 .75 .75 356 255 @b -.75 -.75 348.75 259.75 .75 .75 356 255 @b 348 259.75 -.75 .75 356.75 263 .75 348 259 @a -.75 -.75 391.75 259.75 .75 .75 398 258 @b np 396 255 :M 396 262 :L 404 258 :L 396 255 :L eofill -.75 -.75 396.75 262.75 .75 .75 396 255 @b -.75 -.75 396.75 262.75 .75 .75 404 258 @b 396 255.75 -.75 .75 404.75 258 .75 396 255 @a 429 260.75 -.75 .75 436.75 260 .75 429 260 @a np 434 256 :M 434 264 :L 442 260 :L 434 256 :L eofill -.75 -.75 434.75 264.75 .75 .75 434 256 @b -.75 -.75 434.75 264.75 .75 .75 442 260 @b 434 256.75 -.75 .75 442.75 260 .75 434 256 @a np 431 264 :M 431 256 :L 423 260 :L 431 264 :L eofill -.75 -.75 431.75 264.75 .75 .75 431 256 @b -.75 -.75 423.75 260.75 .75 .75 431 256 @b 423 260.75 -.75 .75 431.75 264 .75 423 260 @a 0 90 112 26 396.5 273.5 @n 90 180 112 26 399.5 273.5 @n -.75 -.75 452.75 274.75 .75 .75 452 272 @b np 448 274 :M 456 274 :L 452 266 :L 448 274 :L eofill 448 274.75 -.75 .75 456.75 274 .75 448 274 @a 452 266.75 -.75 .75 456.75 274 .75 452 266 @a -.75 -.75 448.75 274.75 .75 .75 452 266 @b 100 257.75 -.75 .75 107.75 257 .75 100 257 @a np 105 253 :M 105 261 :L 113 257 :L 105 253 :L eofill -.75 -.75 105.75 261.75 .75 .75 105 253 @b -.75 -.75 105.75 261.75 .75 .75 113 257 @b 105 253.75 -.75 .75 113.75 257 .75 105 253 @a np 102 261 :M 102 253 :L 94 257 :L 102 261 :L eofill -.75 -.75 102.75 261.75 .75 .75 102 253 @b -.75 -.75 94.75 257.75 .75 .75 102 253 @b 94 257.75 -.75 .75 102.75 261 .75 94 257 @a 174 256.75 -.75 .75 181.75 257 .75 174 256 @a np 179 253 :M 179 260 :L 187 257 :L 179 253 :L eofill -.75 -.75 179.75 260.75 .75 .75 179 253 @b -.75 -.75 179.75 260.75 .75 .75 187 257 @b 179 253.75 -.75 .75 187.75 257 .75 179 253 @a 212 257.75 -.75 .75 219.75 257 .75 212 257 @a np 217 253 :M 217 261 :L 225 257 :L 217 253 :L eofill -.75 -.75 217.75 261.75 .75 .75 217 253 @b -.75 -.75 217.75 261.75 .75 .75 225 257 @b 217 253.75 -.75 .75 225.75 257 .75 217 253 @a np 214 261 :M 214 253 :L 206 257 :L 214 261 :L eofill -.75 -.75 214.75 261.75 .75 .75 214 253 @b -.75 -.75 206.75 257.75 .75 .75 214 253 @b 206 257.75 -.75 .75 214.75 261 .75 206 257 @a 1 lw 0 90 112 26 179.5 270.5 @n 90 180 112 26 182.5 270.5 @n -.75 -.75 235.75 271.75 .75 .75 235 269 @b np 231 271 :M 239 271 :L 235 263 :L 231 271 :L .75 lw eofill 231 271.75 -.75 .75 239.75 271 .75 231 271 @a 235 263.75 -.75 .75 239.75 271 .75 235 263 @a -.75 -.75 231.75 271.75 .75 .75 235 263 @b 77.5 227.5 215 96 rS 79 229 212 93 rC 79 250 :M f3_12 sf ( )S 82 250 :M ( )S 85 250 :M ( )S 88 250 :M ( )S 91 250 :M ( )S 94 250 :M ( )S 97 250 :M ( )S 100 250 :M ( )S 103 250 :M ( )S 106 250 :M ( )S 109 250 :M ( )S 112 250 :M ( )S 115 250 :M ( )S 118 250 :M ( )S 121 250 :M ( )S 124 250 :M ( )S 127 250 :M ( )S 130 250 :M ( )S 133 250 :M ( )S 136 250 :M ( )S 139 250 :M ( )S 142 250 :M ( )S 145 250 :M ( )S 148 250 :M ( )S 151 250 :M ( )S 154 250 :M ( )S 157 250 :M ( )S 160 250 :M ( )S 163 250 :M ( )S 166 250 :M ( )S 169 250 :M ( )S 172 250 :M ( )S 175 250 :M ( )S 178 250 :M ( )S 181 250 :M ( )S 184 250 :M ( )S 187 250 :M ( )S 190 250 :M ( )S 193 250 :M ( )S 196 250 :M (o)S 79 262 :M f4_12 sf (X)S 88 265 :M f3_7 sf (1)S 92 262 :M f3_12 sf ( )S 95 262 :M ( )S 98 262 :M ( )S 101 262 :M ( )S 104 262 :M ( )S 107 262 :M ( )S 110 262 :M ( )S 113 262 :M ( )S 116 262 :M f4_12 sf (X)S 125 265 :M f3_7 sf (2)S 129 262 :M f3_12 sf ( )S 132 262 :M ( )S 135 262 :M ( )S 138 262 :M ( )S 141 262 :M ( )S 144 262 :M ( )S 147 262 :M ( )S 150 262 :M (o)S 156 262 :M f4_12 sf (X)S 165 265 :M f3_7 sf (3)S 169 262 :M f3_12 sf (o)S 175 262 :M ( )S 178 262 :M ( )S 181 262 :M ( )S 184 262 :M ( )S 187 262 :M ( )S 190 262 :M ( )S 193 262 :M f4_12 sf (X)S 202 265 :M f3_7 sf (4)S 206 262 :M f3_12 sf ( )S 209 262 :M ( )S 212 262 :M ( )S 215 262 :M ( )S 218 262 :M ( )S 221 262 :M ( )S 224 262 :M ( )S 227 262 :M ( )S 230 262 :M f4_12 sf (X)S 239 265 :M f3_7 sf (5)S 79 274 :M f3_12 sf ( )S 82 274 :M ( )S 85 274 :M ( )S 88 274 :M ( )S 91 274 :M ( )S 94 274 :M ( )S 97 274 :M ( )S 100 274 :M ( )S 103 274 :M ( )S 106 274 :M ( )S 109 274 :M ( )S 112 274 :M ( )S 115 274 :M ( )S 118 274 :M ( )S 121 274 :M (o)S 79 310 :M ( )S 82 310 :M ( )S 85 310 :M ( )S 88 310 :M ( )S 91 310 :M ( )S 94 310 :M ( )S 97 310 :M ( )S 100 310 :M ( )S 103 310 :M ( )S 106 310 :M ( )S 109 310 :M ( )S 112 310 :M ( )S 115 310 :M ( )S 118 310 :M ( )S 121 310 :M ( )S 124 310 :M ( )S 127 310 :M ( )S 130 310 :M ( )S 133 310 :M ( )S 136 310 :M ( )S 139 310 :M ( )S 142 310 :M ( )S 145 310 :M ( )S 148 310 :M ( )S 151 310 :M ( )S 154 310 :M ( )S 157 310 :M <28>S 161 310 :M (i)S 164 310 :M <29>S 168 310 :M ( )S 171 310 :M ( )S 174 310 :M (F)S 181 310 :M (.)S gR gS 77 227 433 99 rC -180 -90 100 26 142.5 244.5 @n -90 0 104 28 145.5 245.5 @n -.75 -.75 89.75 245.75 .75 .75 89 243 @b np 93 243 :M 85 243 :L 89 251 :L 93 243 :L .75 lw eofill 85 243.75 -.75 .75 93.75 243 .75 85 243 @a 85 243.75 -.75 .75 89.75 251 .75 85 243 @a -.75 -.75 89.75 251.75 .75 .75 93 243 @b 141 256.75 -.75 .75 148.75 256 .75 141 256 @a np 143 260 :M 143 252 :L 135 256 :L 143 260 :L eofill -.75 -.75 143.75 260.75 .75 .75 143 252 @b -.75 -.75 135.75 256.75 .75 .75 143 252 @b 135 256.75 -.75 .75 143.75 260 .75 135 256 @a 1 lw -180 -90 100 26 356.5 247.5 @n -90 0 104 28 359.5 248.5 @n -.75 -.75 303.75 248.75 .75 .75 303 246 @b np 307 246 :M 299 246 :L 303 254 :L 307 246 :L .75 lw eofill 299 246.75 -.75 .75 307.75 246 .75 299 246 @a 299 246.75 -.75 .75 303.75 254 .75 299 246 @a -.75 -.75 303.75 254.75 .75 .75 307 246 @b gR gS 0 0 552 730 rC 259 359 :M f0_12 sf (Figure )S 296 359 :M (10)S 77 383 :M f3_12 sf .69 .069(We do not know whether the orientation rules of the FCI algorithm are complete in)J 59 401 :M .315 .032(the sense that if any edge )J 186 401 :M f4_12 sf .411 .041(A )J 197 401 :M f3_12 sf .113(o)A f1_12 sf .225A f3_12 sf .141 .014(* )J f4_12 sf .138(B)A f3_12 sf .295 .03( occurs in the output, then in some member of the set)J 59 419 :M .022 .002(of DAGs represented by the PAG, )J 228 419 :M f4_12 sf (A)S f3_12 sf .019 .002( is an ancestor of )J f4_12 sf (B)S f3_12 sf .025 .002(, and in some other member of the)J 59 437 :M .073 .007(set of DAGs represented by the PAG)J 238 437 :M f4_12 sf .061 .006( A)J f3_12 sf .088 .009( is not an ancestor of )J 352 437 :M f4_12 sf (B)S f3_12 sf .067 .007(. However, we have proved)J 59 455 :M .749 .075(that the FCI algorithm provides enough orientations so that any two DAGs represented)J 59 473 :M 1.157 .116(by the output are in the same )J 213 473 :M f0_12 sf .242(O-equivalence)A f3_12 sf .762 .076( class. See Spirtes and Verma \(1992\) for)J 59 491 :M (details.)S 59 522 :M f0_14 sf (V)S 69 522 :M (I)S 74 522 :M (.)S 77 522 :M ( )S 95 522 :M (Simulation Study)S 77 545 :M f3_12 sf 2.233 .223(We ran some preliminary simulation studies of the FCI algorithm which were)J 59 563 :M 1.835 .184(intended to show how making various variables latent or selection variables would)J 59 581 :M .08 .008(change what information could theoretically be inferred from population information, and)J 59 599 :M (how much sample bias would affect the actual performance of the algorithm.)S 77 623 :M .426 .043(We used 10,000 cases generated pseudo-randomly by Cooper\(1992\) from the Alarm)J 59 641 :M 2.496 .25(network, shown in )J 163 641 :M 2.461 .246(Figure 11. The ALARM network was developed to model an)J 59 659 :M 1.567 .157(emergency medical system \(Beinlich, et al. 1989\). The 37 variables are all discrete,)J 59 677 :M 1.286 .129(taking 2, 3 or 4 distinct values. There are 46 edges in the DAG. In most instances a)J endp %%Page: 25 25 %%BeginPageSetup initializepage (peter; page: 25 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (25)S gR gS 0 0 552 730 rC 59 56 :M f3_12 sf 2.37 .237(directed arrow indicates that one variable is regarded as a cause of another. The)J 59 74 :M .185 .018(physicians who built the network also assigned it a probability distribution: each variable)J 59 92 :M f4_12 sf .13(V)A f3_12 sf .323 .032( is given a probability distribution conditional on each vector of values of the variables)J 59 110 :M .557 .056(having edges directed into )J 192 110 :M f4_12 sf .232(V)A f3_12 sf .56 .056(. This data has been used to test several different discovery)J 59 128 :M 1.443 .144(algorithms. \(Spirtes et al. 1993, Cooper and Herskovits 1992, Chickering 1994\) The)J 59 146 :M (interpretation of the variables is not relevant to the study described here.)S 77 173 1 2 rF 77 173 2 1 rF 78 174 1 1 rF 78 174 1 1 rF 79 173 420 1 rF 79 174 420 1 rF 500 173 1 2 rF 499 173 2 1 rF 499 174 1 1 rF 499 174 1 1 rF 77 175 1 211 rF 78 175 1 211 rF 499 175 1 211 rF 500 175 1 211 rF 77 386 1 2 rF 77 387 2 1 rF 78 386 1 1 rF 78 386 1 1 rF 79 386 420 1 rF 79 387 420 1 rF 500 386 1 2 rF 499 387 2 1 rF 499 386 1 1 rF 499 386 1 1 rF 105 251 9 12 rC 105 259 :M f3_11 sf (6)S gR gS 146 251 10 12 rC 146 259 :M f3_11 sf (5)S gR gS 188 251 10 12 rC 188 259 :M f3_11 sf (4)S gR gS 230 251 16 12 rC 230 259 :M f3_11 sf (2)S 236 259 :M (7)S gR gS 272 251 16 12 rC 272 259 :M f3_11 sf (1)S 278 259 :M (1)S gR gS 314 251 15 12 rC 314 259 :M f3_11 sf (3)S 320 259 :M (2)S gR gS 356 251 15 12 rC 356 259 :M f3_11 sf (3)S 362 259 :M (4)S gR gS 397 251 16 12 rC 397 259 :M f3_11 sf (3)S 403 259 :M (5)S gR gS 439 251 16 12 rC 439 259 :M f3_11 sf (3)S 445 259 :M (6)S gR gS 481 251 16 12 rC 481 259 :M f3_11 sf (3)S 487 259 :M (7)S gR gS 189 212 15 12 rC 189 220 :M f3_11 sf (1)S 195 220 :M (9)S gR gS 230 212 15 12 rC 230 220 :M f3_11 sf (2)S 236 220 :M (0)S gR gS 272 212 15 12 rC 272 220 :M f3_11 sf (3)S 278 220 :M (1)S gR gS 379 213 16 12 rC 379 221 :M f3_11 sf (1)S 385 221 :M (5)S gR gS 421 213 15 12 rC 421 221 :M f3_11 sf (2)S 427 221 :M (3)S gR gS 484 214 15 12 rC 484 222 :M f3_11 sf (1)S 490 222 :M (6)S gR gS 228 177 16 12 rC 228 185 :M f3_11 sf (1)S 234 185 :M (0)S gR gS 272 177 16 12 rC 272 185 :M f3_11 sf (2)S 278 185 :M (1)S gR gS 373 178 16 12 rC 373 186 :M f3_11 sf (2)S 379 186 :M (2)S gR gS 482 180 16 12 rC 482 188 :M f3_11 sf (1)S 488 188 :M (3)S gR gS 82 292 16 12 rC 82 300 :M f3_11 sf (1)S 88 300 :M (7)S gR gS 175 293 15 12 rC 175 301 :M f3_11 sf (2)S 181 301 :M (8)S gR gS 216 293 16 12 rC 216 301 :M f3_11 sf (2)S 222 301 :M (9)S gR gS 314 292 15 12 rC 314 300 :M f3_11 sf (1)S 320 300 :M (2)S gR gS 440 291 16 12 rC 440 299 :M f3_11 sf (2)S 446 299 :M (4)S gR gS 81 334 16 12 rC 81 342 :M f3_11 sf (2)S 87 342 :M (5)S gR gS 125 334 16 12 rC 125 342 :M f3_11 sf (1)S 131 342 :M (8)S gR gS 169 334 15 12 rC 169 342 :M f3_11 sf (2)S 175 342 :M (6)S gR gS 207 335 10 11 rC 207 343 :M f3_11 sf (7)S gR gS 249 335 9 11 rC 249 343 :M f3_11 sf (8)S gR gS 290 335 10 11 rC 290 343 :M f3_11 sf (9)S gR gS 79 374 10 12 rC 79 382 :M f3_11 sf (1)S gR gS 123 374 10 12 rC 123 382 :M f3_11 sf (2)S gR gS 167 374 10 12 rC 167 382 :M f3_11 sf (3)S gR gS 272 374 16 12 rC 272 382 :M f3_11 sf (3)S 278 382 :M (0)S gR gS 79 175 420 211 rC np 143 255 :M 135 259 :L 135 259 :L 134 259 :L 134 259 :L 134 259 :L 134 258 :L 134 258 :L 134 258 :L 134 258 :L 134 258 :L 134 258 :L 134 257 :L 134 257 :L 134 257 :L 134 257 :L 134 257 :L 134 257 :L 134 256 :L 134 256 :L 134 256 :L 134 256 :L 134 256 :L 134 256 :L 134 255 :L 134 255 :L 134 255 :L 134 255 :L 134 255 :L 134 255 :L 134 254 :L 134 254 :L 134 254 :L 134 254 :L 134 254 :L 134 254 :L 134 253 :L 134 253 :L 134 253 :L 134 253 :L 134 253 :L 134 253 :L 134 252 :L 134 252 :L 134 252 :L 134 252 :L 134 252 :L 134 252 :L 135 251 :L 143 255 :L 143 255 :L eofill 121 256.95 -.95 .95 135.95 256 .95 121 256 @a np 162 255 :M 171 251 :L 171 252 :L 171 252 :L 171 252 :L 171 252 :L 171 252 :L 171 252 :L 171 252 :L 171 253 :L 171 253 :L 171 253 :L 171 253 :L 171 253 :L 171 253 :L 171 254 :L 171 254 :L 172 254 :L 172 254 :L 172 254 :L 172 254 :L 172 255 :L 172 255 :L 172 255 :L 172 255 :L 172 255 :L 172 255 :L 172 256 :L 172 256 :L 172 256 :L 172 256 :L 172 256 :L 172 256 :L 172 257 :L 171 257 :L 171 257 :L 171 257 :L 171 257 :L 171 257 :L 171 258 :L 171 258 :L 171 258 :L 171 258 :L 171 258 :L 171 258 :L 171 259 :L 171 259 :L 171 259 :L 171 259 :L 162 255 :L 162 255 :L eofill 169 256.95 -.95 .95 185.95 256 .95 169 256 @a np 227 256 :M 218 260 :L 218 260 :L 218 260 :L 218 260 :L 218 260 :L 218 259 :L 218 259 :L 218 259 :L 218 259 :L 218 259 :L 218 259 :L 218 258 :L 218 258 :L 218 258 :L 218 258 :L 218 258 :L 218 258 :L 218 257 :L 218 257 :L 217 257 :L 217 257 :L 217 257 :L 217 257 :L 217 256 :L 217 256 :L 217 256 :L 217 256 :L 217 256 :L 217 256 :L 217 255 :L 218 255 :L 218 255 :L 218 255 :L 218 255 :L 218 255 :L 218 254 :L 218 254 :L 218 254 :L 218 254 :L 218 254 :L 218 254 :L 218 253 :L 218 253 :L 218 253 :L 218 253 :L 218 253 :L 218 253 :L 218 252 :L 227 256 :L 227 256 :L eofill 204 257.95 -.95 .95 219.95 257 .95 204 257 @a np 245 255 :M 254 251 :L 254 252 :L 254 252 :L 254 252 :L 254 252 :L 254 252 :L 254 252 :L 254 252 :L 254 253 :L 254 253 :L 254 253 :L 254 253 :L 254 253 :L 254 253 :L 254 254 :L 254 254 :L 254 254 :L 254 254 :L 254 254 :L 254 254 :L 254 255 :L 254 255 :L 254 255 :L 254 255 :L 254 255 :L 254 255 :L 254 256 :L 254 256 :L 254 256 :L 254 256 :L 254 256 :L 254 256 :L 254 257 :L 254 257 :L 254 257 :L 254 257 :L 254 257 :L 254 257 :L 254 258 :L 254 258 :L 254 258 :L 254 258 :L 254 258 :L 254 258 :L 254 259 :L 254 259 :L 254 259 :L 254 259 :L 245 255 :L 245 255 :L eofill 252 256.95 -.95 .95 268.95 256 .95 252 256 @a np 287 255 :M 296 251 :L 296 252 :L 296 252 :L 297 252 :L 297 252 :L 297 252 :L 297 252 :L 297 252 :L 297 253 :L 297 253 :L 297 253 :L 297 253 :L 297 253 :L 297 253 :L 297 254 :L 297 254 :L 297 254 :L 297 254 :L 297 254 :L 297 254 :L 297 255 :L 297 255 :L 297 255 :L 297 255 :L 297 255 :L 297 255 :L 297 256 :L 297 256 :L 297 256 :L 297 256 :L 297 256 :L 297 256 :L 297 257 :L 297 257 :L 297 257 :L 297 257 :L 297 257 :L 297 257 :L 297 258 :L 297 258 :L 297 258 :L 297 258 :L 297 258 :L 297 258 :L 297 259 :L 297 259 :L 296 259 :L 296 259 :L 287 255 :L 287 255 :L eofill 295 256.95 -.95 .95 308.95 256 .95 295 256 @a 290 256.95 -.95 .95 291.95 256 .95 290 256 @a np 329 256 :M 338 252 :L 338 252 :L 338 253 :L 338 253 :L 338 253 :L 339 253 :L 339 253 :L 339 253 :L 339 254 :L 339 254 :L 339 254 :L 339 254 :L 339 254 :L 339 254 :L 339 255 :L 339 255 :L 339 255 :L 339 255 :L 339 255 :L 339 255 :L 339 256 :L 339 256 :L 339 256 :L 339 256 :L 339 256 :L 339 256 :L 339 257 :L 339 257 :L 339 257 :L 339 257 :L 339 257 :L 339 257 :L 339 258 :L 339 258 :L 339 258 :L 339 258 :L 339 258 :L 339 258 :L 339 259 :L 339 259 :L 339 259 :L 339 259 :L 339 259 :L 339 259 :L 338 260 :L 338 260 :L 338 260 :L 338 260 :L 329 256 :L 329 256 :L eofill 337 257.95 -.95 .95 351.95 257 .95 337 257 @a np 371 256 :M 380 252 :L 380 252 :L 380 253 :L 380 253 :L 380 253 :L 380 253 :L 380 253 :L 380 253 :L 380 254 :L 380 254 :L 381 254 :L 381 254 :L 381 254 :L 381 254 :L 381 255 :L 381 255 :L 381 255 :L 381 255 :L 381 255 :L 381 255 :L 381 256 :L 381 256 :L 381 256 :L 381 256 :L 381 256 :L 381 256 :L 381 257 :L 381 257 :L 381 257 :L 381 257 :L 381 257 :L 381 257 :L 381 258 :L 381 258 :L 381 258 :L 381 258 :L 381 258 :L 381 258 :L 381 259 :L 380 259 :L 380 259 :L 380 259 :L 380 259 :L 380 259 :L 380 260 :L 380 260 :L 380 260 :L 380 260 :L 371 256 :L 371 256 :L eofill 378 257.95 -.95 .95 394.95 257 .95 378 257 @a np 413 256 :M 422 252 :L 422 252 :L 422 253 :L 422 253 :L 422 253 :L 422 253 :L 422 253 :L 422 253 :L 422 254 :L 422 254 :L 422 254 :L 422 254 :L 422 254 :L 422 254 :L 423 255 :L 423 255 :L 423 255 :L 423 255 :L 423 255 :L 423 255 :L 423 256 :L 423 256 :L 423 256 :L 423 256 :L 423 256 :L 423 256 :L 423 257 :L 423 257 :L 423 257 :L 423 257 :L 423 257 :L 423 257 :L 423 258 :L 423 258 :L 423 258 :L 422 258 :L 422 258 :L 422 258 :L 422 259 :L 422 259 :L 422 259 :L 422 259 :L 422 259 :L 422 259 :L 422 260 :L 422 260 :L 422 260 :L 422 260 :L 413 256 :L 413 256 :L eofill 420 257.95 -.95 .95 436.95 257 .95 420 257 @a np 455 256 :M 464 252 :L 464 252 :L 464 253 :L 464 253 :L 464 253 :L 464 253 :L 464 253 :L 464 253 :L 464 254 :L 464 254 :L 464 254 :L 464 254 :L 464 254 :L 464 254 :L 464 255 :L 464 255 :L 464 255 :L 464 255 :L 464 255 :L 464 255 :L 464 256 :L 464 256 :L 464 256 :L 464 256 :L 465 256 :L 464 256 :L 464 257 :L 464 257 :L 464 257 :L 464 257 :L 464 257 :L 464 257 :L 464 258 :L 464 258 :L 464 258 :L 464 258 :L 464 258 :L 464 258 :L 464 259 :L 464 259 :L 464 259 :L 464 259 :L 464 259 :L 464 259 :L 464 260 :L 464 260 :L 464 260 :L 464 260 :L 455 256 :L 455 256 :L eofill 462 257.95 -.95 .95 478.95 257 .95 462 257 @a 358 335 15 11 rC 358 343 :M f3_11 sf (3)S 364 343 :M (3)S gR gS 399 335 16 11 rC 399 343 :M f3_11 sf (1)S 405 343 :M (4)S gR gS 79 175 420 211 rC np 196 247 :M 192 238 :L 192 238 :L 192 238 :L 193 238 :L 193 237 :L 193 237 :L 193 237 :L 193 237 :L 193 237 :L 194 237 :L 194 237 :L 194 237 :L 194 237 :L 194 237 :L 194 237 :L 195 237 :L 195 237 :L 195 237 :L 195 237 :L 195 237 :L 195 237 :L 196 237 :L 196 237 :L 196 237 :L 196 237 :L 196 237 :L 196 237 :L 197 237 :L 197 237 :L 197 237 :L 197 237 :L 197 237 :L 197 237 :L 198 237 :L 198 237 :L 198 237 :L 198 237 :L 198 237 :L 198 237 :L 199 237 :L 199 237 :L 199 237 :L 199 237 :L 199 237 :L 199 237 :L 199 238 :L 200 238 :L 200 238 :L 196 247 :L 196 247 :L eofill 195 226.95 -.95 .95 196.95 239 .95 195 226 @a np 236 246 :M 232 237 :L 232 237 :L 232 237 :L 232 237 :L 233 236 :L 233 236 :L 233 236 :L 233 236 :L 233 236 :L 233 236 :L 234 236 :L 234 236 :L 234 236 :L 234 236 :L 234 236 :L 234 236 :L 235 236 :L 235 236 :L 235 236 :L 235 236 :L 235 236 :L 235 236 :L 236 236 :L 236 236 :L 236 236 :L 236 236 :L 236 236 :L 236 236 :L 237 236 :L 237 236 :L 237 236 :L 237 236 :L 237 236 :L 237 236 :L 238 236 :L 238 236 :L 238 236 :L 238 236 :L 238 236 :L 238 236 :L 239 236 :L 239 236 :L 239 236 :L 239 236 :L 239 236 :L 239 237 :L 239 237 :L 240 237 :L 236 246 :L 236 246 :L eofill -.95 -.95 235.95 238.95 .95 .95 235 226 @b np 279 247 :M 275 238 :L 275 238 :L 275 238 :L 275 238 :L 275 237 :L 276 237 :L 276 237 :L 276 237 :L 276 237 :L 276 237 :L 276 237 :L 277 237 :L 277 237 :L 277 237 :L 277 237 :L 277 237 :L 277 237 :L 278 237 :L 278 237 :L 278 237 :L 278 237 :L 278 237 :L 278 237 :L 279 237 :L 279 237 :L 279 237 :L 279 237 :L 279 237 :L 279 237 :L 280 237 :L 280 237 :L 280 237 :L 280 237 :L 280 237 :L 280 237 :L 281 237 :L 281 237 :L 281 237 :L 281 237 :L 281 237 :L 281 237 :L 282 237 :L 282 237 :L 282 237 :L 282 237 :L 282 238 :L 282 238 :L 283 238 :L 279 247 :L 279 247 :L eofill -.95 -.95 278.95 239.95 .95 .95 278 226 @b np 384 226 :M 393 230 :L 393 230 :L 393 230 :L 393 230 :L 393 230 :L 392 230 :L 392 231 :L 392 231 :L 392 231 :L 392 231 :L 392 231 :L 392 231 :L 392 231 :L 392 232 :L 392 232 :L 392 232 :L 391 232 :L 391 232 :L 391 232 :L 391 232 :L 391 232 :L 391 233 :L 391 233 :L 391 233 :L 391 233 :L 390 233 :L 390 233 :L 390 233 :L 390 233 :L 390 234 :L 390 234 :L 390 234 :L 390 234 :L 389 234 :L 389 234 :L 389 234 :L 389 234 :L 389 234 :L 389 234 :L 389 235 :L 388 235 :L 388 235 :L 388 235 :L 388 235 :L 388 235 :L 388 235 :L 388 235 :L 387 235 :L 384 226 :L 384 226 :L eofill 389 232.95 -.95 .95 403.95 246 .95 389 232 @a np 479 183 :M 475 191 :L 475 191 :L 475 191 :L 474 191 :L 474 191 :L 474 191 :L 474 191 :L 474 191 :L 474 191 :L 474 190 :L 473 190 :L 473 190 :L 473 190 :L 473 190 :L 473 190 :L 473 190 :L 473 190 :L 473 190 :L 472 190 :L 472 189 :L 472 189 :L 472 189 :L 472 189 :L 472 189 :L 472 189 :L 472 189 :L 472 189 :L 471 188 :L 471 188 :L 471 188 :L 471 188 :L 471 188 :L 471 188 :L 471 188 :L 471 187 :L 471 187 :L 471 187 :L 471 187 :L 470 187 :L 470 187 :L 470 186 :L 470 186 :L 470 186 :L 470 186 :L 470 186 :L 470 186 :L 470 186 :L 470 185 :L 479 183 :L 479 183 :L eofill -.95 -.95 437.95 218.95 .95 .95 473 188 @b np 406 247 :M 410 238 :L 410 238 :L 410 238 :L 411 238 :L 411 238 :L 411 238 :L 411 238 :L 411 238 :L 411 238 :L 411 238 :L 412 239 :L 412 239 :L 412 239 :L 412 239 :L 412 239 :L 412 239 :L 412 239 :L 413 239 :L 413 239 :L 413 240 :L 413 240 :L 413 240 :L 413 240 :L 413 240 :L 413 240 :L 413 240 :L 414 240 :L 414 240 :L 414 241 :L 414 241 :L 414 241 :L 414 241 :L 414 241 :L 414 241 :L 414 241 :L 414 242 :L 415 242 :L 415 242 :L 415 242 :L 415 242 :L 415 242 :L 415 243 :L 415 243 :L 415 243 :L 415 243 :L 415 243 :L 415 243 :L 415 243 :L 406 247 :L 406 247 :L eofill -.95 -.95 411.95 242.95 .95 .95 427 228 @b np 360 247 :M 359 237 :L 360 237 :L 360 237 :L 360 237 :L 360 237 :L 360 237 :L 360 237 :L 361 237 :L 361 237 :L 361 237 :L 361 237 :L 361 237 :L 361 237 :L 362 237 :L 362 237 :L 362 237 :L 362 237 :L 362 237 :L 362 237 :L 363 237 :L 363 237 :L 363 237 :L 363 237 :L 363 237 :L 363 237 :L 364 237 :L 364 237 :L 364 238 :L 364 238 :L 364 238 :L 364 238 :L 364 238 :L 365 238 :L 365 238 :L 365 238 :L 365 238 :L 365 238 :L 365 238 :L 366 238 :L 366 238 :L 366 239 :L 366 239 :L 366 239 :L 366 239 :L 366 239 :L 367 239 :L 367 239 :L 367 239 :L 360 247 :L 360 247 :L eofill -.95 -.95 362.95 239.95 .95 .95 377 194 @b np 407 247 :M 401 239 :L 401 239 :L 401 239 :L 401 239 :L 401 239 :L 402 239 :L 402 239 :L 402 238 :L 402 238 :L 402 238 :L 402 238 :L 402 238 :L 403 238 :L 403 238 :L 403 238 :L 403 238 :L 403 238 :L 403 238 :L 404 238 :L 404 238 :L 404 237 :L 404 237 :L 404 237 :L 404 237 :L 404 237 :L 405 237 :L 405 237 :L 405 237 :L 405 237 :L 405 237 :L 405 237 :L 406 237 :L 406 237 :L 406 237 :L 406 237 :L 406 237 :L 406 237 :L 407 237 :L 407 237 :L 407 237 :L 407 237 :L 407 237 :L 407 237 :L 408 237 :L 408 237 :L 408 237 :L 408 237 :L 408 237 :L 407 247 :L 407 247 :L eofill 388 184.95 -.95 .95 405.95 239 .95 388 184 @a np 387 209 :M 380 202 :L 380 202 :L 380 201 :L 381 201 :L 381 201 :L 381 201 :L 381 201 :L 381 201 :L 381 201 :L 381 201 :L 381 201 :L 382 201 :L 382 200 :L 382 200 :L 382 200 :L 382 200 :L 382 200 :L 382 200 :L 383 200 :L 383 200 :L 383 200 :L 383 200 :L 383 200 :L 383 200 :L 384 200 :L 384 200 :L 384 200 :L 384 199 :L 384 199 :L 384 199 :L 385 199 :L 385 199 :L 385 199 :L 385 199 :L 385 199 :L 385 199 :L 386 199 :L 386 199 :L 386 199 :L 386 199 :L 386 199 :L 386 199 :L 387 199 :L 387 199 :L 387 199 :L 387 199 :L 387 199 :L 387 199 :L 387 209 :L 387 209 :L eofill 380 192.95 -.95 .95 383.95 201 .95 380 192 @a np 490 247 :M 486 238 :L 486 238 :L 486 238 :L 486 238 :L 487 237 :L 487 237 :L 487 237 :L 487 237 :L 487 237 :L 487 237 :L 488 237 :L 488 237 :L 488 237 :L 488 237 :L 488 237 :L 488 237 :L 489 237 :L 489 237 :L 489 237 :L 489 237 :L 489 237 :L 489 237 :L 490 237 :L 490 237 :L 490 237 :L 490 237 :L 490 237 :L 490 237 :L 491 237 :L 491 237 :L 491 237 :L 491 237 :L 491 237 :L 491 237 :L 492 237 :L 492 237 :L 492 237 :L 492 237 :L 492 237 :L 492 237 :L 493 237 :L 493 237 :L 493 237 :L 493 237 :L 493 237 :L 493 238 :L 494 238 :L 494 238 :L 490 247 :L 490 247 :L eofill -.95 -.95 489.95 241.95 .95 .95 489 229 @b np 484 194 :M 482 204 :L 482 204 :L 482 204 :L 482 204 :L 481 204 :L 481 203 :L 481 203 :L 481 203 :L 481 203 :L 481 203 :L 480 203 :L 480 203 :L 480 203 :L 480 203 :L 480 203 :L 480 203 :L 479 203 :L 479 203 :L 479 203 :L 479 203 :L 479 202 :L 479 202 :L 479 202 :L 478 202 :L 478 202 :L 478 202 :L 478 202 :L 478 202 :L 478 202 :L 478 202 :L 478 201 :L 477 201 :L 477 201 :L 477 201 :L 477 201 :L 477 201 :L 477 201 :L 477 201 :L 477 200 :L 477 200 :L 476 200 :L 476 200 :L 476 200 :L 476 200 :L 476 200 :L 476 199 :L 476 199 :L 476 199 :L 484 194 :L 484 194 :L eofill -.95 -.95 446.95 247.95 .95 .95 479 200 @b np 278 209 :M 274 200 :L 274 200 :L 274 200 :L 274 200 :L 274 200 :L 275 200 :L 275 200 :L 275 200 :L 275 199 :L 275 199 :L 275 199 :L 276 199 :L 276 199 :L 276 199 :L 276 199 :L 276 199 :L 276 199 :L 277 199 :L 277 199 :L 277 199 :L 277 199 :L 277 199 :L 277 199 :L 278 199 :L 278 199 :L 278 199 :L 278 199 :L 278 199 :L 278 199 :L 279 199 :L 279 199 :L 279 199 :L 279 199 :L 279 199 :L 279 199 :L 280 199 :L 280 199 :L 280 199 :L 280 199 :L 280 199 :L 280 199 :L 281 200 :L 281 200 :L 281 200 :L 281 200 :L 281 200 :L 281 200 :L 281 200 :L 278 209 :L 278 209 :L eofill -.95 -.95 277.95 201.95 .95 .95 277 191 @b np 246 182 :M 254 178 :L 255 178 :L 255 178 :L 255 178 :L 255 178 :L 255 178 :L 255 179 :L 255 179 :L 255 179 :L 255 179 :L 255 179 :L 255 179 :L 255 180 :L 255 180 :L 255 180 :L 255 180 :L 255 180 :L 255 180 :L 255 181 :L 255 181 :L 255 181 :L 255 181 :L 255 181 :L 255 181 :L 255 182 :L 255 182 :L 255 182 :L 255 182 :L 255 182 :L 255 182 :L 255 183 :L 255 183 :L 255 183 :L 255 183 :L 255 183 :L 255 183 :L 255 184 :L 255 184 :L 255 184 :L 255 184 :L 255 184 :L 255 184 :L 255 185 :L 255 185 :L 255 185 :L 255 185 :L 255 185 :L 255 185 :L 246 182 :L 246 182 :L eofill 253 182.95 -.95 .95 268.95 182 .95 253 182 @a np 322 265 :M 325 274 :L 325 274 :L 325 274 :L 325 274 :L 325 274 :L 325 274 :L 324 274 :L 324 274 :L 324 274 :L 324 274 :L 324 274 :L 324 274 :L 324 274 :L 323 274 :L 323 275 :L 323 275 :L 323 275 :L 323 275 :L 323 275 :L 322 275 :L 322 275 :L 322 275 :L 322 275 :L 322 275 :L 322 275 :L 321 275 :L 321 275 :L 321 275 :L 321 275 :L 321 275 :L 321 275 :L 320 275 :L 320 275 :L 320 275 :L 320 275 :L 320 274 :L 320 274 :L 319 274 :L 319 274 :L 319 274 :L 319 274 :L 319 274 :L 319 274 :L 318 274 :L 318 274 :L 318 274 :L 318 274 :L 318 274 :L 322 265 :L 322 265 :L eofill -.95 -.95 321.95 288.95 .95 .95 321 273 @b np 447 265 :M 451 274 :L 451 274 :L 451 274 :L 450 274 :L 450 274 :L 450 274 :L 450 274 :L 450 274 :L 450 274 :L 450 274 :L 449 274 :L 449 274 :L 449 274 :L 449 274 :L 449 275 :L 449 275 :L 448 275 :L 448 275 :L 448 275 :L 448 275 :L 448 275 :L 448 275 :L 447 275 :L 447 275 :L 447 275 :L 447 275 :L 447 275 :L 447 275 :L 446 275 :L 446 275 :L 446 275 :L 446 275 :L 446 275 :L 446 275 :L 445 275 :L 445 274 :L 445 274 :L 445 274 :L 445 274 :L 445 274 :L 444 274 :L 444 274 :L 444 274 :L 444 274 :L 444 274 :L 444 274 :L 443 274 :L 443 274 :L 447 265 :L 447 265 :L eofill -.95 -.95 446.95 287.95 .95 .95 447 273 @b np 120 257 :M 130 257 :L 130 257 :L 130 257 :L 130 257 :L 130 258 :L 130 258 :L 130 258 :L 130 258 :L 130 258 :L 130 258 :L 130 259 :L 130 259 :L 130 259 :L 130 259 :L 130 259 :L 130 259 :L 130 260 :L 130 260 :L 129 260 :L 129 260 :L 129 260 :L 129 260 :L 129 261 :L 129 261 :L 129 261 :L 129 261 :L 129 261 :L 129 261 :L 129 261 :L 129 262 :L 129 262 :L 129 262 :L 129 262 :L 128 262 :L 128 262 :L 128 262 :L 128 263 :L 128 263 :L 128 263 :L 128 263 :L 128 263 :L 128 263 :L 128 263 :L 127 264 :L 127 264 :L 127 264 :L 127 264 :L 127 264 :L 120 257 :L 120 257 :L eofill 126 261.95 -.95 .95 212.95 295 .95 126 261 @a np 110 265 :M 120 268 :L 119 268 :L 119 268 :L 119 269 :L 119 269 :L 119 269 :L 119 269 :L 119 269 :L 119 269 :L 119 270 :L 119 270 :L 119 270 :L 119 270 :L 119 270 :L 118 270 :L 118 270 :L 118 271 :L 118 271 :L 118 271 :L 118 271 :L 118 271 :L 118 271 :L 118 271 :L 118 271 :L 117 272 :L 117 272 :L 117 272 :L 117 272 :L 117 272 :L 117 272 :L 117 272 :L 117 272 :L 116 272 :L 116 273 :L 116 273 :L 116 273 :L 116 273 :L 116 273 :L 116 273 :L 115 273 :L 115 273 :L 115 273 :L 115 273 :L 115 274 :L 115 274 :L 115 274 :L 114 274 :L 114 274 :L 110 265 :L 110 265 :L eofill 116 270.95 -.95 .95 176.95 328 .95 116 270 @a np 99 338 :M 108 334 :L 108 334 :L 108 334 :L 108 334 :L 108 334 :L 108 335 :L 108 335 :L 108 335 :L 108 335 :L 108 335 :L 108 335 :L 108 336 :L 108 336 :L 108 336 :L 108 336 :L 108 336 :L 108 336 :L 108 337 :L 108 337 :L 108 337 :L 108 337 :L 108 337 :L 108 337 :L 108 338 :L 108 338 :L 108 338 :L 108 338 :L 108 338 :L 108 338 :L 108 338 :L 108 339 :L 108 339 :L 108 339 :L 108 339 :L 108 339 :L 108 340 :L 108 340 :L 108 340 :L 108 340 :L 108 340 :L 108 340 :L 108 340 :L 108 341 :L 108 341 :L 108 341 :L 108 341 :L 108 341 :L 108 341 :L 99 338 :L 99 338 :L eofill 106 338.95 -.95 .95 122.95 338 .95 106 338 @a np 166 338 :M 157 342 :L 157 341 :L 157 341 :L 157 341 :L 157 341 :L 157 341 :L 157 341 :L 157 340 :L 157 340 :L 156 340 :L 156 340 :L 156 340 :L 156 340 :L 156 340 :L 156 339 :L 156 339 :L 156 339 :L 156 339 :L 156 339 :L 156 338 :L 156 338 :L 156 338 :L 156 338 :L 156 338 :L 156 338 :L 156 338 :L 156 337 :L 156 337 :L 156 337 :L 156 337 :L 156 337 :L 156 337 :L 156 336 :L 156 336 :L 156 336 :L 156 336 :L 156 336 :L 156 336 :L 156 335 :L 156 335 :L 157 335 :L 157 335 :L 157 335 :L 157 335 :L 157 334 :L 157 334 :L 157 334 :L 157 334 :L 166 338 :L 166 338 :L eofill 142 338.95 -.95 .95 158.95 338 .95 142 338 @a np 88 329 :M 84 320 :L 84 320 :L 84 320 :L 85 320 :L 85 320 :L 85 320 :L 85 320 :L 85 320 :L 85 320 :L 85 320 :L 86 320 :L 86 320 :L 86 320 :L 86 319 :L 86 319 :L 86 319 :L 87 319 :L 87 319 :L 87 319 :L 87 319 :L 87 319 :L 87 319 :L 88 319 :L 88 319 :L 88 319 :L 88 319 :L 88 319 :L 88 319 :L 89 319 :L 89 319 :L 89 319 :L 89 319 :L 89 319 :L 89 319 :L 90 319 :L 90 319 :L 90 320 :L 90 320 :L 90 320 :L 90 320 :L 91 320 :L 91 320 :L 91 320 :L 91 320 :L 91 320 :L 91 320 :L 92 320 :L 92 320 :L 88 329 :L 88 329 :L eofill 87 305.95 -.95 .95 88.95 322 .95 87 305 @a np 166 333 :M 156 333 :L 156 333 :L 156 333 :L 156 333 :L 156 333 :L 156 333 :L 156 333 :L 156 332 :L 156 332 :L 156 332 :L 156 332 :L 156 332 :L 156 332 :L 156 331 :L 156 331 :L 156 331 :L 156 331 :L 156 331 :L 156 331 :L 156 330 :L 157 330 :L 157 330 :L 157 330 :L 157 330 :L 157 330 :L 157 329 :L 157 329 :L 157 329 :L 157 329 :L 157 329 :L 157 329 :L 157 328 :L 157 328 :L 157 328 :L 157 328 :L 158 328 :L 158 328 :L 158 328 :L 158 327 :L 158 327 :L 158 327 :L 158 327 :L 158 327 :L 158 327 :L 158 327 :L 159 326 :L 159 326 :L 159 326 :L 166 333 :L 166 333 :L eofill 88 305.95 -.95 .95 158.95 330 .95 88 305 @a np 88 370 :M 84 361 :L 84 361 :L 84 361 :L 85 361 :L 85 361 :L 85 361 :L 85 360 :L 85 360 :L 85 360 :L 85 360 :L 86 360 :L 86 360 :L 86 360 :L 86 360 :L 86 360 :L 86 360 :L 87 360 :L 87 360 :L 87 360 :L 87 360 :L 87 360 :L 87 360 :L 88 360 :L 88 360 :L 88 360 :L 88 360 :L 88 360 :L 88 360 :L 89 360 :L 89 360 :L 89 360 :L 89 360 :L 89 360 :L 89 360 :L 90 360 :L 90 360 :L 90 360 :L 90 360 :L 90 360 :L 90 360 :L 91 360 :L 91 360 :L 91 360 :L 91 361 :L 91 361 :L 91 361 :L 92 361 :L 92 361 :L 88 370 :L 88 370 :L eofill 87 348.95 -.95 .95 88.95 362 .95 87 348 @a np 131 371 :M 121 370 :L 121 370 :L 121 370 :L 121 369 :L 121 369 :L 121 369 :L 121 369 :L 121 369 :L 121 369 :L 121 368 :L 121 368 :L 121 368 :L 121 368 :L 122 368 :L 122 368 :L 122 367 :L 122 367 :L 122 367 :L 122 367 :L 122 367 :L 122 367 :L 122 367 :L 122 366 :L 122 366 :L 122 366 :L 122 366 :L 122 366 :L 122 366 :L 123 366 :L 123 365 :L 123 365 :L 123 365 :L 123 365 :L 123 365 :L 123 365 :L 123 365 :L 123 364 :L 123 364 :L 124 364 :L 124 364 :L 124 364 :L 124 364 :L 124 364 :L 124 364 :L 124 363 :L 124 363 :L 125 363 :L 125 363 :L 131 371 :L 131 371 :L eofill 87 348.95 -.95 .95 123.95 367 .95 87 348 @a np 175 370 :M 165 369 :L 165 369 :L 165 369 :L 165 368 :L 165 368 :L 165 368 :L 165 368 :L 165 368 :L 165 368 :L 165 367 :L 165 367 :L 165 367 :L 165 367 :L 165 367 :L 165 367 :L 165 366 :L 166 366 :L 166 366 :L 166 366 :L 166 366 :L 166 366 :L 166 365 :L 166 365 :L 166 365 :L 166 365 :L 166 365 :L 166 365 :L 166 365 :L 166 364 :L 167 364 :L 167 364 :L 167 364 :L 167 364 :L 167 364 :L 167 364 :L 167 363 :L 167 363 :L 167 363 :L 168 363 :L 168 363 :L 168 363 :L 168 363 :L 168 363 :L 168 362 :L 168 362 :L 168 362 :L 168 362 :L 169 362 :L 175 370 :L 175 370 :L eofill 132 347.95 -.95 .95 167.95 366 .95 132 347 @a np 212 329 :M 202 327 :L 202 327 :L 202 326 :L 202 326 :L 202 326 :L 202 326 :L 202 326 :L 202 326 :L 203 325 :L 203 325 :L 203 325 :L 203 325 :L 203 325 :L 203 325 :L 203 325 :L 203 324 :L 203 324 :L 203 324 :L 203 324 :L 203 324 :L 203 324 :L 204 324 :L 204 323 :L 204 323 :L 204 323 :L 204 323 :L 204 323 :L 204 323 :L 204 323 :L 204 322 :L 205 322 :L 205 322 :L 205 322 :L 205 322 :L 205 322 :L 205 322 :L 205 322 :L 205 322 :L 205 321 :L 206 321 :L 206 321 :L 206 321 :L 206 321 :L 206 321 :L 206 321 :L 206 321 :L 207 321 :L 207 321 :L 212 329 :L 212 329 :L eofill 181 306.95 -.95 .95 205.95 325 .95 181 306 @a np 212 329 :M 213 319 :L 213 319 :L 213 319 :L 213 319 :L 214 319 :L 214 319 :L 214 319 :L 214 319 :L 214 319 :L 214 319 :L 215 320 :L 215 320 :L 215 320 :L 215 320 :L 215 320 :L 215 320 :L 215 320 :L 216 320 :L 216 320 :L 216 320 :L 216 320 :L 216 320 :L 216 320 :L 217 320 :L 217 320 :L 217 320 :L 217 320 :L 217 321 :L 217 321 :L 217 321 :L 218 321 :L 218 321 :L 218 321 :L 218 321 :L 218 321 :L 218 321 :L 218 321 :L 219 321 :L 219 322 :L 219 322 :L 219 322 :L 219 322 :L 219 322 :L 219 322 :L 219 322 :L 220 322 :L 220 322 :L 220 323 :L 212 329 :L 212 329 :L eofill -.95 -.95 216.95 323.95 .95 .95 223 307 @b np 253 330 :M 244 327 :L 244 327 :L 244 327 :L 244 327 :L 244 327 :L 244 327 :L 244 326 :L 244 326 :L 245 326 :L 245 326 :L 245 326 :L 245 326 :L 245 326 :L 245 325 :L 245 325 :L 245 325 :L 245 325 :L 245 325 :L 245 325 :L 245 325 :L 246 324 :L 246 324 :L 246 324 :L 246 324 :L 246 324 :L 246 324 :L 246 324 :L 246 323 :L 246 323 :L 246 323 :L 247 323 :L 247 323 :L 247 323 :L 247 323 :L 247 323 :L 247 323 :L 247 322 :L 247 322 :L 248 322 :L 248 322 :L 248 322 :L 248 322 :L 248 322 :L 248 322 :L 248 322 :L 249 322 :L 249 321 :L 249 321 :L 253 330 :L 253 330 :L eofill 224 306.95 -.95 .95 247.95 326 .95 224 306 @a np 297 330 :M 288 331 :L 288 331 :L 288 331 :L 288 330 :L 288 330 :L 288 330 :L 287 330 :L 288 330 :L 288 330 :L 288 329 :L 288 329 :L 288 329 :L 288 329 :L 288 329 :L 288 329 :L 288 328 :L 288 328 :L 288 328 :L 288 328 :L 288 328 :L 288 328 :L 288 327 :L 288 327 :L 288 327 :L 288 327 :L 288 327 :L 288 327 :L 288 326 :L 288 326 :L 288 326 :L 288 326 :L 288 326 :L 288 326 :L 289 326 :L 289 325 :L 289 325 :L 289 325 :L 289 325 :L 289 325 :L 289 325 :L 289 325 :L 289 324 :L 289 324 :L 289 324 :L 290 324 :L 290 324 :L 290 324 :L 290 324 :L 297 330 :L 297 330 :L eofill 224 306.95 -.95 .95 289.95 328 .95 224 306 @a np 236 264 :M 245 265 :L 245 265 :L 245 266 :L 245 266 :L 245 266 :L 245 266 :L 245 266 :L 245 266 :L 245 267 :L 245 267 :L 245 267 :L 245 267 :L 245 267 :L 245 267 :L 244 267 :L 244 268 :L 244 268 :L 244 268 :L 244 268 :L 244 268 :L 244 268 :L 244 268 :L 244 269 :L 244 269 :L 244 269 :L 244 269 :L 244 269 :L 243 269 :L 243 269 :L 243 270 :L 243 270 :L 243 270 :L 243 270 :L 243 270 :L 243 270 :L 243 270 :L 243 270 :L 242 270 :L 242 271 :L 242 271 :L 242 271 :L 242 271 :L 242 271 :L 242 271 :L 242 271 :L 241 271 :L 241 271 :L 241 272 :L 236 264 :L 236 264 :L eofill 242 268.95 -.95 .95 354.95 339 .95 242 268 @a np 395 339 :M 387 343 :L 387 342 :L 387 342 :L 386 342 :L 386 342 :L 386 342 :L 386 342 :L 386 341 :L 386 341 :L 386 341 :L 386 341 :L 386 341 :L 386 341 :L 386 340 :L 386 340 :L 386 340 :L 386 340 :L 386 340 :L 386 340 :L 386 339 :L 386 339 :L 386 339 :L 386 339 :L 386 339 :L 386 339 :L 386 338 :L 386 338 :L 386 338 :L 386 338 :L 386 338 :L 386 338 :L 386 337 :L 386 337 :L 386 337 :L 386 337 :L 386 337 :L 386 337 :L 386 336 :L 386 336 :L 386 336 :L 386 336 :L 386 336 :L 386 336 :L 386 336 :L 386 335 :L 386 335 :L 387 335 :L 387 335 :L 395 339 :L 395 339 :L eofill 374 339.95 -.95 .95 387.95 339 .95 374 339 @a np 255 347 :M 264 351 :L 264 351 :L 264 351 :L 264 351 :L 264 351 :L 264 352 :L 264 352 :L 264 352 :L 264 352 :L 264 352 :L 264 352 :L 264 352 :L 264 353 :L 263 353 :L 263 353 :L 263 353 :L 263 353 :L 263 353 :L 263 353 :L 263 354 :L 263 354 :L 263 354 :L 262 354 :L 262 354 :L 262 354 :L 262 354 :L 262 354 :L 262 355 :L 262 355 :L 262 355 :L 261 355 :L 261 355 :L 261 355 :L 261 355 :L 261 355 :L 261 355 :L 261 355 :L 261 356 :L 260 356 :L 260 356 :L 260 356 :L 260 356 :L 260 356 :L 260 356 :L 259 356 :L 259 356 :L 259 356 :L 259 356 :L 255 347 :L 255 347 :L eofill 261 354.95 -.95 .95 277.95 370 .95 261 354 @a np 298 347 :M 295 357 :L 295 356 :L 295 356 :L 295 356 :L 294 356 :L 294 356 :L 294 356 :L 294 356 :L 294 356 :L 294 356 :L 293 356 :L 293 356 :L 293 356 :L 293 356 :L 293 355 :L 293 355 :L 293 355 :L 292 355 :L 292 355 :L 292 355 :L 292 355 :L 292 355 :L 292 355 :L 292 355 :L 292 354 :L 291 354 :L 291 354 :L 291 354 :L 291 354 :L 291 354 :L 291 354 :L 291 354 :L 291 353 :L 291 353 :L 290 353 :L 290 353 :L 290 353 :L 290 353 :L 290 353 :L 290 352 :L 290 352 :L 290 352 :L 290 352 :L 290 352 :L 290 352 :L 289 352 :L 289 351 :L 289 351 :L 298 347 :L 298 347 :L eofill -.95 -.95 277.95 370.95 .95 .95 293 354 @b np 477 179 :M 468 183 :L 468 182 :L 468 182 :L 468 182 :L 468 182 :L 468 182 :L 468 182 :L 468 181 :L 468 181 :L 468 181 :L 468 181 :L 468 181 :L 468 181 :L 468 180 :L 468 180 :L 468 180 :L 468 180 :L 468 180 :L 468 180 :L 468 179 :L 468 179 :L 468 179 :L 468 179 :L 468 179 :L 467 179 :L 468 179 :L 468 178 :L 468 178 :L 468 178 :L 468 178 :L 468 178 :L 468 178 :L 468 177 :L 468 177 :L 468 177 :L 468 177 :L 468 177 :L 468 177 :L 468 176 :L 468 176 :L 468 176 :L 468 176 :L 468 176 :L 468 176 :L 468 175 :L 468 175 :L 468 175 :L 468 175 :L 477 179 :L 477 179 :L eofill 389 179.95 -.95 .95 469.95 179 .95 389 179 @a np 404 330 :M 400 321 :L 400 321 :L 401 321 :L 401 321 :L 401 321 :L 401 321 :L 401 321 :L 401 321 :L 402 321 :L 402 321 :L 402 321 :L 402 320 :L 402 320 :L 402 320 :L 403 320 :L 403 320 :L 403 320 :L 403 320 :L 403 320 :L 403 320 :L 404 320 :L 404 320 :L 404 320 :L 404 320 :L 404 320 :L 404 320 :L 405 320 :L 405 320 :L 405 320 :L 405 320 :L 405 320 :L 405 320 :L 406 320 :L 406 320 :L 406 320 :L 406 320 :L 406 320 :L 406 320 :L 407 321 :L 407 321 :L 407 321 :L 407 321 :L 407 321 :L 407 321 :L 408 321 :L 408 321 :L 408 321 :L 408 321 :L 404 330 :L 404 330 :L eofill -.95 -.95 404.95 323.95 .95 .95 404 265 @b np 288 216 :M 295 210 :L 295 210 :L 295 210 :L 295 210 :L 296 210 :L 296 210 :L 296 210 :L 296 210 :L 296 211 :L 296 211 :L 296 211 :L 296 211 :L 296 211 :L 297 211 :L 297 211 :L 297 212 :L 297 212 :L 297 212 :L 297 212 :L 297 212 :L 297 212 :L 297 212 :L 297 213 :L 297 213 :L 297 213 :L 297 213 :L 298 213 :L 298 213 :L 298 214 :L 298 214 :L 298 214 :L 298 214 :L 298 214 :L 298 214 :L 298 214 :L 298 215 :L 298 215 :L 298 215 :L 298 215 :L 298 215 :L 298 215 :L 298 216 :L 298 216 :L 298 216 :L 298 216 :L 298 216 :L 298 216 :L 298 217 :L 288 216 :L 288 216 :L eofill -.95 -.95 295.95 214.95 .95 .95 369 184 @b np 223 288 :M 224 279 :L 224 279 :L 224 279 :L 224 279 :L 224 279 :L 224 279 :L 225 279 :L 225 279 :L 225 279 :L 225 279 :L 225 279 :L 225 279 :L 226 279 :L 226 279 :L 226 279 :L 226 279 :L 226 279 :L 226 279 :L 227 279 :L 227 279 :L 227 279 :L 227 279 :L 227 279 :L 227 279 :L 227 280 :L 228 280 :L 228 280 :L 228 280 :L 228 280 :L 228 280 :L 228 280 :L 228 280 :L 229 280 :L 229 280 :L 229 280 :L 229 281 :L 229 281 :L 229 281 :L 229 281 :L 230 281 :L 230 281 :L 230 281 :L 230 281 :L 230 281 :L 230 282 :L 230 282 :L 230 282 :L 231 282 :L 223 288 :L 223 288 :L eofill -.95 -.95 227.95 282.95 .95 .95 234 264 @b np 362 331 :M 358 322 :L 359 322 :L 359 322 :L 359 322 :L 359 322 :L 359 322 :L 359 322 :L 360 322 :L 360 322 :L 360 322 :L 360 321 :L 360 321 :L 360 321 :L 361 321 :L 361 321 :L 361 321 :L 361 321 :L 361 321 :L 361 321 :L 362 321 :L 362 321 :L 362 321 :L 362 321 :L 362 321 :L 362 321 :L 363 321 :L 363 321 :L 363 321 :L 363 321 :L 363 321 :L 363 321 :L 364 321 :L 364 321 :L 364 321 :L 364 321 :L 364 321 :L 364 321 :L 365 321 :L 365 321 :L 365 322 :L 365 322 :L 365 322 :L 365 322 :L 366 322 :L 366 322 :L 366 322 :L 366 322 :L 366 322 :L 362 331 :L 362 331 :L eofill -.95 -.95 362.95 324.95 .95 .95 362 265 @b gR gS 0 0 552 730 rC 259 421 :M f0_12 sf (Figure )S 296 421 :M (11)S 77 445 :M f3_12 sf .372 .037(We scored the PAGs output by the algorithm in the following way. For each ordered)J 59 463 :M .146 .015(pair of variables )J f4_12 sf .064(A)A f3_12 sf .084 .008( and )J 171 463 :M f4_12 sf .053(B)A f3_12 sf .129 .013( the PAG either entails nothing about whether )J f4_12 sf .053(A)A f3_12 sf .104 .01( is an ancestor of)J 59 481 :M f4_12 sf .34(B)A f3_12 sf .557 .056(, or it entails that )J 155 481 :M f4_12 sf .339(A)A f3_12 sf .566 .057( is an ancestor of )J f4_12 sf .339(B)A f3_12 sf .548 .055(, or it entails that )J f4_12 sf .339(A)A f3_12 sf .58 .058( is not an ancestor of )J 460 481 :M f4_12 sf .207(B)A f3_12 sf .493 .049(. We)J 59 499 :M .338 .034(count the number of ordered pairs for which the output PAG entails that )J f4_12 sf .143(A)A f3_12 sf .303 .03( is an ancestor)J 59 517 :M .495 .05(of )J f4_12 sf .363(B)A f3_12 sf .872 .087(, the percentage of times the entailment is correct, the number of ordered pairs for)J 59 535 :M .415 .041(which the output PAG entails that )J 229 535 :M f4_12 sf .257(A)A f3_12 sf .439 .044( not is an ancestor of )J 343 535 :M f4_12 sf .162(B)A f3_12 sf .391 .039(, and the percentage of times)J 59 553 :M .86 .086(the entailment is correct. For purposes of comparison, we note that in the Alarm DAG)J 59 571 :M .15 .015(that there are 122 ordered pairs of distinct variables <)J f4_12 sf .063(A)A f3_12 sf (,)S f4_12 sf .063(B)A f3_12 sf .116 .012(> such that )J 392 571 :M f4_12 sf .087(A)A f3_12 sf .145 .015( is an ancestor of )J f4_12 sf (B)S 59 589 :M f3_12 sf .121 .012(\(18.32% of the ordered pairs\), and 544 ordered pairs of distinct variables <)J 421 589 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf .098 .01(> such that)J 59 607 :M f4_12 sf .271(A)A f3_12 sf .462 .046( is not an ancestor of )J 174 607 :M f4_12 sf .176(B)A f3_12 sf .423 .042( \(81.68% of the ordered pairs.\) Of course, making a variable or)J 59 625 :M -.002(variables latent and conditioning on a value of a selection variable will reduce the number)A 59 643 :M (of ancestor pairs.)S 77 667 :M 2.197 .22(We ran the algorithm on 6 different versions of the Alarm network, variously)J 59 685 :M .385 .039(obtained by treating some of the Alarm variables as latent, and by selecting on values of)J endp %%Page: 26 26 %%BeginPageSetup initializepage (peter; page: 26 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (26)S gR gS 0 0 552 730 rC 59 56 :M f3_12 sf 1.143 .114(some of the Alarm variables. The results are shown in )J 339 56 :M 1.093 .109(Table 1)J 376 56 :M 1.249 .125(. A variable )J 441 56 :M f4_12 sf .542(A)A f3_12 sf 1.095 .109( is made)J 59 74 :M .152 .015(latent by simply removing all of its value from the original data set. A variable )J 444 74 :M f4_12 sf .073(B)A f3_12 sf .147 .015( is made)J 59 92 :M .34 .034(a selection variably by choosing a subpopulation which all share the same )J 424 92 :M f4_12 sf .121(B)A f3_12 sf .313 .031( value Since)J 59 110 :M .631 .063(conditioning on a )J 149 110 :M f4_12 sf .241(B)A f3_12 sf .604 .06( value reduces the sample size, we chose by hand selection variable)J 59 128 :M 1.33 .133(values that did not reduce the sample size too much. In no case was the sample size)J 59 146 :M .129 .013(reduced below 6000. We ran the algorithm with no latent variables or selection variables,)J 59 164 :M .764 .076(with variable 29 made latent \(abbreviated as 29L in Table 1\), with variables 29 and 22)J 59 182 :M .097 .01(made latent, with variable 29 latent and 8 a selection variable \(abbreviated by 8S in Table)J 59 200 :M .386 .039(1\), with 29 latent and 1 a selection variable, and with 29, 22, and 4 latent and 8 and 1 as)J 59 218 :M (selection variables.)S 220 268 :M (Results of Simulation Studies)S 53 245 2 2 rF 53 245 2 2 rF 55 245 473 2 rF 528 245 2 2 rF 528 245 2 2 rF 53 247 2 24 rF 528 247 2 24 rF 95 293 :M (Model)S 187 293 :M (Number of)S 192 305 :M (Ancestor)S 191 317 :M (Relations)S 191 329 :M (Predicted)S 275 293 :M (% Ancestor)S 281 305 :M (Relations)S 281 317 :M (Predicted)S 285 329 :M (Correct)S 367 293 :M (Number of)S 359 305 :M (Non-Ancestor)S 371 317 :M (Relations)S 371 329 :M (Predicted)S 464 293 :M (% Non-)S 462 305 :M (Ancestor)S 461 317 :M (Relations)S 461 329 :M (Predicted)S 465 341 :M (Correct)S 53 271 2 1 rF 55 271 113 1 rF 168 271 1 1 rF 169 271 89 1 rF 258 271 1 1 rF 259 271 89 1 rF 348 271 1 1 rF 349 271 89 1 rF 438 271 1 1 rF 439 271 89 1 rF 528 271 2 1 rF 53 272 2 72 rF 168 272 1 72 rF 258 272 1 72 rF 348 272 1 72 rF 438 272 1 72 rF 528 272 2 72 rF 95 366 :M (Alarm)S 207 366 :M (62)S 287 366 :M (100.00)S 381 366 :M (1088)S 470 366 :M (96.97)S 53 344 2 1 rF 55 344 113 1 rF 168 344 1 1 rF 169 344 89 1 rF 258 344 1 1 rF 259 344 89 1 rF 348 344 1 1 rF 349 344 89 1 rF 438 344 1 1 rF 439 344 89 1 rF 528 344 2 1 rF 53 345 2 24 rF 168 345 1 24 rF 258 345 1 24 rF 348 345 1 24 rF 438 345 1 24 rF 528 345 2 24 rF 101 391 :M (29L)S 207 391 :M (21)S 287 391 :M (100.00)S 381 391 :M (1119)S 470 391 :M (90.80)S 53 369 2 1 rF 55 369 113 1 rF 168 369 1 1 rF 169 369 89 1 rF 258 369 1 1 rF 259 369 89 1 rF 348 369 1 1 rF 349 369 89 1 rF 438 369 1 1 rF 439 369 89 1 rF 528 369 2 1 rF 53 370 2 24 rF 168 370 1 24 rF 258 370 1 24 rF 348 370 1 24 rF 438 370 1 24 rF 528 370 2 24 rF 88 416 :M (29L, 22L)S 207 416 :M (25)S 287 416 :M (100.00)S 381 416 :M (1059)S 470 416 :M (91.60)S 53 394 2 1 rF 55 394 113 1 rF 168 394 1 1 rF 169 394 89 1 rF 258 394 1 1 rF 259 394 89 1 rF 348 394 1 1 rF 349 394 89 1 rF 438 394 1 1 rF 439 394 89 1 rF 528 394 2 1 rF 53 395 2 24 rF 168 395 1 24 rF 258 395 1 24 rF 348 395 1 24 rF 438 395 1 24 rF 528 395 2 24 rF 92 441 :M (29L, 8S)S 207 441 :M (43)S 290 441 :M (97.67)S 381 441 :M (1082)S 470 441 :M (92.05)S 53 419 2 1 rF 55 419 113 1 rF 168 419 1 1 rF 169 419 89 1 rF 258 419 1 1 rF 259 419 89 1 rF 348 419 1 1 rF 349 419 89 1 rF 438 419 1 1 rF 439 419 89 1 rF 528 419 2 1 rF 53 420 2 24 rF 168 420 1 24 rF 258 420 1 24 rF 348 420 1 24 rF 438 420 1 24 rF 528 420 2 24 rF 92 466 :M (29L, 1S)S 207 466 :M (31)S 290 466 :M (83.87)S 381 466 :M (1117)S 470 466 :M (86.75)S 53 444 2 1 rF 55 444 113 1 rF 168 444 1 1 rF 169 444 89 1 rF 258 444 1 1 rF 259 444 89 1 rF 348 444 1 1 rF 349 444 89 1 rF 438 444 1 1 rF 439 444 89 1 rF 528 444 2 1 rF 53 445 2 25 rF 168 445 1 25 rF 258 445 1 25 rF 348 445 1 25 rF 438 445 1 25 rF 528 445 2 25 rF 60 491 :M (29L, 22L, 4L, 8S, 1S)S 207 491 :M (23)S 290 491 :M (86.96)S 384 491 :M (861)S 470 491 :M (91.52)S 53 469 2 1 rF 55 469 113 1 rF 168 469 1 1 rF 169 469 89 1 rF 258 469 1 1 rF 259 469 89 1 rF 348 469 1 1 rF 349 469 89 1 rF 438 469 1 1 rF 439 469 89 1 rF 528 469 2 1 rF 53 470 2 24 rF 53 494 2 2 rF 53 494 2 2 rF 55 494 113 2 rF 168 470 1 24 rF 168 494 2 2 rF 170 494 88 2 rF 258 470 1 24 rF 258 494 2 2 rF 260 494 88 2 rF 348 470 1 24 rF 348 494 2 2 rF 350 494 88 2 rF 438 470 1 24 rF 438 494 2 2 rF 440 494 88 2 rF 528 470 2 24 rF 528 494 2 2 rF 528 494 2 2 rF 265 517 :M f0_12 sf (Table 1)S 77 541 :M f3_12 sf 1.576 .158(Often, when the output PAG incorrectly states that )J f4_12 sf .615(A)A f3_12 sf .51 .051( is )J 367 541 :M f4_12 sf .505(not)A f3_12 sf 1.292 .129( an ancestor of )J f4_12 sf .724(B)A f3_12 sf 1.342 .134( the)J 59 559 :M .337 .034(mistake would produce only small errors in predicting the effects on )J 396 559 :M f4_12 sf .129(B)A f3_12 sf .321 .032( of intervening on)J 59 577 :M f4_12 sf .194(A)A f3_12 sf .436 .044(, because the influence of )J 195 577 :M f4_12 sf .214(A)A f3_12 sf .219 .022( on )J f4_12 sf .214(B)A f3_12 sf .485 .048( is very weak. For example, in the last simulation test)J 59 595 :M .175 .018(we did, the output PAG incorrectly stated that 32 is not an ancestor of 6. However, 6 and)J 59 613 :M .371 .037(32 are almost independent; they pass a test of independence at the .01 significance level.)J 59 631 :M .876 .088(So for the purposes of predicting the effects of intervention, this particular error is not)J 59 649 :M .486 .049(important. However, we have not yet systematically calculated how important the errors)J 59 667 :M (that algorithm makes are for prediction.)S endp %%Page: 27 27 %%BeginPageSetup initializepage (peter; page: 27 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (27)S gR gS 0 0 552 730 rC 59 51 :M f0_14 sf (V)S 69 51 :M (I)S 74 51 :M (I)S 79 51 :M (.)S 82 51 :M ( )S 95 51 :M (Future Work)S 77 74 :M f3_12 sf .475 .047(The FCI algorithm could be improved in several ways. First, when the results of the)J 59 92 :M .336 .034(statistical tests that it performs are conflict in the sense that they are not compatible with)J 59 110 :M .24 .024(any DAG, it could make more intelligent compromises based on what the preponderance)J 59 128 :M 1.222 .122(of the evidence is. Second, there has been some progress recently in heuristic greedy)J 59 146 :M 2.978 .298(DAG searches based upon maximizing some model score such as the posterior)J 59 164 :M .066 .007(probability of the minimum description length. \(Cooper and Herskovits 1992, Heckerman)J 59 182 :M .258 .026(et al. 1994, Chickering et al. 1995.\) Combining an independence test algorithm for DAG)J 59 200 :M 2.609 .261(search \(essentially a special case of FCI\) and greedy DAG searches based upon)J 59 218 :M .335 .034(maximizing a score has proved successful in simulation studies \(Spirtes and Meek 1995,)J 59 236 :M .525 .053(Singh and Valorta 1993\). An analogous strategy might improve the accuracy of the FCI)J 59 254 :M (algorithm, although the task of calculating scores for a PAG faces a number of obstacles.)S 77 278 :M 1.171 .117(The FCI algorithm will also be tested on a wider variety of DAGs, and empirical)J 59 296 :M (examples.)S 59 327 :M f0_14 sf (V)S 69 327 :M (I)S 74 327 :M (I)S 79 327 :M (I)S 84 327 :M (.)S 87 327 :M ( )S 95 327 :M (Appendix)S 95 350 :M f8_12 sf (A)S 103 350 :M (.)S 106 350 :M ( )S 131 350 :M (The Assumptions)S 77 374 :M f3_12 sf 1.411 .141(The Causal Markov Assumption does not hold for every set of variables, nor for)J 59 392 :M .382 .038(every subpopulation. Consider the following example. Suppose that the DAG in \(i\) is an)J 59 410 :M .405 .041(accurate description of the causal relations in a population )J f4_12 sf .14(Pop)A f3_12 sf .289 .029( for the set of variables )J 482 410 :M f0_12 sf (V)S 59 428 :M f3_12 sf (= {)S 75 428 :M f4_12 sf (age)S f3_12 sf (, )S f4_12 sf (intelligence)S 154 428 :M f3_12 sf (, )S f4_12 sf (sex drive)S f3_12 sf (, )S f4_12 sf .03 .003(student status)J 275 428 :M f3_12 sf .019 .002(}and the Causal Markov Assumption and the)J 59 446 :M (Causal Faithfulness Assumption hold in )S 254 446 :M f4_12 sf (Pop)S f3_12 sf ( for )S 293 446 :M f0_12 sf (V)S 302 446 :M f3_12 sf (.)S 77 470 :M .735 .073(Let )J 97 470 :M f0_12 sf <56D5>S 110 470 :M f3_12 sf .831 .083( = {)J 131 470 :M f4_12 sf (intelligence)S 187 470 :M f3_12 sf .195 .02(, )J f4_12 sf .931 .093(sex drive)J 237 470 :M f3_12 sf .641 .064(}. The causal DAGs for the sets of variables )J 461 470 :M f0_12 sf (V)S 470 470 :M f3_12 sf .735 .073( and)J 59 488 :M f0_12 sf <56D5>S 72 488 :M f3_12 sf ( are shown in \(i\) and \(ii\) respectively of )S 266 488 :M (Figure 12.)S .75 lw 111 521 345 159 rC 111.5 521.5 172 157 rS 113 523 169 154 rC 131 546 :M f4_12 sf ( )S 134 546 :M ( )S 137 546 :M (a)S 142 546 :M (g)S 148 546 :M (e)S 153 546 :M ( )S 156 546 :M ( )S 159 546 :M ( )S 162 546 :M ( )S 165 546 :M ( )S 168 546 :M ( )S 171 546 :M ( )S 174 546 :M ( )S 177 546 :M ( )S 180 546 :M ( )S 183 546 :M ( )S 186 546 :M ( )S 189 546 :M ( )S 192 546 :M ( )S 195 546 :M ( )S 198 546 :M (i)S 201 546 :M (n)S 207 546 :M (t)S 210 546 :M (e)S 215 546 :M (l)S 218 546 :M (l)S 221 546 :M (i)S 224 546 :M (g)S 230 546 :M (e)S 235 546 :M (n)S 241 546 :M (c)S 246 546 :M (e)S 131 600 :M (s)S 136 600 :M (e)S 141 600 :M (x)S 147 600 :M ( )S 150 600 :M (d)S 156 600 :M (r)S 160 600 :M (i)S 163 600 :M (v)S 169 600 :M (e)S 174 600 :M ( )S 177 600 :M ( )S 180 600 :M ( )S 183 600 :M ( )S 186 600 :M ( )S 189 600 :M ( )S 192 600 :M ( )S 195 600 :M ( )S 198 600 :M (s)S 203 600 :M (t)S 206 600 :M (u)S 212 600 :M (d)S 218 600 :M (e)S 223 600 :M (n)S 229 600 :M (t)S 232 600 :M ( )S 235 600 :M (s)S 240 600 :M (t)S 243 600 :M (a)S 248 600 :M (t)S 251 600 :M (u)S 257 600 :M (s)S 131 627 :M ( )S 134 627 :M ( )S 137 627 :M ( )S 140 627 :M ( )S 143 627 :M ( )S 146 627 :M ( )S 149 627 :M ( )S 152 627 :M ( )S 155 627 :M ( )S 158 627 :M ( )S 161 627 :M ( )S 164 627 :M ( )S 167 627 :M ( )S 170 627 :M ( )S 173 627 :M ( )S 176 627 :M ( )S 179 627 :M ( )S 182 627 :M ( )S 185 627 :M ( )S 188 627 :M f3_12 sf <28>S 192 627 :M (i)S 195 627 :M <29>S 145 654 :M (C)S 153 654 :M (a)S 158 654 :M (u)S 164 654 :M (s)S 169 654 :M (a)S 174 654 :M (l)S 177 654 :M ( )S 180 654 :M (D)S 188 654 :M (A)S 196 654 :M (G)S 205 654 :M ( )S 208 654 :M (f)S 212 654 :M (o)S 218 654 :M (r)S 222 654 :M ( )S 225 654 :M f0_12 sf (V)S 234 654 :M f3_12 sf ( )S 237 654 :M (i)S 240 654 :M (n)S 246 654 :M ( )S 249 654 :M f4_12 sf (P)S 256 654 :M (o)S 262 654 :M (p)S 185 667 :M f3_12 sf (a)S 190 667 :M (n)S 196 667 :M (d)S 202 667 :M ( )S 205 667 :M f4_12 sf (P)S 212 667 :M (o)S 218 667 :M (p)S 224 667 :M S gR gS 111 521 345 159 rC -.75 -.75 148.75 581.75 .75 .75 148 552 @b np 152 579 :M 144 579 :L 148 587 :L 152 579 :L .75 lw eofill 144 579.75 -.75 .75 152.75 579 .75 144 579 @a 144 579.75 -.75 .75 148.75 587 .75 144 579 @a -.75 -.75 148.75 587.75 .75 .75 152 579 @b 149 551.75 -.75 .75 219.75 583 .75 149 551 @a np 218 579 :M 215 585 :L 224 585 :L 218 579 :L eofill -.75 -.75 215.75 585.75 .75 .75 218 579 @b 215 585.75 -.75 .75 224.75 585 .75 215 585 @a 218 579.75 -.75 .75 224.75 585 .75 218 579 @a -.75 -.75 229.75 579.75 .75 .75 229 550 @b np 233 577 :M 225 577 :L 229 585 :L 233 577 :L eofill 225 577.75 -.75 .75 233.75 577 .75 225 577 @a 225 577.75 -.75 .75 229.75 585 .75 225 577 @a -.75 -.75 229.75 585.75 .75 .75 233 577 @b 282.5 521.5 172 157 rS 284 523 169 154 rC 302 546 :M f4_12 sf ( )S 305 546 :M ( )S 308 546 :M ( )S 311 546 :M ( )S 314 546 :M ( )S 317 546 :M ( )S 320 546 :M ( )S 323 546 :M ( )S 326 546 :M ( )S 329 546 :M ( )S 332 546 :M ( )S 335 546 :M ( )S 338 546 :M ( )S 341 546 :M ( )S 344 546 :M ( )S 347 546 :M ( )S 350 546 :M ( )S 353 546 :M ( )S 356 546 :M ( )S 359 546 :M ( )S 362 546 :M ( )S 365 546 :M (i)S 368 546 :M (n)S 374 546 :M (t)S 377 546 :M (e)S 382 546 :M (l)S 385 546 :M (l)S 388 546 :M (i)S 391 546 :M (g)S 397 546 :M (e)S 402 546 :M (n)S 408 546 :M (c)S 413 546 :M (e)S 302 600 :M (s)S 307 600 :M (e)S 312 600 :M (x)S 318 600 :M ( )S 321 600 :M (d)S 327 600 :M (r)S 331 600 :M (i)S 334 600 :M (v)S 340 600 :M (e)S 302 627 :M f3_12 sf ( )S 305 627 :M ( )S 308 627 :M ( )S 311 627 :M ( )S 314 627 :M ( )S 317 627 :M ( )S 320 627 :M ( )S 323 627 :M ( )S 326 627 :M ( )S 329 627 :M ( )S 332 627 :M ( )S 335 627 :M ( )S 338 627 :M ( )S 341 627 :M ( )S 344 627 :M ( )S 347 627 :M ( )S 350 627 :M ( )S 353 627 :M ( )S 356 627 :M <28>S 360 627 :M (i)S 363 627 :M (i)S 366 627 :M <29>S 313 654 :M (C)S 321 654 :M (a)S 326 654 :M (u)S 332 654 :M (s)S 337 654 :M (a)S 342 654 :M (l)S 345 654 :M ( )S 348 654 :M (D)S 356 654 :M (A)S 364 654 :M (G)S 373 654 :M ( )S 376 654 :M (f)S 380 654 :M (o)S 386 654 :M (r)S 390 654 :M ( )S 393 654 :M f0_12 sf (V)S 402 654 :M S 407 654 :M f3_12 sf ( )S 410 654 :M (i)S 413 654 :M (n)S 419 654 :M ( )S 422 654 :M f4_12 sf (P)S 429 654 :M (o)S 435 654 :M (p)S 356 667 :M f3_12 sf (a)S 361 667 :M (n)S 367 667 :M (d)S 373 667 :M ( )S 376 667 :M f4_12 sf (P)S 383 667 :M (o)S 389 667 :M (p)S 395 667 :M S endp %%Page: 28 28 %%BeginPageSetup initializepage (peter; page: 28 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (28)S gR gS 0 0 552 730 rC 259 56 :M f0_12 sf (Figure )S 296 56 :M (12)S 77 80 :M f3_12 sf .241 .024(There is no edge between )J f4_12 sf .073(intelligence)A 261 80 :M f3_12 sf .403 .04( and)J 282 80 :M f4_12 sf .274 .027( sex drive)J f3_12 sf .232 .023( in \(ii\) because )J f4_12 sf .091(intelligence)A 461 80 :M f3_12 sf .403 .04( is not)J 59 98 :M .123 .012(a cause of )J 111 98 :M f4_12 sf .149 .015(sex drive)J f3_12 sf .062 .006( and )J f4_12 sf .149 .015(sex drive)J f3_12 sf .07 .007( is not a cause of )J f4_12 sf .034(intelligence)A 360 98 :M f3_12 sf .107 .011(. Note that although \(i\) and)J 59 116 :M .24 .024(\(ii\) are different DAGs, the causal facts represented in \(ii\) are a subset of the causal facts)J 59 134 :M (represented in \(i\).)S 77 158 :M 2.996 .3(It follows from the Causal Markov Assumption and the Causal Faithfulness)J 59 176 :M 2.803 .28(Assumption for )J 147 176 :M f0_12 sf (V)S 156 176 :M f3_12 sf 1.975 .198( in )J f4_12 sf 1.992(Pop)A f3_12 sf 3.21 .321(, that )J 236 176 :M f4_12 sf 4.877 .488(sex drive)J f3_12 sf 2.096 .21( and )J 316 176 :M f4_12 sf (intelligence)S 372 176 :M f3_12 sf 3.177 .318( are dependent in the)J 59 194 :M 1.337 .134(subpopulation )J f4_12 sf .264(Pop\325)A f3_12 sf .856 .086( of college students. \(Causal Faithfulness entails that )J 424 194 :M f4_12 sf .976 .098(sex drive)J 469 194 :M f3_12 sf 1.351 .135( and)J 59 212 :M f4_12 sf (intelligence)S 115 212 :M f3_12 sf 1.85 .185( are dependent conditional on some value of )J 352 212 :M f4_12 sf 1.237 .124(student status)J 421 212 :M f3_12 sf 1.778 .178(, and because)J 59 230 :M .626 .063(student status is binary, it follows that )J f4_12 sf .856 .086(sex drive)J f3_12 sf .353 .035( and )J f4_12 sf .193(intelligence)A 377 230 :M f3_12 sf .763 .076( are dependent on both)J 59 248 :M (values of )S 106 248 :M f4_12 sf (student status)S 172 248 :M f3_12 sf (\).)S 77 272 :M 1.221 .122(The Causal Markov Assumption is not true in )J f4_12 sf .397(Pop\325)A f3_12 sf .566 .057( for )J 361 272 :M f0_12 sf <56D5>S 374 272 :M f3_12 sf .945 .094(, because )J f4_12 sf 1.44 .144(sex drive)J 469 272 :M f3_12 sf 1.392 .139( and)J 59 290 :M f4_12 sf (intelligence)S 115 290 :M f3_12 sf .304 .03( are dependent in the subpopulation )J 292 290 :M f4_12 sf .1(Pop\325)A f3_12 sf .275 .028( of college students, i.e. they are not)J 59 308 :M .515 .051(independent given the parents in \(ii\) of sex drive or of intelligence \(i.e. not independent)J 59 326 :M 2.31 .231(given the empty set\). Indeed because the Causal Markov and Causal Faithfulness)J 59 344 :M .045 .005(Assumptions hold in )J f4_12 sf (Pop)S f3_12 sf ( for )S f0_12 sf (V)S 209 344 :M f3_12 sf .043 .004(, the Causal Markov Assumption is )J 382 344 :M f4_12 sf (entailed)S 421 344 :M f3_12 sf .052 .005( to fail in )J 468 344 :M f4_12 sf (Pop\325)S 59 362 :M f3_12 sf (for )S 76 362 :M f0_12 sf <56D5>S 89 362 :M f3_12 sf (.)S 77 386 :M 2.124 .212(As a consequence, in general we will )J f4_12 sf .641(not)A f3_12 sf 2.057 .206( assume that for the set of measured)J 59 404 :M 1.526 .153(variables, and the subpopulation from which the sample was drawn, that the Causal)J 59 422 :M 1.156 .116(Markov and Causal Faithfulness Assumptions hold. Rather we will assume, as in this)J 59 440 :M .254 .025(example, that there is a larger set of variables that includes the measured variables, and a)J 59 458 :M .188 .019(larger population that includes the subpopulation from which the sample was drawn such)J 59 476 :M (that:)S 77 500 :M f1_12 sf S 82 500 :M 8 .8( )J 95 500 :M f3_12 sf 1.401 .14(the causal DAG in the expanded set of variables for the expanded population)J 95 518 :M (satisfies the Causal Markov and Causal Faithfulness Assumptions;)S 77 542 :M f1_12 sf S 82 542 :M 10 1( )J 95 542 :M f3_12 sf .292 .029(the causal structure in the subpopulation is the same as the causal structure in the)J 95 560 :M (expanded population.)S 77 584 :M .334 .033(In general, there will be more than one way of expanding the set of variables and the)J 59 602 :M .962 .096(population so that the conditions described above are satisfied; corresponding to these)J 59 620 :M .538 .054(different sets of variables and populations will be different causal DAGs. \(For example,)J 59 638 :M .164 .016(instead of adding just )J 166 638 :M f4_12 sf .039(age)A f3_12 sf .063 .006( and )J f4_12 sf .039(student)A 241 638 :M f3_12 sf ( )S f4_12 sf .077(status)A f3_12 sf .11 .011( to )J 288 638 :M f0_12 sf <56D5>S 301 638 :M f3_12 sf .188 .019(, we could add )J 375 638 :M f4_12 sf .048(age)A f3_12 sf .033 .003( , )J f4_12 sf .048(student)A 436 638 :M f3_12 sf .254 .025( )J 440 638 :M f4_12 sf .023(status)A f3_12 sf .083 .008(, and)J 59 656 :M .555 .055(some irrelevant variable such as )J 221 656 :M f4_12 sf .813 .081(eye color)J f3_12 sf .559 .056(, with no edges between )J f4_12 sf .813 .081(eye color)J f3_12 sf .468 .047( and any of)J endp %%Page: 29 29 %%BeginPageSetup initializepage (peter; page: 29 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (29)S gR gS 0 0 552 730 rC 59 56 :M f3_12 sf .197 .02(the other variables.\) This does not matter, because the conclusions that we will draw will)J 59 74 :M (be true of )S 108 74 :M f4_12 sf (all)S 121 74 :M f3_12 sf ( of the causal DAGs that satisfy the conditions we have laid down.)S 77 98 :M .453 .045(The assumptions that \(i\) the )J f0_12 sf (S)S 224 98 :M f3_12 sf .683 .068( = )J 239 98 :M f0_12 sf .147(1)A f3_12 sf .462 .046( subpopulation from which the sample is drawn is)J 59 116 :M .197 .02(part of a population )J f4_12 sf .079(Pop)A f3_12 sf .255 .025( in which the Causal Mark and Causal Faithfulness Assumptions)J 59 134 :M .388 .039(hold for some causally sufficient set of variables )J f0_12 sf (V)S 309 134 :M f3_12 sf .4 .04( and \(ii\) the causal structures relative)J 59 152 :M .324 .032(to )J f0_12 sf (V)S 81 152 :M f3_12 sf .278 .028( in )J f4_12 sf .281(Pop)A f3_12 sf .458 .046( and the )J f0_12 sf (S)S 166 152 :M f3_12 sf .652 .065( = )J 181 152 :M f0_12 sf .128(1)A f3_12 sf .416 .042( subpopulation are the same, are sufficient \(but not necessary\))J 59 170 :M (conditions for the following assumption:)S 77 194 :M f0_12 sf -.003(Selection Bias Causal Assumption:)A 255 194 :M f3_12 sf -.005( For each set of variables )A f0_12 sf (O)S f3_12 sf -.006(, and each population)A 59 212 :M f4_12 sf .074 .007(Pop )J f3_12 sf .056 .006( such that )J f0_12 sf (S)S 137 212 :M f3_12 sf .083 .008( = )J 151 212 :M f0_12 sf (1)S f3_12 sf .055 .005(, there is a causally sufficient set of variables )J f0_12 sf (V)S 385 212 :M f3_12 sf .068 .007( such that )J 435 212 :M f0_12 sf .054(O)A f3_12 sf ( )S f1_12 sf .054A f3_12 sf ( )S f0_12 sf (S)S 466 212 :M f3_12 sf ( )S f1_12 sf S 478 212 :M f3_12 sf ( )S f0_12 sf (V)S 59 230 :M f3_12 sf .165 .016(and for all )J f0_12 sf (A)S 121 230 :M f3_12 sf .094 .009(, )J f0_12 sf .15(B)A f3_12 sf .094 .009(, )J f0_12 sf (C)S 150 230 :M f3_12 sf .269 .027( )J 154 230 :M f1_12 sf S 163 230 :M f3_12 sf .054 .005( )J f0_12 sf .186(O)A f3_12 sf .1 .01(, )J f0_12 sf (A)S 190 230 :M f3_12 sf .269 .027( )J 193 217 16 16 rC -1 -1 200 230 1 1 199 220 @b -1 -1 204 230 1 1 203 220 @b 196 231 -1 1 207 230 1 196 230 @a gR gS 0 0 552 730 rC 209 230 :M f3_12 sf .269 .027( )J 213 230 :M f0_12 sf .148(B)A f3_12 sf .096 .01( | \()J f0_12 sf (C)S 242 230 :M f3_12 sf .269 .027( )J 246 230 :M f1_12 sf .108A f3_12 sf .069 .007( \()J f0_12 sf (S)S 269 230 :M f3_12 sf .142 .014( = )J f0_12 sf .154(1)A f3_12 sf .239 .024(\)\) in )J 312 230 :M f4_12 sf .079(Pop)A f3_12 sf .163 .016( if and only if the causal DAG )J f4_12 sf (G)S 59 248 :M f3_12 sf (relative to )S f0_12 sf (V)S 119 248 :M f3_12 sf ( in )S f4_12 sf (Pop)S f3_12 sf ( entails that )S f0_12 sf (A)S 223 248 :M f3_12 sf ( )S 226 235 16 16 rC -1 -1 233 248 1 1 232 238 @b -1 -1 237 248 1 1 236 238 @b 229 249 -1 1 240 248 1 229 248 @a gR gS 0 0 552 730 rC 242 248 :M f3_12 sf ( )S f0_12 sf (B)S f3_12 sf ( | \()S f0_12 sf (C)S 274 248 :M f3_12 sf ( )S f1_12 sf S f3_12 sf ( \()S 293 248 :M f0_12 sf (S)S 300 248 :M f3_12 sf ( = )S 313 248 :M f0_12 sf (1)S f3_12 sf (\)\) in )S f4_12 sf (Pop)S 77 272 :M f3_12 sf .355 .036(The Selection Bias Causal Assumption is sufficient for the asymptotic correctness of)J 59 290 :M (the methods of inference described in this paper.)S 95 314 :M f8_12 sf (B)S 103 314 :M (.)S 106 314 :M ( )S 131 314 :M (Proofs)S 77 338 :M f3_12 sf 1.039 .104(In this section we will prove all of the theorems in the main body of the paper. In)J 59 356 :M (order to simplify the proofs, the theorems are not proved in the order they were stated.)S 77 380 :M .244 .024(In the usual graph theoretic definition, a graph is an ordered pair <)J f0_12 sf (V)S 409 380 :M f3_12 sf (,)S f0_12 sf .119(E)A f3_12 sf .232 .023(> where )J 463 380 :M f0_12 sf (V)S 472 380 :M f3_12 sf .329 .033( is a)J 59 398 :M .087 .009(set of vertices, and )J f0_12 sf (E)S f3_12 sf .076 .008( is a set of edges. The members of )J 328 398 :M f0_12 sf (E )S f3_12 sf .08 .008(are pairs of vertices \(an ordered)J 59 416 :M .498 .05(pair in a directed graph and an unordered pair in an undirected graph\). For example, the)J 59 434 :M .477 .048(edge )J 86 434 :M f4_12 sf .474(A)A f3_12 sf .194 .019( )J 97 434 :M f1_12 sf S 109 434 :M f3_12 sf .079 .008( )J f4_12 sf .214(B)A f3_12 sf .476 .048( is represented by the ordered pair <)J 297 434 :M f4_12 sf .154(A)A f3_12 sf .063(,)A f4_12 sf .154(B)A f3_12 sf .379 .038(>. In directed graphs the ordering of)J 59 452 :M .598 .06(the pair of vertices representing an edge in effect marks an arrowhead at one end of the)J 59 470 :M .072 .007(edge. For our purposes we need to represent a larger variety of marks attached to the ends)J 59 488 :M .303 .03(of undirected edges. In general, we allow that the end of an edge can be marked \322out of\323)J 59 506 :M (by \322)S f1_12 sf (-)S 86 506 :M f3_12 sf (\323, or can be marked with \322>\323, or can be marked with an \322o\323.)S 77 530 :M .618 .062(In order to specify completely the type of an edge, therefore, we need to specify the)J 59 548 :M 1.965 .197(variables and )J 133 548 :M f0_12 sf (marks)S 166 548 :M f3_12 sf 2.176 .218( at each end. For example, the left end of ")J f4_12 sf 1.125(A)A f3_12 sf 1.151 .115( o)J f1_12 sf S 433 548 :M f3_12 sf 3.242 .324( )J 440 548 :M f4_12 sf 1.029(B)A f3_12 sf 2.02 .202(" can be)J 59 566 :M .058 .006(represented as the ordered pair [)J 214 566 :M f4_12 sf (A)S f3_12 sf .051 .005(, o])J 237 563 :M f3_8 sf (3)S 241 566 :M f3_12 sf .063 .006(, and the right end can be represented as the ordered)J 59 584 :M 1.033 .103(pair [)J 87 584 :M f4_12 sf .631(B)A f3_12 sf 1.12 .112(, >]. We will also call [)J f4_12 sf .631(A)A f3_12 sf .83 .083(, o] the )J 263 584 :M f4_12 sf .524(A)A f3_12 sf 1.044 .104( end of the edge between )J 402 584 :M f4_12 sf .535(A)A f3_12 sf .68 .068( and )J f4_12 sf .535(B)A f3_12 sf 1.142 .114(. The first)J 59 602 :M .247 .025(member of the ordered pair is called an endpoint of an edge, e.g. in [)J 394 602 :M f4_12 sf .112(A)A f3_12 sf .231 .023(, o] the endpoint is)J 59 620 :M f4_12 sf .206(A)A f3_12 sf .466 .047(. The entire edge consists of a set of ordered pairs that represent both of the endpoints,)J 59 661 :M ( )S 59 658.48 -.48 .48 203.48 658 .48 59 658 @a 77 673 :M f3_10 sf (3)S 82 676 :M .327 .033(It is customary to represent the ordered pair )J f4_10 sf .142(A)A f3_10 sf .097 .01(, )J f4_10 sf .142(B)A f3_10 sf .312 .031( with angle brackets as <)J 382 676 :M f4_10 sf .14(A)A f3_10 sf .095 .01(, )J f4_10 sf .14(B)A f3_10 sf .331 .033(>, but for endpoints of)J 59 687 :M (an edge we use square brackets so that the angle brackets will not be misread as arrowheads.)S endp %%Page: 30 30 %%BeginPageSetup initializepage (peter; page: 30 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (30)S gR gS 0 0 552 730 rC 59 56 :M f3_12 sf .097 .01(e.g. {[)J f4_12 sf (A)S f3_12 sf .053 .005(, o], [)J f4_12 sf (B)S f3_12 sf .082 .008(, >]}. The edge {[)J 216 56 :M f4_12 sf (B)S f3_12 sf .062 .006(, >],[)J 247 56 :M f4_12 sf (A)S f3_12 sf .076 .008(, o]} is the same as {[)J f4_12 sf (A)S f3_12 sf .093 .009(, o],[)J 389 56 :M f4_12 sf (B)S f3_12 sf .075 .007(, >]} since it doesn't)J 59 74 :M (matter which end of the edge is listed first.)S 77 98 :M (Note that a directed edge such as )S 238 98 :M f4_12 sf (A)S f3_12 sf ( )S f1_12 sf S 260 98 :M f3_12 sf ( )S f4_12 sf (B)S f3_12 sf ( has a mark \322)S f1_12 sf (-)S 340 98 :M f3_12 sf (\323at the )S 375 98 :M f4_12 sf (A)S f3_12 sf ( end.)S 77 122 :M .056 .006(We say a )J f0_12 sf .036(graph)A 155 122 :M f3_12 sf .084 .008( is an ordered triple <)J 259 122 :M f0_12 sf (V)S 268 122 :M f3_12 sf (,)S f0_12 sf .024(M,E)A f3_12 sf .048 .005(> where )J f0_12 sf (V)S 344 122 :M f3_12 sf .08 .008( is a non-empty set of vertices,)J 59 140 :M f0_12 sf 1.252(M)A f3_12 sf 1.587 .159( is a non-empty set of marks, and )J 252 140 :M f0_12 sf .989(E)A f3_12 sf 1.638 .164( is a set of sets of ordered pairs of the form)J 59 158 :M ({[)S 69 158 :M f4_12 sf (V)S f3_10 sf 0 2 rm (1)S 0 -2 rm f3_12 sf (,)S f4_12 sf (M)S 94 160 :M f3_10 sf (1)S f3_12 sf 0 -2 rm (],[)S 0 2 rm 110 158 :M f4_12 sf (V)S f3_10 sf 0 2 rm (2)S 0 -2 rm f3_12 sf (,)S f4_12 sf (M)S 135 160 :M f3_10 sf .12(2)A f3_12 sf 0 -2 rm .399 .04(]}, where )J 0 2 rm 190 158 :M f4_12 sf .289(V)A f3_10 sf 0 2 rm .197(1)A 0 -2 rm f3_12 sf .368 .037( and )J f4_12 sf .289(V)A f3_10 sf 0 2 rm .197(2)A 0 -2 rm f3_12 sf .351 .035( are in )J 273 158 :M f0_12 sf (V)S 282 158 :M f3_12 sf .299 .03(, )J f4_12 sf .439(V)A f3_10 sf 0 2 rm .299(1)A 0 -2 rm f3_12 sf .179 .018( )J 304 158 :M f1_12 sf S 311 158 :M f3_12 sf .638 .064( )J 315 158 :M f4_12 sf .192(V)A f3_10 sf 0 2 rm .131(2)A 0 -2 rm f3_12 sf .266 .027(, and )J f4_12 sf (M)S 364 160 :M f3_10 sf .167(1)A f3_12 sf 0 -2 rm .311 .031( and )J 0 2 rm f4_12 sf 0 -2 rm (M)S 0 2 rm 403 160 :M f3_10 sf .252(2)A f3_12 sf 0 -2 rm .449 .045( are in )J 0 2 rm 443 158 :M f0_12 sf .391(M)A f3_12 sf .235 .023(. If )J f4_12 sf (G)S 481 158 :M f3_12 sf .58 .058( =)J 59 176 :M (<)S 66 176 :M f0_12 sf (V)S 75 176 :M f3_12 sf (,)S f0_12 sf (M,E)S f3_12 sf (> we say that )S 167 176 :M f4_12 sf (G)S 176 176 :M f3_12 sf ( is )S f0_12 sf (over)S 213 176 :M f3_12 sf ( )S f0_12 sf (V)S 225 176 :M f3_12 sf (.)S 77 200 :M 1.111 .111(We distinguish the following kinds of edges. An edge {[)J f4_12 sf .436(A)A f3_12 sf .324 .032(, )J 378 200 :M f1_12 sf (-)S 385 200 :M f3_12 sf (],[)S 396 200 :M f4_12 sf .756(B)A f3_12 sf 1.106 .111(,>]} is a )J 450 200 :M f0_12 sf (directed)S 59 218 :M .793(edge)A f3_12 sf 1.534 .153( from )J f4_12 sf .997(A)A f3_12 sf .869 .087( to )J f4_12 sf .997(B)A f3_12 sf 1.773 .177(, and is written )J f4_12 sf .997(A)A f3_12 sf .408 .041( )J 247 218 :M f1_12 sf S 259 218 :M f3_12 sf .752 .075( )J f4_12 sf 2.022(B)A f3_12 sf 1.838 .184( or )J f4_12 sf 2.022(A)A f3_12 sf .827 .083( )J 305 218 :M f1_12 sf S 317 218 :M f3_12 sf .397 .04( )J f4_12 sf 1.068(B)A f3_12 sf 2.099 .21(. An edge {[)J 396 218 :M f4_12 sf .93(A)A f3_12 sf .634 .063(, )J f1_12 sf (-)S 418 218 :M f3_12 sf (],[)S 429 218 :M f4_12 sf 1.39(B)A f3_12 sf 1.034 .103(, )J 445 218 :M f1_12 sf (-)S 452 218 :M f3_12 sf 1.872 .187(]} is an)J 59 236 :M f0_12 sf .529 .053(undirected edge)J 142 236 :M f3_12 sf .907 .091( between )J 191 236 :M f4_12 sf .497(A)A f3_12 sf .633 .063( and )J f4_12 sf .497(B)A f3_12 sf .9 .09(, and is written )J 310 236 :M f4_12 sf .507(A)A f3_12 sf .189 .019( )J f1_12 sf .83A f3_12 sf .189 .019( )J f4_12 sf .507(B)A f3_12 sf .997 .1(. An edge {[)J 407 236 :M f4_12 sf .464(A)A f3_12 sf .939 .094(, >],[)J f4_12 sf .464(B)A f3_12 sf .679 .068(, >]} is a)J 59 254 :M f0_12 sf .621 .062(bidirected edge)J 139 254 :M f3_12 sf 1.065 .107( between )J 188 254 :M f4_12 sf .566(A)A f3_12 sf .721 .072( and )J f4_12 sf .566(B)A f3_12 sf 1.007 .101(, and is written )J f4_12 sf .566(A)A f3_12 sf .211 .021( )J f1_12 sf S 332 254 :M f3_12 sf 1.491 .149( )J 337 254 :M f4_12 sf .452(B)A f3_12 sf .868 .087(. An edge {[)J f4_12 sf .452(A)A f3_12 sf .945 .095(, o],[)J 439 254 :M f4_12 sf .667(B)A f3_12 sf .976 .098(, >]} is a)J 59 272 :M f0_12 sf .862 .086(partially directed edge)J f3_12 sf .528 .053( between )J 225 272 :M f4_12 sf .507(A)A f3_12 sf .646 .065( and )J f4_12 sf .507(B)A f3_12 sf .918 .092(, and is written )J 344 272 :M f4_12 sf .337(A)A f3_12 sf .345 .034( o)J f1_12 sf S 373 272 :M f3_12 sf .403 .04( )J f4_12 sf 1.082(B)A f3_12 sf 1.026 .103( or )J 403 272 :M f4_12 sf .445(A)A f3_12 sf .165 .017( )J f1_12 sf .559A f3_12 sf .165 .017( )J f4_12 sf .445(B)A f3_12 sf 1.003 .1(. An edge)J 59 290 :M ({[)S 69 290 :M f4_12 sf .157(A)A f3_12 sf .309 .031(, o],[)J f4_12 sf .157(B)A f3_12 sf .189 .019(, o]} is a )J 152 290 :M f0_12 sf .141 .014(nondirected edge)J 240 290 :M f3_12 sf .251 .025( between )J 287 290 :M f4_12 sf .122(A)A f3_12 sf .156 .016( and )J f4_12 sf .122(B)A f3_12 sf .218 .022(, and is written )J f4_12 sf .122(A)A f3_12 sf .125 .013( o)J f1_12 sf .31A 435 290 :M f3_12 sf .351 .035( )J 439 290 :M f4_12 sf (B)S f3_12 sf .159 .016(. Vertices)J 59 308 :M .394 .039(X, Y, and Z are in a )J 161 308 :M f0_12 sf (triangle)S 201 308 :M f3_12 sf .319 .032( in graph )J f4_12 sf (G)S 257 308 :M f3_12 sf .389 .039( if and only if )J 328 308 :M f4_12 sf .185(A)A f3_12 sf .235 .023( and )J f4_12 sf .185(B)A f3_12 sf .373 .037( are adjacent, )J f4_12 sf .185(B)A f3_12 sf .235 .023( and )J f4_12 sf .202(C)A f3_12 sf .342 .034( are)J 59 326 :M (adjacent, and )S f4_12 sf (A)S f3_12 sf ( and )S f4_12 sf (C)S f3_12 sf ( are adjacent.)S 77 350 :M .113 .011(For a directed edge )J 173 350 :M f4_12 sf .055(A)A f3_12 sf ( )S f1_12 sf S 195 350 :M f3_12 sf .056 .006( )J f4_12 sf .15(B)A f3_12 sf .111 .011(, )J 212 350 :M f4_12 sf .066(A)A f3_12 sf .075 .007( is the )J f0_12 sf .05(tail)A 268 350 :M f3_12 sf .126 .013( of the edge and )J 349 350 :M f4_12 sf .057(B)A f3_12 sf .065 .007( is the )J f0_12 sf .064(head)A 413 350 :M f3_12 sf .126 .013(; the edge is )J 475 350 :M f0_12 sf (out)S 59 368 :M (of)S 69 368 :M f3_12 sf .37 .037( )J f4_12 sf .994(A)A f3_12 sf 1.266 .127( and )J f0_12 sf .904(into)A 129 368 :M f3_12 sf 2.426 .243( )J 135 368 :M f4_12 sf .799(B)A f3_12 sf 1.104 .11(, and )J f4_12 sf .799(A)A f3_12 sf .636 .064( is )J f0_12 sf .741(parent)A 232 368 :M f3_12 sf 2.11 .211( of )J 253 368 :M f4_12 sf 1.225(B)A f3_12 sf 1.559 .156( and )J f4_12 sf 1.225(B)A f3_12 sf 1.036 .104( is a )J f0_12 sf .847(child)A f3_12 sf 1.162 .116( of )J 370 368 :M f4_12 sf .634(A)A f3_12 sf 1.512 .151(. A sequence of edges)J 59 386 :M (<)S 66 386 :M f4_12 sf (E)S f3_9 sf 0 2 rm (1)S 0 -2 rm 78 386 :M f3_12 sf (,...,)S f4_12 sf (E)S f4_9 sf 0 2 rm (n)S 0 -2 rm 105 386 :M f3_12 sf .547 .055(> in )J f4_12 sf (G)S 138 386 :M f3_12 sf .299 .03( is an )J f0_12 sf .918 .092(undirected path )J f3_12 sf .641 .064(if and only if there exists a sequence of vertices)J 59 404 :M (<)S 66 404 :M f4_12 sf (V)S f3_10 sf 0 2 rm (1)S 0 -2 rm f3_12 sf (,...,)S f4_12 sf (V)S f4_10 sf 0 2 rm (n)S 0 -2 rm f3_10 sf 0 2 rm (+1)S 0 -2 rm 116 404 :M f3_12 sf .177 .018(> such that for 1 )J cF f1_12 sf .018A sf .177 .018( )J 209 404 :M f4_12 sf (i)S f3_12 sf .072 .007( )J cF f1_12 sf .007A sf .072 .007( )J f4_12 sf .079(n)A f3_12 sf ( )S f4_12 sf .097(E)A f4_8 sf 0 3 rm (i)S 0 -3 rm f3_12 sf .214 .021( has endpoints )J 316 404 :M f4_12 sf (V)S f4_10 sf 0 2 rm (i)S 0 -2 rm 326 404 :M f3_12 sf .194 .019( and )J 350 404 :M f4_12 sf (V)S f4_10 sf 0 2 rm (i)S 0 -2 rm 360 406 :M f3_10 sf (+1)S 371 404 :M f3_12 sf .155 .016(, and )J f4_12 sf .113(E)A f4_8 sf 0 3 rm (i)S 0 -3 rm f3_12 sf ( )S f1_12 sf S 416 404 :M f3_12 sf .233 .023( )J 420 404 :M f4_12 sf (E)S f4_8 sf 0 3 rm (i)S 0 -3 rm f3_8 sf 0 3 rm (+1)S 0 -3 rm 438 404 :M f3_12 sf .161 .016(. An empty)J 59 422 :M 2.188 .219(sequence of edges with an associated sequence of vertices <)J f4_12 sf .829(V)A f3_8 sf 0 3 rm .452(1)A 0 -3 rm f3_12 sf 1.072 .107(> is an )J 431 422 :M f0_12 sf 1.66 .166(empty path)J 59 440 :M f3_12 sf 1.415 .142(between )J 105 440 :M f4_12 sf 1.018(V)A f3_8 sf 0 3 rm .555(1)A 0 -3 rm f3_12 sf 1.349 .135( and )J 143 440 :M f4_12 sf 1.005(V)A f3_8 sf 0 3 rm .548(1)A 0 -3 rm f3_12 sf 1.491 .149(. A path )J 203 440 :M f4_12 sf (U)S 212 440 :M f3_12 sf .577 .058( is )J f0_12 sf .571(acyclic)A 265 440 :M f3_12 sf 1.64 .164( if no vertex appears more than once in the)J 59 458 :M 1.036 .104(corresponding sequence of vertices. We will assume that an undirected path is acyclic)J 59 476 :M 1.732 .173(unless specifically mentioned otherwise. A sequence of edges <)J f4_12 sf .589(E)A f3_9 sf 0 2 rm (1)S 0 -2 rm 402 476 :M f3_12 sf (,...,)S f4_12 sf (E)S f4_9 sf 0 2 rm (n)S 0 -2 rm 429 476 :M f3_12 sf 2.661 .266(> in )J 458 476 :M f4_12 sf (G)S 467 476 :M f3_12 sf 2.661 .266( is a)J 59 494 :M f0_12 sf .463 .046(directed path )J f4_12 sf (D)S 141 494 :M f0_12 sf 1.064 .106( from)J f3_12 sf .169 .017( )J 174 494 :M f4_12 sf .468(V)A f3_10 sf 0 2 rm .319(1)A 0 -2 rm f3_12 sf .425 .043( to )J 202 494 :M f4_12 sf .422(V)A f4_10 sf 0 2 rm .288(n)A 0 -2 rm f0_12 sf .173 .017( )J 218 494 :M f3_12 sf .584 .058(if and only if there exists a sequence of vertices <)J 465 494 :M f4_12 sf (V)S f3_10 sf 0 2 rm (1)S 0 -2 rm f3_12 sf (,...,)S 59 512 :M f4_12 sf (V)S f4_10 sf 0 2 rm (n)S 0 -2 rm f3_10 sf 0 2 rm (+1)S 0 -2 rm 82 512 :M f3_12 sf .049 .005(> such that for 1 )J cF f1_12 sf .005A sf .049 .005( )J f4_12 sf (i )S f3_12 sf cF f1_12 sf S sf ( )S 189 512 :M f4_12 sf (n)S f3_12 sf .038 .004(, there is a directed edge )J 316 512 :M f4_12 sf (V)S f4_10 sf 0 2 rm (i)S 0 -2 rm 326 512 :M f3_12 sf ( )S f1_12 sf S 341 512 :M f4_12 sf (V)S f4_10 sf 0 2 rm (i)S 0 -2 rm 351 514 :M f3_10 sf (+1)S 362 512 :M f3_12 sf .032 .003( on )J f4_12 sf (D)S 389 512 :M f3_12 sf .039 .004(. If there is an acyclic)J 59 530 :M .454 .045(directed path from )J 153 530 :M f4_12 sf .47(A)A f3_12 sf .428 .043( to )J 177 530 :M f4_12 sf .491(B)A f3_12 sf .446 .045( or )J f4_12 sf .491(B)A f3_12 sf .389 .039( = )J 223 530 :M f4_12 sf .289(A)A f3_12 sf .405 .04( then )J f4_12 sf .289(A)A f3_12 sf .302 .03( is an )J f0_12 sf .214(ancestor)A f3_12 sf .274 .027( of )J 356 530 :M f4_12 sf .239(B)A f3_12 sf .33 .033(, and )J f4_12 sf .239(B)A f3_12 sf .202 .02( is a )J f0_12 sf .187(descendant)A f3_12 sf .353 .035( of)J 59 548 :M f4_12 sf .855(A)A f3_12 sf .825 .083(. If )J 86 548 :M f0_12 sf .625(Z)A f3_12 sf .949 .095( is a set of variables, )J 204 548 :M f4_12 sf .547(A)A f3_12 sf .571 .057( is an )J f0_12 sf .404(ancestor)A f3_12 sf .519 .052( of )J 305 548 :M f0_12 sf .657(Z)A f3_12 sf .987 .099( if and only if it is an ancestor of a)J 59 566 :M .813 .081(member of )J f0_12 sf .353(Z)A f3_12 sf .665 .066(, and similarly for )J 218 566 :M f0_12 sf .155(descendant)A f3_12 sf .183 .018(. If )J f0_12 sf (X)S 303 566 :M f3_12 sf .746 .075( is a set of vertices in directed acyclic)J 59 584 :M 1.192 .119(graph )J f4_12 sf (G)S 100 584 :M f3_12 sf <28>S 104 584 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 123 584 :M f3_12 sf .409(,)A f0_12 sf 1.091(L)A f3_12 sf 1.31 .131(\), let )J 164 584 :M f0_12 sf (Ancestors)S 215 584 :M f3_12 sf <28>S 219 584 :M f4_12 sf (G)S 228 584 :M f3_12 sf (,)S f0_12 sf (X)S 240 584 :M f3_12 sf 1.715 .171(\) be the set of all ancestors of members of )J 469 584 :M f0_12 sf (X)S 478 584 :M f3_12 sf 2.044 .204( in)J 59 602 :M f4_12 sf (G)S 68 602 :M f3_12 sf <28>S 72 602 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 91 602 :M f3_12 sf .164(,)A f0_12 sf .439(L)A f3_12 sf .887 .089(\). \(If the context makes clear what graph is being referred to, we will simply)J 59 620 :M (write )S 87 620 :M f0_12 sf (Ancestors)S 138 620 :M f3_12 sf <28>S 142 620 :M f0_12 sf (X)S 151 620 :M f3_12 sf (\).\))S 77 644 :M .071 .007(In a )J f0_12 sf .275 .027(directed graph)J 174 644 :M f3_12 sf .182 .018(, all of the edges are directed edges. A directed graph is )J 447 644 :M f0_12 sf (acyclic)S 482 644 :M f3_12 sf .225 .023( if)J 59 662 :M .388 .039(and only if it contains no directed cyclic paths. A vertex )J f4_12 sf .177(V)A f3_12 sf .15 .015( is a )J f0_12 sf .133(collider)A 407 662 :M f3_12 sf .382 .038( on an undirected)J 59 680 :M .526 .053(path )J 84 680 :M f4_12 sf (U)S 93 680 :M f3_12 sf .495 .05( if and only if )J f4_12 sf (U)S 173 680 :M f3_12 sf .494 .049( contains a pair of distinct edges adjacent on the path and into )J 482 680 :M f4_12 sf (V)S f3_12 sf (.)S endp %%Page: 31 31 %%BeginPageSetup initializepage (peter; page: 31 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (31)S gR gS 0 0 552 730 rC 59 56 :M f3_12 sf .111 .011(The )J 81 56 :M f0_12 sf (orientation)S 138 56 :M f3_12 sf .093 .009( of an acyclic undirected path between )J 327 56 :M f4_12 sf .051(A)A f3_12 sf .065 .007( and )J f4_12 sf .051(B)A f3_12 sf .101 .01( is the set consisting of the)J 59 74 :M f4_12 sf (A)S f3_12 sf ( end of the edge on )S 161 74 :M f4_12 sf (U)S 170 74 :M f3_12 sf ( that contains )S 237 74 :M f4_12 sf (A)S f3_12 sf (, and the )S 288 74 :M f4_12 sf (B)S f3_12 sf ( end of the edge on )S 390 74 :M f4_12 sf (U)S 399 74 :M f3_12 sf ( that contains )S 466 74 :M f4_12 sf (B)S f3_12 sf (.)S 77 98 :M 1.353 .135(If )J f4_12 sf (U)S 100 98 :M f3_12 sf 1.955 .195( is an undirected path that is a sequence of edges <)J f4_12 sf .955(E)A f3_8 sf 0 3 rm .521(1)A 0 -3 rm f3_12 sf .391(,...,)A f4_12 sf .955(E)A f4_8 sf 0 3 rm .521(n)A 0 -3 rm f3_12 sf 1.659 .166(>, and )J 454 98 :M f4_12 sf <55D5>S 467 98 :M f3_12 sf 2.621 .262( is a)J 59 116 :M .253 .025(subsequence of the edges in )J 198 116 :M f4_12 sf (U)S 207 116 :M f3_12 sf .264 .026( that is also an undirected path, then )J 386 116 :M f4_12 sf <55D5>S 399 116 :M f3_12 sf .151 .015( is a )J f0_12 sf .144(subpath)A f3_12 sf .17 .017( of )J 480 116 :M f4_12 sf (U)S 489 116 :M f3_12 sf (.)S 59 134 :M .272 .027(Note that if )J 118 134 :M f4_12 sf (U)S 127 134 :M f3_12 sf .259 .026( is a cyclic undirected path, and )J 285 134 :M f4_12 sf (U)S 294 134 :M f3_12 sf .32 .032( contains edges )J f4_12 sf .142(E)A f4_8 sf 0 3 rm (i)S 0 -3 rm f3_12 sf .112 .011( = )J 394 134 :M f4_12 sf .126(X)A f3_12 sf .047 .005( )J f1_12 sf S 416 134 :M f3_12 sf .107 .011( )J f4_12 sf (Y)S 426 134 :M f3_12 sf .311 .031( and )J f4_12 sf .245(E)A f4_8 sf 0 3 rm .074(j)A 0 -3 rm f3_12 sf .194 .019( = )J 473 134 :M f4_12 sf (Y)S 480 134 :M f1_12 sf S 59 152 :M f4_12 sf (Z)S 66 152 :M f3_12 sf .239 .024( that are not adjacent on )J f4_12 sf (U)S 195 152 :M f3_12 sf .263 .026(, then a subpath )J 275 152 :M f4_12 sf <55D5>S 288 152 :M f3_12 sf .243 .024( may leave out all of the edges between )J f4_12 sf .12(E)A f4_8 sf 0 3 rm (i)S 0 -3 rm 59 170 :M f3_12 sf .186 .019(and )J f4_12 sf .094(E)A f4_8 sf 0 3 rm (j)S 0 -3 rm f3_12 sf .09 .009(. If )J 106 170 :M f4_12 sf (U)S 115 170 :M f3_12 sf .13 .013( is acyclic, then there for any two vertices on )J f4_12 sf (U)S 344 170 :M f3_12 sf .132 .013( there is a unique subpath of )J f4_12 sf (U)S 59 188 :M f3_12 sf 1.115 .112(between the two vertices. If )J f4_12 sf (U)S 213 188 :M f3_12 sf 1.215 .122( is an acyclic undirected path between )J 412 188 :M f4_12 sf .736(X)A f3_12 sf .937 .094( and )J f4_12 sf (Y)S 453 188 :M f3_12 sf 1.085 .109(, and )J f4_12 sf (U)S 59 206 :M f3_12 sf .245 .025(contains distinct vertices )J 182 206 :M f4_12 sf .198(A)A f3_12 sf .253 .025( and )J f4_12 sf .198(B)A f3_12 sf .309 .031(, then )J 251 206 :M f4_12 sf (U)S 260 206 :M f3_12 sf <28>S 264 206 :M f4_12 sf .141(A)A f3_12 sf .058(,)A f4_12 sf .141(B)A f3_12 sf .28 .028(\) is the unique subpath of )J 409 206 :M f4_12 sf (U)S 418 206 :M f3_12 sf .327 .033( between )J 465 206 :M f4_12 sf .121(A)A f3_12 sf .259 .026( and)J 59 224 :M f4_12 sf (B)S f3_12 sf (.)S 77 248 :M .323 .032(For three disjoint sets of variables )J f0_12 sf (A)S 255 248 :M f3_12 sf .499 .05(, )J 262 248 :M f0_12 sf .204(B)A f3_12 sf .259 .026(, and )J f0_12 sf (C)S 306 248 :M f3_12 sf .187 .019(, )J f0_12 sf (A)S 321 248 :M f3_12 sf .086 .009( is )J f0_12 sf .089(d-separated)A 397 248 :M f3_12 sf .354 .035( from )J f0_12 sf .251(B)A f3_12 sf .394 .039( given )J 469 248 :M f0_12 sf (C)S 478 248 :M f3_12 sf .457 .046( in)J 59 266 :M .469 .047(DAG )J 89 266 :M f4_12 sf (G)S 98 266 :M f3_12 sf .405 .04(, if and only if there is an undirected path from some member of )J f0_12 sf (A)S 426 266 :M f3_12 sf .446 .045( to a member)J 59 284 :M .658 .066(of )J f0_12 sf .527(B)A f3_12 sf .947 .095( such that every collider on that path is either in )J 328 284 :M f0_12 sf (C)S 337 284 :M f3_12 sf 1.002 .1( or has a descendant in )J 458 284 :M f0_12 sf (C)S 467 284 :M f3_12 sf .99 .099(, and)J 59 302 :M .035 .003(every non-collider on the path is not in )J f0_12 sf (C)S 257 302 :M f3_12 sf .036 .004(. For three disjoint sets of variables )J 430 302 :M f0_12 sf (A)S 439 302 :M f3_12 sf (, )S f0_12 sf (B)S f3_12 sf .034 .003(, and )J f0_12 sf (C)S 488 302 :M f3_12 sf (,)S 59 320 :M f0_12 sf (A)S 68 320 :M f3_12 sf 1.236 .124( is )J 85 320 :M f0_12 sf (d-connected)S 147 320 :M f3_12 sf .648 .065( to )J f0_12 sf .812(B)A f3_12 sf 1.275 .127( given )J 209 320 :M f0_12 sf (C)S 218 320 :M f3_12 sf .938 .094( in DAG )J f4_12 sf (G)S 275 320 :M f3_12 sf .955 .096( if and only if )J f0_12 sf (A)S 359 320 :M f3_12 sf .887 .089( is not d-separated from )J f0_12 sf (B)S 59 338 :M f3_12 sf 1.083 .108(given )J 91 338 :M f0_12 sf (C)S 100 338 :M f3_12 sf 1.049 .105(. Geiger, Pearl, and Verma have shown that )J f4_12 sf (G)S 338 338 :M f3_12 sf .91 .091( entails )J f0_12 sf (A)S 387 338 :M f3_12 sf 1.121 .112( is independent of )J 483 338 :M f0_12 sf (B)S 59 356 :M f3_12 sf (given )S 89 356 :M f0_12 sf (C)S 98 356 :M f3_12 sf ( if and only if )S f0_12 sf (A)S 175 356 :M f3_12 sf ( is d-separated from )S 274 356 :M f0_12 sf (B)S f3_12 sf ( given )S 315 356 :M f0_12 sf (C)S 324 356 :M f3_12 sf ( in)S f4_12 sf ( G)S 348 356 :M f3_12 sf (. See Pearl\(1988\).)S 77 398 :M f0_12 sf .089 .009(Lemma 1: )J 133 398 :M f3_12 sf .075 .008(In a directed acyclic graph )J f4_12 sf (G)S 272 398 :M f3_12 sf .073 .007( over )J f0_12 sf (V)S 308 398 :M f3_12 sf .104 .01(, if )J 325 398 :M f4_12 sf (X)S f3_12 sf .063 .006( and )J f4_12 sf (Y)S 362 398 :M f3_12 sf .077 .008( are not in )J f0_12 sf .062(Z)A f3_12 sf .098 .01(, and there is a)J 59 416 :M .378 .038(sequence )J f4_12 sf .162 .016(H )J 119 416 :M f3_12 sf .276 .028( of distinct vertices in )J f0_12 sf (V)S 237 416 :M f3_12 sf .344 .034( from )J f4_12 sf .224(X)A f3_12 sf .195 .02( to )J f4_12 sf (Y)S 297 416 :M f3_12 sf .336 .034(, and there is a set )J 389 416 :M f4_12 sf (T)S 396 416 :M f3_12 sf .274 .027( of undirected paths)J 59 434 :M (such that)S 77 458 :M .055 .005(\(i\) for each pair of adjacent vertices )J 252 458 :M f4_12 sf (V)S f3_12 sf .058 .006( and )J f4_12 sf .067 .007(W )J f3_12 sf .059 .006(in )J f4_12 sf .066 .007(H )J 319 458 :M f3_12 sf .056 .006( there is a unique undirected path in)J 59 470 :M f4_12 sf (T)S 66 470 :M f3_12 sf ( that d-connects )S 145 470 :M f4_12 sf (V)S f3_12 sf ( and )S f4_12 sf (W )S 188 470 :M f3_12 sf (given )S 218 470 :M f0_12 sf (Z)S f3_12 sf (\\{)S f4_12 sf (V)S f3_12 sf (,)S f4_12 sf (W)S 255 470 :M f3_12 sf (}, and)S 77 488 :M .148 .015(\(ii\) if a vertex )J f4_12 sf (Q)S 156 488 :M f3_12 sf .126 .013( in )J f4_12 sf (H)S 180 488 :M f3_12 sf .104 .01( is in )J f0_12 sf .117(Z)A f3_12 sf .184 .018(, then the paths in )J f4_12 sf (T)S 311 488 :M f3_12 sf .141 .014( that contain )J f4_12 sf (Q)S 383 488 :M f3_12 sf .152 .015( as an endpoint collide)J 59 500 :M (at )S 71 500 :M f4_12 sf (Q)S 80 500 :M f3_12 sf (, and)S 77 518 :M .075 .007(\(iii\) if for three vertices )J 194 518 :M f4_12 sf (V)S f3_12 sf (, )S f4_12 sf (W)S 217 518 :M f3_12 sf (, )S f4_12 sf (Q)S 232 518 :M f3_12 sf .079 .008( occurring in that order in )J 359 518 :M f4_12 sf (H)S 368 518 :M f3_12 sf .071 .007( the d-connecting paths in)J 59 530 :M f4_12 sf (T)S 66 530 :M f3_12 sf ( between )S 112 530 :M f4_12 sf (V)S f3_12 sf ( and )S f4_12 sf (W)S 152 530 :M f3_12 sf (, and )S f4_12 sf (W )S 191 530 :M f3_12 sf (and )S f4_12 sf (Q)S 220 530 :M f3_12 sf ( collide at )S f4_12 sf (W )S 283 530 :M f3_12 sf (then )S 307 530 :M f4_12 sf (W )S 320 530 :M f3_12 sf (has a descendant in )S 416 530 :M f0_12 sf (Z)S f3_12 sf (,)S 77 548 :M .461 .046(then there is a path )J f4_12 sf (U)S 183 548 :M f3_12 sf .621 .062( in )J 200 548 :M f4_12 sf (G)S 209 548 :M f3_12 sf .516 .052( that d-connects )J f4_12 sf .225(X)A f3_12 sf .287 .029( and )J f4_12 sf (Y)S 328 548 :M f3_12 sf .55 .055( given )J 363 548 :M f0_12 sf .274(Z)A f3_12 sf .461 .046(. In addition, if all of the)J 59 566 :M .584 .058(edges in all of the paths in )J 194 566 :M f4_12 sf (T)S 201 566 :M f3_12 sf .613 .061( that contain )J f4_12 sf .317(X)A f3_12 sf .519 .052( are into \(out of\) )J 358 566 :M f4_12 sf .353(X)A f3_12 sf .513 .051( then )J 394 566 :M f4_12 sf (U)S 403 566 :M f3_12 sf .529 .053( is into \(out of\) )J f4_12 sf .352(X)A f3_12 sf (,)S 59 584 :M (and similarly for )S 142 584 :M f4_12 sf (Y)S 149 584 :M f3_12 sf (.)S 77 614 :M (Proof.)S 107 614 :M f0_12 sf ( )S f3_12 sf .04 .004(Let )J f4_12 sf <55D5>S 142 614 :M f3_12 sf .036 .004( be the concatenation of all of the paths in )J 347 614 :M f4_12 sf (T)S 354 614 :M f3_12 sf .035 .004( in the order of the sequence)J 59 632 :M f4_12 sf (H)S 68 632 :M f3_12 sf .353 .035(. )J 75 632 :M f4_12 sf .16 .016<55D520>J f3_12 sf .25 .025(may not be an acyclic undirected path, because it may contain some vertices more)J 59 650 :M .537 .054(than once. Let )J f4_12 sf (U)S 142 650 :M f3_12 sf .598 .06( be the result of removing all of the cycles from )J f4_12 sf (U)S 392 650 :M f3_12 sf .664 .066('. If each edge in )J 479 650 :M f4_12 sf (U)S 488 650 :M f3_12 sf (')S 59 668 :M .586 .059(that contains )J f4_12 sf .228(X)A f3_12 sf .343 .034( is into \(out of\) )J f4_12 sf .228(X)A f3_12 sf .355 .036(, then )J 247 668 :M f4_12 sf (U)S 256 668 :M f3_12 sf .473 .047( is into \(out of\) )J 335 668 :M f4_12 sf .215(X)A f3_12 sf .442 .044(, because each edge in )J f4_12 sf .285 .029(U )J f3_12 sf .468 .047(is an)J 59 686 :M .286 .029(edge in )J f4_12 sf .267<55D5>A 111 686 :M f3_12 sf .364 .036(. Similarly, if each edge in )J 245 686 :M f4_12 sf (U)S 254 686 :M f3_12 sf .317 .032(' that contains )J f4_12 sf (Y)S 331 686 :M f3_12 sf .398 .04( is into \(out of\) )J 409 686 :M f4_12 sf (Y)S 416 686 :M f3_12 sf .41 .041(, then )J 447 686 :M f4_12 sf (U)S 456 686 :M f3_12 sf .395 .04( is into)J endp %%Page: 32 32 %%BeginPageSetup initializepage (peter; page: 32 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (32)S gR gS 0 0 552 730 rC 59 56 :M f3_12 sf .434 .043(\(out of\) )J f4_12 sf (Y)S 106 56 :M f3_12 sf .512 .051(, because each edge in )J 220 56 :M f4_12 sf (U)S 229 56 :M f3_12 sf .547 .055( is an edge in)J f4_12 sf .609 .061<2055D5>J 312 56 :M f3_12 sf .526 .053(. We will prove that )J 415 56 :M f4_12 sf (U)S 424 56 :M f3_12 sf .409 .041( d-connects )J f4_12 sf (X)S 59 74 :M f3_12 sf (and )S f4_12 sf (Y)S 86 74 :M f3_12 sf ( given )S 119 74 :M f0_12 sf (Z)S f3_12 sf (.)S 77 98 :M .19 .019(We will call an edge in )J 193 98 :M f4_12 sf (U)S 202 98 :M f3_12 sf .173 .017( containing a given vertex )J 331 98 :M f4_12 sf .06(V)A f3_12 sf .059 .006( an )J f0_12 sf .267 .027(endpoint edge)J 428 98 :M f3_12 sf .118 .012( if )J f4_12 sf .156(V)A f3_12 sf .224 .022( is in the)J 59 116 :M 1.516 .152(sequence )J 109 116 :M f4_12 sf (H)S 118 116 :M f3_12 sf 1.819 .182(, and the edge containing )J 255 116 :M f4_12 sf 1(V)A f3_12 sf 1.742 .174( occurs on the path in )J f4_12 sf (T)S 391 116 :M f3_12 sf 1.949 .195( between )J 443 116 :M f4_12 sf .953(V)A f3_12 sf 1.668 .167( and its)J 59 134 :M (predecessor or successor in )S 193 134 :M f4_12 sf (H)S 202 134 :M f3_12 sf (; otherwise the edge is an internal edge.)S 77 158 :M .25 .025(First we prove that every member )J f4_12 sf .103(R)A f3_12 sf .094 .009( of )J f0_12 sf .113(Z)A f3_12 sf .135 .014( that is on )J f4_12 sf (U)S 335 158 :M f3_12 sf .224 .022( is a collider on )J 414 158 :M f4_12 sf (U)S 423 158 :M f3_12 sf .218 .022(. If there is an)J 59 176 :M .49 .049(endpoint edge containing )J 186 176 :M f4_12 sf .381(R)A f3_12 sf .389 .039( on )J f4_12 sf (U)S 221 176 :M f3_12 sf .668 .067( then it is into )J 295 176 :M f4_12 sf .24(R)A f3_12 sf .539 .054( because by assumption the paths in )J f4_12 sf (T)S 59 194 :M f3_12 sf .482 .048(containing )J 114 194 :M f4_12 sf .389(R)A f3_12 sf .635 .063( collide at )J f4_12 sf .389(R)A f3_12 sf .578 .058(. If an edge on )J f4_12 sf (U)S 266 194 :M f3_12 sf .649 .065( is an internal edge with endpoint )J 436 194 :M f4_12 sf .411(R)A f3_12 sf .621 .062( then it is)J 59 212 :M .363 .036(into )J 82 212 :M f4_12 sf .168(R)A f3_12 sf .349 .035( because it is an edge on a path that d-connects two variables )J f4_12 sf .168(A)A f3_12 sf .213 .021( and )J f4_12 sf .168(B)A f3_12 sf .325 .032( not equal to)J 59 230 :M f4_12 sf .196(R)A f3_12 sf .323 .032( given )J f0_12 sf .214(Z)A f3_12 sf .121(\\{)A f4_12 sf .196(A)A f3_12 sf .08(,)A f4_12 sf .196(B)A f3_12 sf .329 .033(}, and )J 168 230 :M f4_12 sf .315(R)A f3_12 sf .305 .031( is in )J f0_12 sf .344(Z)A f3_12 sf .55 .055(. All of the edges on paths in )J 358 230 :M f4_12 sf (T)S 365 230 :M f3_12 sf .372 .037( are into )J f4_12 sf .258(R)A f3_12 sf .575 .057(, and hence the)J 59 248 :M (subset of those edges that occur on )S f4_12 sf (U)S 238 248 :M f3_12 sf ( are into )S f4_12 sf (R)S f3_12 sf (.)S 77 272 :M .929 .093(Next we show that every collider )J 248 272 :M f4_12 sf .825(R)A f3_12 sf .881 .088( on )J 276 272 :M f4_12 sf (U)S 285 272 :M f3_12 sf 1.04 .104( has a descendant in )J f0_12 sf .597(Z)A f3_12 sf .373 .037(. )J f4_12 sf .547(R)A f3_12 sf .959 .096( is not equal to)J 59 290 :M .268 .027(either of the endpoints )J f4_12 sf .114(X)A f3_12 sf .108 .011( or )J 195 290 :M f4_12 sf (Y)S 202 290 :M f3_12 sf .224 .022(, because the endpoints of a path are not colliders along the)J 59 308 :M .216 .022(path. If )J 98 308 :M f4_12 sf .126(R)A f3_12 sf .211 .021( is a collider on any of the paths in )J 277 308 :M f4_12 sf (T)S 284 308 :M f3_12 sf .241 .024( then )J 312 308 :M f4_12 sf .101(R)A f3_12 sf .191 .019( has a descendant in )J f0_12 sf .11(Z)A f3_12 sf .197 .02( because it is)J 59 326 :M .246 .025(an edge on a path that d-connects two variables )J 293 326 :M f4_12 sf .179(A)A f3_12 sf .238 .024( and )J 323 326 :M f4_12 sf .157(B)A f3_12 sf .253 .025( not equal to )J 395 326 :M f4_12 sf .137(R)A f3_12 sf .235 .024( given )J 436 326 :M f0_12 sf .055(Z)A f3_12 sf .031(\\{)A f4_12 sf .05(A)A f3_12 sf (,)S f4_12 sf .05(B)A f3_12 sf .097 .01(}. If)J 59 344 :M f4_12 sf .855(R)A f3_12 sf 1.731 .173( is a collider on two endpoint edges then it has a descendant in )J 405 344 :M f0_12 sf .515(Z)A f3_12 sf 1.368 .137( by hypothesis.)J 59 362 :M .299 .03(Suppose then that )J f4_12 sf .119(R)A f3_12 sf .197 .02( is not a collider on the path in )J 307 362 :M f4_12 sf (T)S 314 362 :M f3_12 sf .237 .024( between )J 361 362 :M f4_12 sf .159(A)A f3_12 sf .21 .021( and )J 392 362 :M f4_12 sf .1(B)A f3_12 sf .208 .021(, and not a collider)J 59 380 :M .744 .074(on the path in )J f4_12 sf (T)S 139 380 :M f3_12 sf .921 .092( between )J f4_12 sf .465(C)A f3_12 sf .564 .056( and )J 221 380 :M f4_12 sf (D)S 230 380 :M f3_12 sf .742 .074(, but after cycles have been removed from )J 444 380 :M f4_12 sf (U)S 453 380 :M f3_12 sf .607 .061(', )J f4_12 sf .709(R)A f3_12 sf .779 .078( is a)J 59 398 :M .86 .086(collider on )J 117 398 :M f4_12 sf (U)S 126 398 :M f3_12 sf .988 .099(. In that case )J 195 398 :M f4_12 sf (U)S 204 398 :M f3_12 sf .814 .081(' contains an undirected cycle containing )J f4_12 sf .307(R)A f3_12 sf .708 .071(. Because )J 469 398 :M f4_12 sf (G)S 478 398 :M f3_12 sf 1.111 .111( is)J 59 416 :M 1.613 .161(acyclic, the undirected cycle contains a collider. Hence )J f4_12 sf .606(R)A f3_12 sf 1.192 .119( has a descendant that is a)J 59 434 :M .071 .007(collider on )J 114 434 :M f4_12 sf (U)S 123 434 :M f3_12 sf .075 .008('. Each collider on )J 214 434 :M f4_12 sf (U)S 223 434 :M f3_12 sf .073 .007(' has a descendant in )J f0_12 sf (Z)S f3_12 sf .077 .008(. Hence )J 372 434 :M f4_12 sf (R)S f3_12 sf .074 .007( has a descendant in )J 479 434 :M f0_12 sf (Z)S f3_12 sf (.)S 59 443 9 9 rC gS 1.286 1 scale 45.889 452 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 77 476 :M f0_12 sf 2.143 .214(Lemma 2: )J 138 476 :M f3_12 sf 2.5 .25(If )J 152 476 :M f4_12 sf (G)S 161 476 :M f3_12 sf 1.949 .195( is a directed acyclic graph, )J f4_12 sf .958(R)A f3_12 sf 1.982 .198( is d-connected to )J 420 476 :M f4_12 sf (Y)S 427 476 :M f3_12 sf 2.308 .231( given )J 466 476 :M f0_12 sf 1.044(Z)A f3_12 sf 1.63 .163( by)J 59 494 :M 1.058 .106(undirected path )J f4_12 sf (U)S 149 494 :M f3_12 sf 1.635 .164(, and )J f4_12 sf 1.908 .191(W )J 195 494 :M f3_12 sf 1.224 .122(and )J f4_12 sf .618(X)A f3_12 sf 1.268 .127( are distinct vertices on )J f4_12 sf (U)S 357 494 :M f3_12 sf 1.716 .172( not in )J 398 494 :M f0_12 sf .9(Z)A f3_12 sf 1.282 .128(, then )J 440 494 :M f4_12 sf (U)S 449 494 :M f3_12 sf <28>S 453 494 :M f4_12 sf (W)S 463 494 :M f3_12 sf .232(,)A f4_12 sf .568(X)A f3_12 sf 1.013 .101(\) d-)J 59 512 :M (connects )S 104 512 :M f4_12 sf (W )S 117 512 :M f3_12 sf (and )S f4_12 sf (X)S f3_12 sf ( given )S 177 512 :M f0_12 sf (Z)S f3_12 sf (.)S 77 536 :M 1.232 .123(Proof. Suppose )J f4_12 sf (G)S 167 536 :M f3_12 sf 1.56 .156( is a directed acyclic graph, )J f4_12 sf .766(R)A f3_12 sf 1.586 .159( is d-connected to )J 420 536 :M f4_12 sf (Y)S 427 536 :M f3_12 sf 1.846 .185( given )J 465 536 :M f0_12 sf .835(Z)A f3_12 sf 1.304 .13( by)J 59 554 :M .195 .019(undirected path )J f4_12 sf (U)S 146 554 :M f3_12 sf .264 .026(, and )J f4_12 sf .283 .028(W )J f3_12 sf .408 .041(and )J 207 554 :M f4_12 sf .11(X)A f3_12 sf .225 .022( are distinct vertices on )J f4_12 sf (U)S 339 554 :M f3_12 sf .316 .032( not in )J 374 554 :M f0_12 sf .082(Z)A f3_12 sf .213 .021(. Each non-collider on)J 59 572 :M f4_12 sf (U)S 68 572 :M f3_12 sf <28>S 72 572 :M f4_12 sf (W)S 82 572 :M f3_12 sf .277(,)A f4_12 sf .676(X)A f3_12 sf 1.492 .149(\) except for the endpoints is a non-collider on )J 336 572 :M f4_12 sf (U)S 345 572 :M f3_12 sf 1.747 .175(, and hence not in )J 445 572 :M f0_12 sf .446(Z)A f3_12 sf 1.207 .121(. Every)J 59 590 :M .515 .051(collider on )J 116 590 :M f4_12 sf (U)S 125 590 :M f3_12 sf <28>S 129 590 :M f4_12 sf (W)S 139 590 :M f3_12 sf .106(,)A f4_12 sf .258(X)A f3_12 sf .511 .051(\) has a descendant in )J 256 590 :M f0_12 sf .243(Z)A f3_12 sf .5 .05( because each collider on )J 392 590 :M f4_12 sf (U)S 401 590 :M f3_12 sf <28>S 405 590 :M f4_12 sf (W)S 415 590 :M f3_12 sf .098(,)A f4_12 sf .239(X)A f3_12 sf .467 .047(\) is a collider)J 59 608 :M 1.302 .13(on )J 76 608 :M f4_12 sf (U)S 85 608 :M f3_12 sf 1.139 .114(, which d-connects )J f4_12 sf .438(R)A f3_12 sf .558 .056( and )J f4_12 sf (Y)S 222 608 :M f3_12 sf 1.202 .12( given )J 258 608 :M f0_12 sf .593(Z)A f3_12 sf .992 .099(. It follows that )J 349 608 :M f4_12 sf (U)S 358 608 :M f3_12 sf <28>S 362 608 :M f4_12 sf (W)S 372 608 :M f3_12 sf .178(,)A f4_12 sf .434(X)A f3_12 sf 1.111 .111(\) d-connects )J f4_12 sf .641 .064(W )J f3_12 sf .925 .093(and )J 483 608 :M f4_12 sf (X)S 59 626 :M f3_12 sf (given )S 89 626 :M f0_12 sf (Z)S f3_12 sf ( = )S 110 626 :M f0_12 sf (Z)S f3_12 sf (\\{)S f4_12 sf (W)S 137 626 :M f3_12 sf (,)S f4_12 sf (X)S f3_12 sf (}. )S 159 617 9 9 rC gS 1.286 1 scale 123.668 626 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 77 650 :M f0_12 sf 3.514 .351(Lemma 3)J f3_12 sf .857 .086(: If )J f4_12 sf (G)S 160 650 :M f3_12 sf 2.064 .206( is a directed acyclic graph, )J 313 650 :M f4_12 sf .881(R)A f3_12 sf 1.791 .179( is d-connected to )J f4_12 sf (Y)S 427 650 :M f3_12 sf 2.356 .236( given )J 466 650 :M f0_12 sf 1.066(Z)A f3_12 sf 1.664 .166( by)J 59 668 :M 1.31 .131(undirected path )J 141 668 :M f4_12 sf (U)S 150 668 :M f3_12 sf 1.472 .147(, there is a directed path )J f4_12 sf (D)S 290 668 :M f3_12 sf 1.834 .183( from )J 324 668 :M f4_12 sf 1.483(R)A f3_12 sf 1.348 .135( to )J 351 668 :M f4_12 sf .645(X)A f3_12 sf 1.442 .144( that does not contain any)J endp %%Page: 33 33 %%BeginPageSetup initializepage (peter; page: 33 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (33)S gR gS 0 0 552 730 rC 59 56 :M f3_12 sf .566 .057(member of )J f0_12 sf .246(Z)A f3_12 sf .323 .032(, and )J 151 56 :M f4_12 sf .369(X)A f3_12 sf .477 .048( is not on )J 207 56 :M f4_12 sf (U)S 216 56 :M f3_12 sf .53 .053(, then )J 248 56 :M f4_12 sf .198(X)A f3_12 sf .403 .04( is d-connected to )J f4_12 sf (Y)S 352 56 :M f3_12 sf .53 .053( given )J 387 56 :M f0_12 sf .38(Z)A f3_12 sf .475 .048( by a path )J 448 56 :M f4_12 sf (U)S 457 56 :M f3_12 sf .492 .049(' that is)J 59 74 :M (into )S 81 74 :M f4_12 sf (X)S f3_12 sf (. If )S 105 74 :M f4_12 sf (D)S 114 74 :M f3_12 sf ( does not contain )S 199 74 :M f4_12 sf (Y)S 206 74 :M f3_12 sf (, then )S 236 74 :M f4_12 sf (U)S 245 74 :M f3_12 sf (' is into )S 283 74 :M f4_12 sf (Y)S 290 74 :M f3_12 sf ( if and only if )S f4_12 sf (U)S 367 74 :M f3_12 sf ( is.)S 77 98 :M .518 .052(Proof. Let )J 131 98 :M f4_12 sf (D)S 140 98 :M f3_12 sf .613 .061( be a directed path from )J f4_12 sf .326(R)A f3_12 sf .296 .03( to )J 285 98 :M f4_12 sf .229(X)A f3_12 sf .483 .048( that does not contain any member of )J f0_12 sf .25(Z)A f3_12 sf (,)S 59 116 :M .195 .019(and )J f4_12 sf (U)S 88 116 :M f3_12 sf .245 .025( an undirected path that d-connects )J 262 116 :M f4_12 sf .153(R)A f3_12 sf .194 .019( and )J f4_12 sf (Y)S 300 116 :M f3_12 sf .226 .023( given )J f0_12 sf .15(Z)A f3_12 sf .276 .028( and does not contain )J 448 116 :M f4_12 sf .176(X)A f3_12 sf .24 .024(. Let )J 481 116 :M f4_12 sf (Q)S 59 134 :M f3_12 sf -.003(be the point of intersection of )A 204 134 :M f4_12 sf (D)S 213 134 :M f3_12 sf ( and )S f4_12 sf (U)S 245 134 :M f3_12 sf -.004( that is closest to )A 328 134 :M f4_12 sf (Y)S 335 134 :M f3_12 sf ( on )S 353 134 :M f4_12 sf (U)S 362 134 :M f3_12 sf (. )S 368 134 :M f4_12 sf (Q)S 377 134 :M f3_12 sf -.006( is not in )A 422 134 :M f0_12 sf (Z)S f3_12 sf ( because it is)S 59 152 :M (on )S f4_12 sf (D)S 83 152 :M f3_12 sf (.)S 77 176 :M .215 .021(If )J f4_12 sf (D)S 97 176 :M f3_12 sf .349 .035( does contain )J 165 176 :M f4_12 sf (Y)S 172 176 :M f3_12 sf .385 .038(, then )J 203 176 :M f4_12 sf (Y)S 210 176 :M f3_12 sf .454 .045( = )J 224 176 :M f4_12 sf (Q)S 233 176 :M f3_12 sf .289 .029(, and )J f4_12 sf (D)S 269 176 :M f3_12 sf <28>S 273 176 :M f4_12 sf (Y)S 280 176 :M f3_12 sf .071(,)A f4_12 sf .174(X)A f3_12 sf .254 .025(\) is a path into )J f4_12 sf .174(X)A f3_12 sf .408 .041( that d-connects )J 452 176 :M f4_12 sf .196(X)A f3_12 sf .25 .025( and )J f4_12 sf (Y)S 59 194 :M f3_12 sf (given )S 89 194 :M f0_12 sf (Z)S f3_12 sf ( because it contains no colliders and no members of )S f0_12 sf (Z)S f3_12 sf (.)S 77 218 :M (If )S f4_12 sf (D)S 97 218 :M f3_12 sf .02 .002( does not contain )J 182 218 :M f4_12 sf (Y)S 189 218 :M f3_12 sf .023 .002( then )J 217 218 :M f4_12 sf (Q)S 226 218 :M f3_12 sf ( )S f1_12 sf S 236 218 :M f3_12 sf ( )S f4_12 sf (Y)S 246 218 :M f3_12 sf (. )S f4_12 sf (X)S f3_12 sf ( )S f1_12 sf S 269 218 :M f3_12 sf ( )S f4_12 sf (Q)S 281 218 :M f3_12 sf .024 .002( because )J f4_12 sf (X)S f3_12 sf .015 .001( is not on )J f4_12 sf (U)S 388 218 :M f3_12 sf ( and )S f4_12 sf (Q)S 420 218 :M f3_12 sf .02 .002( is. By Lemma)J 59 236 :M (2)S f4_12 sf (U)S 74 236 :M f3_12 sf <28>S 78 236 :M f4_12 sf (Q)S 87 236 :M f3_12 sf (,)S f4_12 sf (Y)S 97 236 :M f3_12 sf .023 .002(\) d-connects )J f4_12 sf (Q)S 168 236 :M f3_12 sf .037 .004( and )J 192 236 :M f4_12 sf (Y)S 199 236 :M f3_12 sf .034 .003( given )J 232 236 :M f0_12 sf (Z)S f3_12 sf (\\{)S f4_12 sf (Q)S 258 236 :M f3_12 sf (,)S f4_12 sf (Y)S 268 236 :M f3_12 sf .038 .004(} = )J 287 236 :M f0_12 sf (Z)S f3_12 sf (. )S f4_12 sf (A)S f3_12 sf (lso, )S f4_12 sf (D)S 337 236 :M f3_12 sf <28>S 341 236 :M f4_12 sf (Q)S 350 236 :M f3_12 sf (,)S f4_12 sf (X)S f3_12 sf .02 .002(\) d-connects )J f4_12 sf (Q)S 431 236 :M f3_12 sf .02 .002( and )J f4_12 sf (X)S f3_12 sf .043 .004( given)J 59 254 :M f0_12 sf (Z)S f3_12 sf (\\{)S f4_12 sf (Q)S 85 254 :M f3_12 sf .085(,)A f4_12 sf .208(X)A f3_12 sf .228 .023(} = )J 115 254 :M f0_12 sf .144(Z)A f3_12 sf .09 .009(. )J f4_12 sf (D)S 138 254 :M f3_12 sf <28>S 142 254 :M f4_12 sf (Q)S 151 254 :M f3_12 sf .082(,)A f4_12 sf .201(X)A f3_12 sf .265 .027(\) is out of )J 212 254 :M f4_12 sf (Q)S 221 254 :M f3_12 sf .237 .024(, and )J f4_12 sf (Q)S 257 254 :M f3_12 sf .284 .028( is not in )J f0_12 sf .259(Z)A f3_12 sf .311 .031(. By )J 335 254 :M .255 .026(Lemma 1)J 382 254 :M .251 .025( there is a path )J f4_12 sf .277 .028<55D520>J f3_12 sf .138(that)A 59 272 :M 1.29 .129(d-connects )J 117 272 :M f4_12 sf 1.014(X)A f3_12 sf 1.29 .129( and )J f4_12 sf (Y)S 159 272 :M f3_12 sf 1.985 .199( given )J 197 272 :M f0_12 sf 1.278(Z)A f3_12 sf 1.707 .171( that is into )J 273 272 :M f4_12 sf 1.221(X)A f3_12 sf 1.132 .113(. If )J f4_12 sf (Y)S 309 272 :M f3_12 sf 1.711 .171( is not on )J f4_12 sf (D)S 375 272 :M f3_12 sf 1.792 .179(, then all of the edges)J 59 290 :M .095 .01(containing )J 113 290 :M f4_12 sf (Y)S 120 290 :M f3_12 sf .166 .017( in )J 136 290 :M f4_12 sf <55D5>S 149 290 :M f3_12 sf .121 .012( are on )J f4_12 sf (U)S 194 290 :M f3_12 sf <28>S 198 290 :M f4_12 sf .041(Q,Y)A f3_12 sf .115 .011(\), and hence by Lemma 1)J 339 290 :M ( )S f4_12 sf .154<55D5>A 355 290 :M f3_12 sf .15 .015( is into )J 392 290 :M f4_12 sf (Y)S 399 290 :M f3_12 sf .128 .013( if and only if )J f4_12 sf (U)S 477 290 :M f3_12 sf .146 .015( is.)J 59 299 9 9 rC gS 1.286 1 scale 45.889 308 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 77 332 :M f4_12 sf (U)S 86 332 :M f3_12 sf .222 .022( is an )J f0_12 sf .894 .089(inducing path)J f3_12 sf .475 .048( between )J 235 332 :M f4_12 sf .283(A)A f3_12 sf .359 .036( and )J f4_12 sf .283(B)A f3_12 sf .45 .045( in DAG )J f4_12 sf (G)S 328 332 :M f3_12 sf <28>S 332 332 :M f0_12 sf .524(O)A f3_12 sf .168(,)A f0_12 sf .323 .032( S )J 359 332 :M f3_12 sf .089(,)A f0_12 sf .238(L)A f3_12 sf .394 .039(\) relative to )J f0_12 sf .277(O)A f3_12 sf .359 .036( given )J f0_12 sf (S)S 480 332 :M f3_12 sf .584 .058( if)J 59 350 :M .824 .082(and only if there is an undirected path from )J 281 350 :M f4_12 sf .441(A)A f3_12 sf .384 .038( to )J f4_12 sf .518 .052(B )J f3_12 sf 1.017 .102(such that every collider on )J 453 350 :M f4_12 sf (U)S 462 350 :M f3_12 sf .982 .098( has a)J 59 368 :M .432 .043(descendant in {)J 135 368 :M f4_12 sf .279(A)A f3_12 sf .114(,)A f4_12 sf .279(B)A f3_12 sf .278 .028(} )J f1_12 sf .35A f3_12 sf .114 .011( )J 174 368 :M f0_12 sf (S)S 181 368 :M f3_12 sf .496 .05(, and no non-collider on )J 303 368 :M f4_12 sf (U)S 312 368 :M f3_12 sf .541 .054( is in )J f0_12 sf .709(O)A f3_12 sf .228 .023( )J 353 368 :M f1_12 sf .358A f3_12 sf .106 .011( )J f0_12 sf (S)S 372 368 :M f3_12 sf .496 .05(. For example, all of the)J 59 386 :M 1.881 .188(paths between )J 137 386 :M f4_12 sf 1.047(A)A f3_12 sf 1.333 .133( and )J f4_12 sf 1.047(B)A f3_12 sf 2.159 .216( in Figure 4 are inducing paths relative to )J 408 386 :M f0_12 sf .743(O)A f3_12 sf 1.574 .157( given)J 450 386 :M f0_12 sf 2.821 .282( S)J 463 386 :M f3_12 sf 2.217 .222(. The)J 59 404 :M (following theorems generalize Verma and Pearl \(1990\).)S 77 428 :M f0_12 sf .814 .081(Lemma 4)J f3_12 sf .334 .033(: In)J f4_12 sf .078 .008( )J f3_12 sf .606 .061(directed graph )J 220 428 :M f4_12 sf (G)S 229 428 :M f3_12 sf <28>S 233 428 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 252 428 :M f3_12 sf .106(,)A f0_12 sf .283(L)A f3_12 sf .544 .054(\), if there is an inducing path between )J f4_12 sf .26(A)A f3_12 sf .344 .034( and )J 485 428 :M f4_12 sf (B)S 59 446 :M f3_12 sf .209 .021(that is out of )J f4_12 sf .133(A)A f3_12 sf .209 .021( and into )J 176 446 :M f4_12 sf .102(B)A f3_12 sf .197 .02(, then for any subset )J 285 446 :M f0_12 sf .093(Z)A f3_12 sf .077 .008( of )J f0_12 sf .109(O)A f3_12 sf .053(\\{)A f4_12 sf .085(A)A f3_12 sf (,)S f4_12 sf .085(B)A f3_12 sf .179 .018(} there is an undirected path )J f4_12 sf (C)S 59 464 :M f3_12 sf (that d-connects )S 135 464 :M f4_12 sf (A)S f3_12 sf ( and )S f4_12 sf (B)S f3_12 sf ( given )S 205 464 :M f0_12 sf (Z)S f3_12 sf ( )S f1_12 sf S f3_12 sf ( )S f0_12 sf (S)S 235 464 :M f3_12 sf ( that is out of )S f4_12 sf (A)S f3_12 sf ( and into )S 353 464 :M f4_12 sf (B)S f3_12 sf (.)S 77 488 :M (Proof)S 104 488 :M f0_12 sf .653 .065(. )J 111 488 :M f3_12 sf .553 .055(Let )J 131 488 :M f4_12 sf (U)S 140 488 :M f3_12 sf .532 .053( be an inducing path between )J f4_12 sf .242(A)A f3_12 sf .321 .032( and )J 319 488 :M f4_12 sf .282(B)A f3_12 sf .387 .039( that is out of )J f4_12 sf .282(A)A f3_12 sf .433 .043( and into )J f4_12 sf .282(B)A f3_12 sf .834 .083(. Every)J 59 506 :M .902 .09(member of )J 117 506 :M f0_12 sf .995(O)A f3_12 sf .291 .029( )J f1_12 sf .982A f3_12 sf .32 .032( )J 144 506 :M f0_12 sf (S)S 151 506 :M f3_12 sf .736 .074( on )J f4_12 sf (U)S 180 506 :M f3_12 sf .853 .085( except for the endpoints is a collider, and every collider is an)J 59 524 :M (ancestor of either )S f4_12 sf (A)S f3_12 sf ( or )S 168 524 :M f4_12 sf (B)S f3_12 sf ( or a member of )S 255 524 :M f0_12 sf (S)S 262 524 :M f3_12 sf (.)S 77 548 :M .25 .025(If every collider on )J f4_12 sf (U)S 182 548 :M f3_12 sf .317 .032( has a descendant in )J f0_12 sf .182(Z)A f3_12 sf .068 .007( )J 295 548 :M f1_12 sf .203A f3_12 sf .06 .006( )J f0_12 sf (S)S 314 548 :M f3_12 sf .312 .031(, then let )J 361 548 :M f4_12 sf .226(C)A f3_12 sf .157 .016( = )J f4_12 sf (U)S 391 548 :M f3_12 sf .378 .038(. )J 398 548 :M f4_12 sf .101(C)A f3_12 sf .236 .024( d-connects )J 465 548 :M f4_12 sf .11(A)A f3_12 sf .235 .024( and)J 59 566 :M f4_12 sf (B)S f3_12 sf .043 .004( given )J 99 566 :M f0_12 sf (Z)S f3_12 sf ( )S f1_12 sf S f3_12 sf ( )S f0_12 sf (S)S 129 566 :M f3_12 sf .045 .004( because every collider has a descendant in )J 339 566 :M f0_12 sf (Z)S f3_12 sf ( )S f1_12 sf S f3_12 sf ( )S f0_12 sf (S)S 369 566 :M f3_12 sf .045 .005(, and no non-collider is in)J 59 584 :M f0_12 sf (Z)S f3_12 sf ( )S f1_12 sf S f3_12 sf ( )S f0_12 sf (S)S 89 584 :M f3_12 sf (. )S f4_12 sf (C)S f3_12 sf ( is out of )S f4_12 sf (A)S f3_12 sf ( and into )S 200 584 :M f4_12 sf (B)S f3_12 sf (.)S 77 608 :M .188 .019(Suppose that not every collider on )J f4_12 sf (U)S 255 608 :M f3_12 sf .223 .022( has a descendant in )J 356 608 :M f0_12 sf .168(Z)A f3_12 sf .057 .006( )J f1_12 sf .193A f3_12 sf .057 .006( )J f0_12 sf (S)S 386 608 :M f3_12 sf .25 .025(. Let )J 412 608 :M f4_12 sf .087(R)A f3_12 sf .189 .019( be the collider)J 59 626 :M .635 .063(on )J 75 626 :M f4_12 sf (U)S 84 626 :M f3_12 sf .464 .046( closest to )J f4_12 sf .28(A)A f3_12 sf .56 .056( that does not have a descendant in )J f0_12 sf .306(Z)A f3_12 sf .115 .011( )J 332 626 :M f1_12 sf .575A f3_12 sf .187 .019( )J 345 626 :M f0_12 sf (S)S 352 626 :M f3_12 sf .589 .059(, and )J f4_12 sf .687 .069(W )J 393 626 :M f3_12 sf .525 .053(be the collider on )J 483 626 :M f4_12 sf (U)S 59 644 :M f3_12 sf (closest to )S 107 644 :M f4_12 sf (A)S f3_12 sf (. )S f4_12 sf (R)S f3_12 sf ( )S f1_12 sf S 137 644 :M f3_12 sf ( )S f4_12 sf (A)S f3_12 sf ( and )S f4_12 sf (R)S f3_12 sf ( )S f1_12 sf S 187 644 :M f3_12 sf ( )S f4_12 sf (B)S f3_12 sf ( because )S 241 644 :M f4_12 sf (A)S f3_12 sf ( and )S f4_12 sf (B)S f3_12 sf ( are not colliders on )S f4_12 sf (U)S 385 644 :M f3_12 sf (.)S 77 668 :M .595 .06(Suppose first that )J 167 668 :M f4_12 sf .695(R)A f3_12 sf .55 .055( = )J 189 668 :M f4_12 sf (W)S 199 668 :M f3_12 sf .866 .087(. )J 206 668 :M f4_12 sf .537(R)A f3_12 sf .654 .065( is not in )J 262 668 :M f0_12 sf .657(Z)A f3_12 sf .224 .022( )J f1_12 sf .756A f3_12 sf .246 .025( )J 286 668 :M f0_12 sf (S)S 293 668 :M f3_12 sf .68 .068( because )J 339 668 :M f4_12 sf .309(R)A f3_12 sf .627 .063( has no descendant in )J 457 668 :M f0_12 sf .693(Z)A f3_12 sf .26 .026( )J 469 668 :M f1_12 sf .718A f3_12 sf .234 .023( )J 482 668 :M f0_12 sf (S)S 489 668 :M f3_12 sf (.)S 59 686 :M .334 .033(There is a directed path from )J 204 686 :M f4_12 sf .187(R)A f3_12 sf .163 .016( to )J f4_12 sf .187(B)A f3_12 sf .375 .038( that does not contain )J 343 686 :M f4_12 sf .123(A)A f3_12 sf .298 .03(, because otherwise there is a)J endp %%Page: 34 34 %%BeginPageSetup initializepage (peter; page: 34 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (34)S gR gS 0 0 552 730 rC 59 56 :M f3_12 sf .767 .077(cycle in )J 102 56 :M f4_12 sf (G)S 111 56 :M f3_12 sf <28>S 115 56 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 134 56 :M f3_12 sf .191(,)A f0_12 sf .51(L)A f3_12 sf .489 .049(\). )J f4_12 sf .467(B)A f3_12 sf .591 .059( is not on )J f4_12 sf (U)S 223 56 :M f3_12 sf <28>S 227 56 :M f4_12 sf .217(A)A f3_12 sf .089(,)A f4_12 sf .217(R)A f3_12 sf .227 .023(\). )J f4_12 sf (U)S 264 56 :M f3_12 sf <28>S 268 56 :M f4_12 sf .285(A)A f3_12 sf .117(,)A f4_12 sf .285(R)A f3_12 sf .731 .073(\) d-connects )J f4_12 sf .285(A)A f3_12 sf .378 .038( and )J 382 56 :M f4_12 sf .394(R)A f3_12 sf .674 .067( given )J 424 56 :M f0_12 sf .575(Z)A f3_12 sf .196 .02( )J f1_12 sf .662A f3_12 sf .196 .02( )J f0_12 sf (S)S 456 56 :M f3_12 sf .795 .08(, and is)J 59 74 :M .026 .003(out of )J f4_12 sf (A)S f3_12 sf .037 .004(. By Lemma 3)J 166 74 :M .032 .003( there is a d-connecting path )J 306 74 :M f4_12 sf (C)S f3_12 sf .036 .004( between )J f4_12 sf (A)S f3_12 sf .021 .002( and )J f4_12 sf (B)S f3_12 sf .029 .003( given )J 430 74 :M f0_12 sf (Z)S f3_12 sf ( )S f1_12 sf S f3_12 sf ( )S f0_12 sf (S)S 460 74 :M f3_12 sf .035 .003( that is)J 59 92 :M (out of )S f4_12 sf (A)S f3_12 sf ( and into )S 142 92 :M f4_12 sf (B)S f3_12 sf (.)S 77 116 :M 1.094 .109(Suppose then that )J f4_12 sf .436(R)A f3_12 sf .178 .018( )J 181 116 :M f1_12 sf S 188 116 :M f3_12 sf .308 .031( )J f4_12 sf (W)S 202 116 :M f3_12 sf 1.011 .101(. Because )J 254 116 :M f4_12 sf (U)S 263 116 :M f3_12 sf 1.091 .109( is out of )J f4_12 sf .9(A)A f3_12 sf .614 .061(, )J f4_12 sf 1.45 .145(W )J 343 116 :M f3_12 sf 1.011 .101(is a descendant of )J 438 116 :M f4_12 sf .816(A)A f3_12 sf .557 .056(. )J f4_12 sf 1.315 .132(W )J 467 116 :M f3_12 sf 1.047 .105(has a)J 59 134 :M 1.281 .128(descendant in )J f0_12 sf .508(Z)A f3_12 sf .19 .019( )J 144 134 :M f1_12 sf .822A f3_12 sf .243 .024( )J f0_12 sf (S)S 164 134 :M f3_12 sf 1.182 .118( by definition of )J 252 134 :M f4_12 sf .515(R)A f3_12 sf 1.074 .107(. It follows that every collider on )J 432 134 :M f4_12 sf (U)S 441 134 :M f3_12 sf 1.263 .126( that is an)J 59 152 :M .514 .051(ancestor of )J f4_12 sf .215(A)A f3_12 sf .41 .041( has a descendant in )J f0_12 sf .235(Z)A f3_12 sf .088 .009( )J 237 152 :M f1_12 sf .48A f3_12 sf .156 .016( )J 250 152 :M f0_12 sf (S)S 257 152 :M f3_12 sf .471 .047(. Hence )J 298 152 :M f4_12 sf .268(R)A f3_12 sf .447 .045( is an ancestor of )J f4_12 sf .268(B)A f3_12 sf .405 .041(, and not of )J 460 152 :M f4_12 sf .295(A)A f3_12 sf .201 .02(. )J f4_12 sf .295(B)A f3_12 sf .368 .037( is)J 59 170 :M .599 .06(not on )J f4_12 sf (U)S 103 170 :M f3_12 sf <28>S 107 170 :M f4_12 sf .206(A)A f3_12 sf .084(,)A f4_12 sf .206(R)A f3_12 sf .215 .022(\). )J f4_12 sf (U)S 144 170 :M f3_12 sf <28>S 148 170 :M f4_12 sf .298(A)A f3_12 sf .122(,)A f4_12 sf .298(R)A f3_12 sf .764 .076(\) d-connects )J f4_12 sf .298(A)A f3_12 sf .379 .038( and )J f4_12 sf .298(R)A f3_12 sf .511 .051( given )J 303 170 :M f0_12 sf .545(Z)A f3_12 sf .186 .019( )J f1_12 sf .628A f3_12 sf .186 .019( )J f0_12 sf (S)S 335 170 :M f3_12 sf .796 .08( and is out of )J 406 170 :M f4_12 sf .176(A)A f3_12 sf .532 .053(. By hypothesis,)J 59 188 :M .515 .051(there is a directed path )J 175 188 :M f4_12 sf (D)S 184 188 :M f3_12 sf .661 .066( from )J f4_12 sf .43(R)A f3_12 sf .391 .039( to )J 238 188 :M f4_12 sf .247(B)A f3_12 sf .495 .049( that does not contain )J 355 188 :M f4_12 sf .305(A)A f3_12 sf .56 .056( or any member of )J f0_12 sf .333(Z)A f3_12 sf .125 .012( )J 469 188 :M f1_12 sf .575A f3_12 sf .187 .019( )J 482 188 :M f0_12 sf (S)S 489 188 :M f3_12 sf (.)S 59 206 :M .094 .009(By Lemma 3)J 122 206 :M .092 .009(, there is a path that d-connects )J 275 206 :M f4_12 sf .065(A)A f3_12 sf .086 .009( and )J 306 206 :M f4_12 sf (B)S f3_12 sf .085 .009( given )J 346 206 :M f0_12 sf .073(Z)A f3_12 sf ( )S f1_12 sf .084A f3_12 sf ( )S f0_12 sf (S)S 376 206 :M f3_12 sf .083 .008( that is out of )J f4_12 sf .06(A)A f3_12 sf .123 .012( and into)J 59 224 :M f4_12 sf (B)S f3_12 sf (. )S 72 215 9 9 rC gS 1.286 1 scale 56 224 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 77 248 :M f0_12 sf .667 .067(Lemma 5:)J 130 248 :M f3_12 sf .583 .058( If )J f4_12 sf (G)S 155 248 :M f3_12 sf <28>S 159 248 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 178 248 :M f3_12 sf .14(,)A f0_12 sf .375(L)A f3_12 sf .727 .073(\) is a directed acyclic graph, and there is an inducing path )J 483 248 :M f4_12 sf (U)S 59 266 :M f3_12 sf .414 .041(between )J 103 266 :M f4_12 sf .327(A)A f3_12 sf .416 .042( and )J f4_12 sf .327(B)A f3_12 sf .477 .048( that is into )J 201 266 :M f4_12 sf .271(A)A f3_12 sf .417 .042( and into )J f4_12 sf .271(B)A f3_12 sf .539 .054( then for every subset )J f0_12 sf .296(Z)A f3_12 sf .257 .026( of )J 397 266 :M f0_12 sf .209(O)A f3_12 sf .102(\\{)A f4_12 sf .164(A)A f3_12 sf .067(,)A f4_12 sf .164(B)A f3_12 sf .31 .031(} there is an)J 59 284 :M (undirected path )S f4_12 sf (C)S f3_12 sf ( that d-connects )S 223 284 :M f4_12 sf (A)S f3_12 sf ( and )S f4_12 sf (B)S f3_12 sf ( given )S 293 284 :M f0_12 sf (Z)S f3_12 sf ( )S f1_12 sf S f3_12 sf ( )S f0_12 sf (S)S 323 284 :M f3_12 sf ( that is into )S 380 284 :M f4_12 sf (A)S f3_12 sf ( and into )S 432 284 :M f4_12 sf (B)S f3_12 sf (.)S 77 308 :M (Proof)S 104 308 :M f0_12 sf (. )S f3_12 sf .027 .003(If every collider on )J 206 308 :M f4_12 sf (U)S 215 308 :M f3_12 sf .026 .003( has a descendant in )J f0_12 sf (Z)S f3_12 sf ( )S f1_12 sf S f3_12 sf ( )S f0_12 sf (S)S 344 308 :M f3_12 sf .027 .003(, then )J 374 308 :M f4_12 sf (U)S 383 308 :M f3_12 sf .022 .002( is a d-connecting path)J 59 326 :M 1.048 .105(between )J 104 326 :M f4_12 sf .852(A)A f3_12 sf 1.129 .113( and )J 138 326 :M f4_12 sf .768(B)A f3_12 sf 1.268 .127( given )J f0_12 sf .839(Z)A f3_12 sf .314 .031( )J 194 326 :M f1_12 sf 1.344A f3_12 sf .437 .044( )J 208 326 :M f0_12 sf (S)S 215 326 :M f3_12 sf 1.344 .134( that is into )J 279 326 :M f4_12 sf .596(A)A f3_12 sf .914 .091( and into )J f4_12 sf .596(B)A f3_12 sf 1.257 .126(. Suppose then that there is a)J 59 344 :M .13 .013(collider that does not have a descendant in )J 268 344 :M f0_12 sf .107(Z)A f3_12 sf ( )S f1_12 sf .123A f3_12 sf ( )S f0_12 sf (S)S 298 344 :M f3_12 sf .152 .015(. Let )J f4_12 sf .186 .019(W )J 337 344 :M f3_12 sf .123 .012(be the collider on )J f4_12 sf (U)S 433 344 :M f3_12 sf .146 .015( closest to )J 485 344 :M f4_12 sf (A)S 59 362 :M f3_12 sf .163 .016(that does not have a descendant in )J 228 362 :M f0_12 sf .167(Z)A f3_12 sf .057 .006( )J f1_12 sf .192A f3_12 sf .062 .006( )J 252 362 :M f0_12 sf (S)S 259 362 :M f3_12 sf .141 .014(. Suppose that )J f4_12 sf (W)S 340 362 :M f3_12 sf .166 .017( is the source of a directed path)J 59 380 :M f4_12 sf (D)S 68 380 :M f3_12 sf .367 .037( to )J f4_12 sf .421(B)A f3_12 sf .843 .084( that does not contain )J 203 380 :M f4_12 sf .567(A)A f3_12 sf .387 .039(. )J f4_12 sf .567(B)A f3_12 sf .733 .073( is not on )J 276 380 :M f4_12 sf (U)S 285 380 :M f3_12 sf <28>S 289 380 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (W)S 309 380 :M f3_12 sf .434 .043(\). )J f4_12 sf (U)S 329 380 :M f3_12 sf <28>S 333 380 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (W)S 353 380 :M f3_12 sf .678 .068(\) is a path that d-connects )J f4_12 sf (A)S 59 398 :M f3_12 sf 1.176 .118(and )J f4_12 sf .957 .096(W )J 95 398 :M f3_12 sf 1.419 .142(given )J f0_12 sf .613(Z)A f3_12 sf .209 .021( )J f1_12 sf .705A f3_12 sf .23 .023( )J 152 398 :M f0_12 sf (S)S 159 398 :M f3_12 sf .99 .099(, and is into )J f4_12 sf .664(A)A f3_12 sf .869 .087(. By )J 256 398 :M .843 .084(Lemma 3)J 303 398 :M .91 .091(, there is an undirected path )J 447 398 :M f4_12 sf .514(C)A f3_12 sf .809 .081( that d-)J 59 416 :M 1.658 .166(connects )J f4_12 sf .513(A)A f3_12 sf .681 .068( and )J 139 416 :M f4_12 sf .647(B)A f3_12 sf 1.109 .111( given )J 183 416 :M f0_12 sf 1.217(Z)A f3_12 sf .415 .041( )J f1_12 sf 1.401A f3_12 sf .456 .046( )J 209 416 :M f0_12 sf (S)S 216 416 :M f3_12 sf 1.021 .102( and is into )J f4_12 sf .709(A)A f3_12 sf 1.087 .109( and into )J f4_12 sf .709(B)A f3_12 sf 1.439 .144(. Similarly, if )J 416 416 :M f4_12 sf .542(R)A f3_12 sf 1.081 .108( is the closest)J 59 434 :M .93 .093(collider to )J 114 434 :M f4_12 sf .622(B)A f3_12 sf .636 .064( on )J f4_12 sf (U)S 150 434 :M f3_12 sf 1.003 .1( that does not have a descendant in )J 332 434 :M f0_12 sf .995(Z)A f3_12 sf .339 .034( )J f1_12 sf 1.145A f3_12 sf .373 .037( )J 358 434 :M f0_12 sf (S)S 365 434 :M f3_12 sf .867 .087(, and )J f4_12 sf .628(R)A f3_12 sf 1.061 .106( is the source of a)J 59 452 :M .325 .033(directed path )J f4_12 sf (D)S 134 452 :M f3_12 sf .218 .022( to )J f4_12 sf .251(A)A f3_12 sf .501 .05( that does not contain )J 266 452 :M f4_12 sf .265(B)A f3_12 sf .414 .041(, then by )J 320 452 :M .391 .039(Lemma 3)J 367 452 :M .256 .026(, )J f4_12 sf .376(A)A f3_12 sf .498 .05( and )J 405 452 :M f4_12 sf .106(B)A f3_12 sf .328 .033( are d-connected)J 59 470 :M (given )S 89 470 :M f0_12 sf (Z)S f3_12 sf ( )S f1_12 sf S f3_12 sf ( )S f0_12 sf (S)S 119 470 :M f3_12 sf ( by an undirected path into )S f4_12 sf (A)S f3_12 sf ( and into )S 302 470 :M f4_12 sf (B)S f3_12 sf (.)S 77 494 :M .255 .026(Suppose then that the collider )J 225 494 :M f4_12 sf .371 .037(W )J 239 494 :M f3_12 sf .199 .02(on )J f4_12 sf (U)S 263 494 :M f3_12 sf .22 .022( closest to )J f4_12 sf .133(A)A f3_12 sf .291 .029( that does not have a descendant in)J 59 512 :M f0_12 sf 1.381(Z)A f3_12 sf .518 .052( )J 72 512 :M f1_12 sf 1.433A f3_12 sf .466 .047( )J 86 512 :M f0_12 sf (S)S 93 512 :M f3_12 sf 1.215 .122( is not the source of a directed path to )J f4_12 sf .685(B)A f3_12 sf 1.353 .135( that does not contain )J f4_12 sf .685(A)A f3_12 sf 1.348 .135(, and that the)J 59 530 :M .193 .019(collider )J 99 530 :M f4_12 sf .201(R)A f3_12 sf .215 .021( on )J 125 530 :M f4_12 sf (U)S 134 530 :M f3_12 sf .203 .02( closest to )J f4_12 sf .123(B)A f3_12 sf .247 .025( that does not have a descendant in )J 366 530 :M f0_12 sf .186(Z)A f3_12 sf .063 .006( )J f1_12 sf .214A f3_12 sf .063 .006( )J f0_12 sf (S)S 396 530 :M f3_12 sf .248 .025( is not the source of)J 59 548 :M 1.21 .121(a directed path to )J 152 548 :M f4_12 sf .575(A)A f3_12 sf 1.135 .113( that does not contain )J f4_12 sf .575(B)A f3_12 sf 1.134 .113(. The subpath of )J 368 548 :M f4_12 sf (U)S 377 548 :M f3_12 sf 1.403 .14( from )J 410 548 :M f4_12 sf (W)S 420 548 :M f3_12 sf .679 .068( to )J f4_12 sf .779(A)A f3_12 sf 1.645 .165( does not)J 59 566 :M .417 .042(contain )J f4_12 sf .143(B)A f3_12 sf .24 .024( or a member of )J 186 566 :M f0_12 sf .297(Z)A f3_12 sf .111 .011( )J 198 566 :M f1_12 sf .199A f3_12 sf .059 .006( )J f0_12 sf (S)S 217 566 :M f3_12 sf .264 .026(, and the subpath of )J f4_12 sf (U)S 325 566 :M f3_12 sf .256 .026( from )J f4_12 sf .167(R)A f3_12 sf .145 .015( to )J f4_12 sf .167(B)A f3_12 sf .337 .034( does not contain )J 472 566 :M f4_12 sf .147(A)A f3_12 sf .218 .022( or)J 59 584 :M .291 .029(a member of )J 124 584 :M f0_12 sf .29(Z)A f3_12 sf .108 .011( )J 136 584 :M f1_12 sf .194A f3_12 sf .057 .006( )J f0_12 sf (S)S 155 584 :M f3_12 sf .299 .03(. It follows that there exist two colliders )J f4_12 sf .14(E)A f3_12 sf .178 .018( and )J f4_12 sf .14(F)A f3_12 sf .149 .015( on )J 409 584 :M f4_12 sf (U)S 418 584 :M f3_12 sf .265 .027( such that there)J 59 602 :M .228 .023(is a directed path from )J 172 602 :M f4_12 sf .216(E)A f3_12 sf .196 .02( to )J 194 602 :M f4_12 sf .108(A)A f3_12 sf .216 .022( that does not contain )J 309 602 :M f4_12 sf .121(B)A f3_12 sf .233 .023(, there is a directed path from )J f4_12 sf .121(F)A f3_12 sf .106 .011( to )J f4_12 sf (B)S 59 620 :M f3_12 sf .529 .053(that does not contain )J 165 620 :M f4_12 sf .444(A)A f3_12 sf .33 .033(, )J 179 620 :M f4_12 sf .311(F)A f3_12 sf .576 .058( is between )J f4_12 sf .311(E)A f3_12 sf .413 .041( and )J 277 620 :M f4_12 sf .203(B)A f3_12 sf .491 .049(, and every collider between )J 427 620 :M f4_12 sf .374(E)A f3_12 sf .475 .048( and )J f4_12 sf .374(F)A f3_12 sf .516 .052( is an)J 59 638 :M .15 .015(ancestor of a member of )J 180 638 :M f0_12 sf .16(Z)A f3_12 sf .06 .006( )J 192 638 :M f1_12 sf .107A f3_12 sf ( )S f0_12 sf (S)S 211 638 :M f3_12 sf .075 .007(. )J f4_12 sf (U)S 226 638 :M f3_12 sf <28>S 230 638 :M f4_12 sf .058(E)A f3_12 sf (,)S f4_12 sf .058(F)A f3_12 sf .15 .015(\) d-connects )J f4_12 sf .058(E)A f3_12 sf .077 .008( and )J 340 638 :M f4_12 sf .088(F)A f3_12 sf .157 .016( given \()J f0_12 sf .096(Z)A f3_12 sf ( )S 396 638 :M f1_12 sf .107A f3_12 sf ( )S f0_12 sf (S)S 415 638 :M f3_12 sf (\)\\{)S f4_12 sf (E)S f3_12 sf (,)S f4_12 sf (F)S f3_12 sf .074 .007(} because)J 59 656 :M 1.315 .132(no member of )J f0_12 sf .751(O)A f3_12 sf .241 .024( )J 148 656 :M f1_12 sf 1.253A f3_12 sf .408 .041( )J 162 656 :M f0_12 sf (S)S 169 656 :M f3_12 sf 1.061 .106( is a non-collider on )J f4_12 sf (U)S 285 656 :M f3_12 sf <28>S 289 656 :M f4_12 sf .378(E)A f3_12 sf .155(,)A f4_12 sf .378(F)A f3_12 sf .943 .094(\) except for the endpoints, and every)J 59 674 :M .071 .007(collider on )J 114 674 :M f4_12 sf (U)S 123 674 :M f3_12 sf <28>S 127 674 :M f4_12 sf (E)S f3_12 sf (,)S f4_12 sf (F)S f3_12 sf .067 .007(\) has a descendant in )J 248 674 :M f0_12 sf .059(Z)A f3_12 sf ( )S f1_12 sf .068A f3_12 sf ( )S f0_12 sf (S)S 278 674 :M f3_12 sf .075 .007(. The directed path from )J 398 674 :M f4_12 sf (E)S f3_12 sf .03 .003( to )J f4_12 sf (A)S f3_12 sf .085 .008( d-connects )J f4_12 sf (E)S endp %%Page: 35 35 %%BeginPageSetup initializepage (peter; page: 35 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (35)S gR gS 0 0 552 730 rC 59 56 :M f3_12 sf .627 .063(and )J f4_12 sf .317(A)A f3_12 sf .586 .059( given \()J 126 56 :M f0_12 sf .673(Z)A f3_12 sf .252 .025( )J 138 56 :M f1_12 sf .698A f3_12 sf .227 .023( )J 151 56 :M f0_12 sf (S)S 158 56 :M f3_12 sf .143(\)\\{)A f4_12 sf .241(E)A f3_12 sf .099(,)A f4_12 sf .241(A)A f3_12 sf .516 .052(} and the directed path from )J 332 56 :M f4_12 sf .599(F)A f3_12 sf .544 .054( to )J 355 56 :M f4_12 sf .208(B)A f3_12 sf .53 .053( d-connects )J 422 56 :M f4_12 sf .301(F)A f3_12 sf .383 .038( and )J f4_12 sf .301(B)A f3_12 sf .812 .081( given)J 59 74 :M <28>S 63 74 :M f0_12 sf .284(Z)A f3_12 sf .097 .01( )J f1_12 sf .327A f3_12 sf .106 .011( )J 87 74 :M f0_12 sf (S)S 94 74 :M f3_12 sf .06(\)\\{)A f4_12 sf .101(F)A f3_12 sf (,)S f4_12 sf .101(B)A f3_12 sf .159 .016(}. By )J 154 74 :M .263 .026(Lemma 1 there is an undirected path that d-connects )J 412 74 :M f4_12 sf .171(A)A f3_12 sf .217 .022( and )J f4_12 sf .171(B)A f3_12 sf .293 .029( given )J 484 74 :M f0_12 sf (Z)S 59 92 :M f1_12 sf S f3_12 sf ( )S f0_12 sf (S)S 78 92 :M f3_12 sf ( that is into )S 135 92 :M f4_12 sf (A)S f3_12 sf ( and into )S 187 92 :M f4_12 sf (B)S f3_12 sf (. )S 200 83 9 9 rC gS 1.286 1 scale 155.557 92 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 77 116 :M f0_12 sf .6 .06(Lemma 6:)J 130 116 :M f3_12 sf .524 .052( If )J f4_12 sf (G)S 155 116 :M f3_12 sf <28>S 159 116 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 178 116 :M f3_12 sf .128(,)A f0_12 sf .343(L)A f3_12 sf .647 .065(\) is a directed acyclic graph and )J 352 116 :M f4_12 sf (U)S 361 116 :M f3_12 sf .711 .071( is an inducing path out of)J 59 134 :M (both )S f4_12 sf (A)S f3_12 sf ( and )S f4_12 sf (B)S f3_12 sf ( then every collider on )S 231 134 :M f4_12 sf (U)S 240 134 :M f3_12 sf ( is an ancestor of a member of )S 388 134 :M f0_12 sf (S)S 395 134 :M f3_12 sf (.)S 77 158 :M (Proof)S 104 158 :M f0_12 sf (.)S f3_12 sf .1 .01( Let )J f4_12 sf (W)S 139 158 :M f3_12 sf .145 .014( be the collider on )J 230 158 :M f4_12 sf (U)S 239 158 :M f3_12 sf .146 .015( closest to )J 291 158 :M f4_12 sf .096(A)A f3_12 sf .132 .013( and let )J 337 158 :M f4_12 sf .075(R)A f3_12 sf .134 .013( be the collider on )J 435 158 :M f4_12 sf (U)S 444 158 :M f3_12 sf .133 .013( closest to)J 59 176 :M f4_12 sf .114(B)A f3_12 sf .205 .02(. Since )J 103 176 :M f4_12 sf (U)S 112 176 :M f3_12 sf .237 .024( is out of )J f4_12 sf .195(A)A f3_12 sf .195 .019( and )J f4_12 sf (G)S 201 176 :M f3_12 sf <28>S 205 176 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 224 176 :M f3_12 sf (,)S f0_12 sf .116(L)A f3_12 sf .195 .02(\) is acyclic, )J 294 176 :M f4_12 sf (W)S 304 176 :M f3_12 sf .244 .024( is not an ancestor of )J 409 176 :M f4_12 sf .076(A)A f3_12 sf .184 .018(. Similarly, )J 474 176 :M f4_12 sf .133(R)A f3_12 sf .167 .017( is)J 59 194 :M .556 .056(not an ancestor of )J 151 194 :M f4_12 sf .366(B)A f3_12 sf .499 .05(. Let )J 185 194 :M f4_12 sf (Q)S 194 194 :M f3_12 sf .585 .058( be the collider on )J 288 194 :M f4_12 sf (U)S 297 194 :M f3_12 sf .523 .052( closest to )J f4_12 sf .316(A)A f3_12 sf .567 .057( that is not an ancestor of a)J 59 212 :M (member of )S f0_12 sf (S)S 121 212 :M f3_12 sf (.)S 77 236 :M .138 .014(Suppose that )J 143 236 :M f4_12 sf (W)S 153 236 :M f3_12 sf .114 .011( = )J f4_12 sf (Q)S 175 236 :M f3_12 sf .138 .014(. Hence)J 212 236 :M f4_12 sf .2 .02( Q)J 225 236 :M f3_12 sf .152 .015( is an ancestor of )J f4_12 sf .091(B)A f3_12 sf .062 .006(. )J f4_12 sf (Q)S 332 236 :M f3_12 sf .095 .01( )J cF f1_12 sf .01A sf .095 .01( )J f4_12 sf .128(R)A f3_12 sf .205 .021( since )J 384 236 :M f4_12 sf .076(R)A f3_12 sf .145 .014( is not an ancestor of)J 59 254 :M f4_12 sf .147(B.)A f3_12 sf .077 .008( )J f4_12 sf .208(R)A f3_12 sf .355 .036( is not an ancestor of )J 185 254 :M f4_12 sf .154(A)A f3_12 sf .327 .033(, since otherwise there would be a cycle in )J 404 254 :M f4_12 sf (G)S 413 254 :M f3_12 sf <28>S 417 254 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 436 254 :M f3_12 sf .063(,)A f0_12 sf .167(L)A f3_12 sf .285 .029(\). Thus )J 485 254 :M f4_12 sf (R)S 59 272 :M f3_12 sf 1.649 .165(is an ancestor of a member of )J 221 272 :M f0_12 sf (S)S 228 272 :M f3_12 sf 1.361 .136( and )J f4_12 sf (Q)S 264 272 :M f3_12 sf 1.696 .17( is an ancestor of a member of )J 431 272 :M f0_12 sf (S)S 438 272 :M f3_12 sf 1.75 .175(. This is a)J 59 290 :M (contradiction. The argument for )S f4_12 sf (R)S f3_12 sf ( = )S 235 290 :M f4_12 sf (Q)S 244 290 :M f3_12 sf ( is similar.)S 77 314 :M .983 .098(Suppose that )J 145 314 :M f4_12 sf (W)S 155 314 :M f3_12 sf 1.431 .143( )J cF f1_12 sf .143A sf 1.431 .143( )J 171 314 :M f4_12 sf (Q)S 180 314 :M f3_12 sf 1.358 .136( and )J f4_12 sf 1.067(R)A f3_12 sf .833 .083( )J cF f1_12 sf .083A sf .833 .083( )J 229 314 :M f4_12 sf (Q)S 238 314 :M f3_12 sf .891 .089(. Because )J f4_12 sf (G)S 299 314 :M f3_12 sf <28>S 303 314 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 322 314 :M f3_12 sf .19(,)A f0_12 sf .507(L)A f3_12 sf .933 .093(\) is acyclic, either )J 428 314 :M f4_12 sf (W)S 438 314 :M f3_12 sf .862 .086( or )J f4_12 sf .948(R)A f3_12 sf 1.311 .131( is an)J 59 332 :M .63 .063(ancestor of )J f0_12 sf (S)S 124 332 :M f3_12 sf .706 .071(. Without loss of generality, suppose that )J 332 332 :M f4_12 sf (W)S 342 332 :M f3_12 sf .758 .076( is an ancestor of )J f0_12 sf (S)S 438 332 :M f3_12 sf .941 .094(. If )J 458 332 :M f4_12 sf .507(R)A f3_12 sf .701 .07( is an)J 59 350 :M .048 .005(ancestor of )J 116 350 :M f4_12 sf (A)S f3_12 sf .047 .005( then )J 150 350 :M f4_12 sf (R)S f3_12 sf .046 .005( is an ancestor of )J f0_12 sf (S)S 248 350 :M f3_12 sf .057 .006( since )J 279 350 :M f4_12 sf (A)S f3_12 sf .05 .005( is an ancestor of )J 371 350 :M f4_12 sf (W)S 381 350 :M f3_12 sf .027 .003(. If )J f4_12 sf (R)S f3_12 sf .06 .006( is not an ancestor)J 59 368 :M 1.009 .101(of )J f4_12 sf .74(A)A f3_12 sf 1.152 .115(, then )J 114 368 :M f4_12 sf .609(R)A f3_12 sf 1.107 .111( is an ancestor of a member of )J f0_12 sf (S)S 289 368 :M f3_12 sf 1.07 .107( from the definition of inducing path. In)J 59 386 :M .3 .03(either case )J 114 386 :M f4_12 sf .173(R)A f3_12 sf .289 .029( is an ancestor of )J f0_12 sf (S)S 214 386 :M f3_12 sf .344 .034(. Since )J 251 386 :M f4_12 sf (W)S 261 386 :M f3_12 sf .219 .022( and )J f4_12 sf .172(R)A f3_12 sf .329 .033( are ancestors of )J f0_12 sf (S)S 382 386 :M f3_12 sf .106 .011(, )J f4_12 sf .156(A)A f3_12 sf .198 .02( and )J f4_12 sf .156(B)A f3_12 sf .42 .042( are ancestors)J 59 404 :M (of )S 72 404 :M f0_12 sf (S)S 79 404 :M f3_12 sf ( and therefore )S 149 404 :M f4_12 sf (Q)S 158 404 :M f3_12 sf ( is an ancestor of )S f0_12 sf (S)S 249 395 9 9 rC gS 1.286 1 scale 193.668 404 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 77 428 :M f0_12 sf .795 .079(Lemma 7: )J f3_12 sf .352 .035(If )J f4_12 sf (G)S 155 428 :M f3_12 sf <28>S 159 428 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 178 428 :M f3_12 sf .128(,)A f0_12 sf .342(L)A f3_12 sf .656 .066(\) is a directed acyclic graph over )J 357 428 :M f0_12 sf (V)S 366 428 :M f3_12 sf .694 .069(, and there is an inducing)J 59 446 :M .366 .037(path )J 84 446 :M f4_12 sf (U)S 93 446 :M f3_12 sf .497 .05( between )J f4_12 sf .23(A)A f3_12 sf .293 .029( and )J f4_12 sf .23(B)A f3_12 sf .315 .032( that is out of )J f4_12 sf .23(A)A f3_12 sf .336 .034( and out of )J 310 446 :M f4_12 sf .165(B)A f3_12 sf .333 .033( then for every subset )J 426 446 :M f0_12 sf .125(Z)A f3_12 sf .104 .01( of )J f0_12 sf .145(O)A f3_12 sf .071(\\{)A f4_12 sf .114(A)A f3_12 sf (,)S f4_12 sf .114(B)A f3_12 sf (})S 59 464 :M .113 .011(there is an undirected path )J 190 464 :M f4_12 sf (C)S f3_12 sf .103 .01( that d-connects )J 278 464 :M f4_12 sf .078(A)A f3_12 sf .099 .01( and )J f4_12 sf .078(B)A f3_12 sf .129 .013( given )J f0_12 sf .085(Z)A f3_12 sf ( )S 360 464 :M f1_12 sf .085A f3_12 sf ( )S f0_12 sf (S)S 379 464 :M f3_12 sf .109 .011( that is out of )J f4_12 sf .08(A)A f3_12 sf .155 .016( and out)J 59 482 :M (of )S 72 482 :M f4_12 sf (B)S f3_12 sf (.)S 77 506 :M .307 .031(Proof. Let )J 130 506 :M f4_12 sf (U)S 139 506 :M f3_12 sf .352 .035( be an inducing path out of both )J f4_12 sf .187(A)A f3_12 sf .238 .024( and )J f4_12 sf .187(B)A f3_12 sf .244 .024(. By )J 360 506 :M .288 .029(Lemma 6)J 407 506 :M .294 .029( every collider on)J 59 524 :M f4_12 sf (U)S 68 524 :M f3_12 sf .259 .026( has a descendant in )J 169 524 :M f0_12 sf .25(Z)A f3_12 sf .085 .009( )J f1_12 sf .288A f3_12 sf .094 .009( )J 193 524 :M f0_12 sf (S)S 200 524 :M f3_12 sf .213 .021(. Thus )J f4_12 sf (U)S 242 524 :M f3_12 sf .234 .023( d-connects )J 301 524 :M f4_12 sf .163(A)A f3_12 sf .207 .021( and )J f4_12 sf .163(B)A f3_12 sf .269 .027( given )J f0_12 sf .178(Z)A f3_12 sf .067 .007( )J 384 524 :M f1_12 sf .177A f3_12 sf .052 .005( )J f0_12 sf (S)S 403 524 :M f3_12 sf .267 .027( and is out of both)J 59 542 :M (endpoints. )S 112 533 9 9 rC gS 1.286 1 scale 87.112 542 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 77 566 :M f0_12 sf .064 .006(Lemma 8: )J f3_12 sf (If )S f4_12 sf (G)S 152 566 :M f3_12 sf <28>S 156 566 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 175 566 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf .056 .006(\) is a directed acyclic graph over )J f0_12 sf (V )S 359 566 :M f3_12 sf .049 .005(and an undirected path )J f4_12 sf (U)S 480 566 :M f3_12 sf .069 .007( in)J 59 584 :M f4_12 sf (G)S 68 584 :M f3_12 sf 1.078 .108( d-connects )J f4_12 sf .436(A)A f3_12 sf .555 .056( and )J f4_12 sf .436(B)A f3_12 sf .864 .086( given \(\()J 212 584 :M f0_12 sf (Ancestors)S 263 584 :M f3_12 sf <28>S 267 584 :M f4_12 sf (G)S 276 584 :M f3_12 sf (,{)S 285 584 :M f4_12 sf .219(A)A f3_12 sf .09(,)A f4_12 sf .219(B)A f3_12 sf .218 .022(} )J f1_12 sf .276A f3_12 sf .09A f0_12 sf (S)S 331 584 :M f3_12 sf <29>S 335 584 :M f1_12 sf 1.232 .123<20C7>J 349 584 :M f3_12 sf .351 .035( )J f0_12 sf 1.203(O)A f3_12 sf .751 .075(\) )J f1_12 sf 1.188A f3_12 sf .387 .039( )J 384 584 :M f0_12 sf (S)S 391 584 :M f3_12 sf .187(\)\\{)A f4_12 sf .315(A)A f3_12 sf .129(,)A f4_12 sf .315(B)A f3_12 sf .535 .053(} then )J 455 584 :M f4_12 sf (U)S 464 584 :M f3_12 sf 1.084 .108( is an)J 59 602 :M (inducing path between )S 171 602 :M f4_12 sf (A)S f3_12 sf ( and )S f4_12 sf (B)S f3_12 sf (.)S 77 626 :M (Proof.)S 107 626 :M f0_12 sf ( )S f3_12 sf .051 .005(If there is a path )J f4_12 sf (U)S 200 626 :M f3_12 sf .043 .004( that d-connects )J 280 626 :M f4_12 sf (A)S f3_12 sf .032 .003( and )J f4_12 sf (B)S f3_12 sf .049 .005( given \(\()J 358 626 :M f0_12 sf (Ancestors)S 409 626 :M f3_12 sf <28>S 413 626 :M f4_12 sf (G)S 422 626 :M f3_12 sf (,{)S 431 626 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf (} )S 457 626 :M f1_12 sf S f3_12 sf S f0_12 sf (S)S 476 626 :M f3_12 sf <29>S 480 626 :M f1_12 sf .061 .006<20C7>J 59 644 :M f0_12 sf 3.958(O)A f3_12 sf 2.696 .27(\) )J 82 644 :M f1_12 sf 5.223A f3_12 sf 1.7 .17( )J 101 644 :M f0_12 sf (S)S 108 644 :M f3_12 sf 1.024(\)\\{)A f4_12 sf 1.721(A)A f3_12 sf .704(,)A f4_12 sf 1.721(B)A f3_12 sf 3.703 .37(} then every collider on )J 290 644 :M f4_12 sf (U)S 299 644 :M f3_12 sf 4.895 .49( is an ancestor of a member of)J 59 662 :M <2828>S 67 662 :M f0_12 sf (Ancestors)S 118 662 :M f3_12 sf <28>S 122 662 :M f4_12 sf (G)S 131 662 :M f3_12 sf (,{)S 140 662 :M f4_12 sf .213(A)A f3_12 sf .087(,)A f4_12 sf .213(B)A f3_12 sf .232 .023(} )J 167 662 :M f1_12 sf S f3_12 sf S f0_12 sf (S)S 186 662 :M f3_12 sf <29>S 190 662 :M f1_12 sf .699 .07<20C7>J 203 662 :M f3_12 sf .217 .022( )J f0_12 sf .743(O)A f3_12 sf .506 .051(\) )J 223 662 :M f1_12 sf .58A f3_12 sf .189 .019( )J 236 662 :M f0_12 sf (S)S 243 662 :M f3_12 sf .121(\)\\{)A f4_12 sf .203(A)A f3_12 sf .083(,)A f4_12 sf .203(B)A f3_12 sf .439 .044(}, and hence has a descendant in {)J f4_12 sf .203(A)A f3_12 sf .083(,)A f4_12 sf .203(B)A f3_12 sf .221 .022(} )J 470 662 :M f1_12 sf S f3_12 sf S f0_12 sf (S)S 489 662 :M f3_12 sf (.)S 59 680 :M .945 .095(Every vertex on )J 143 680 :M f4_12 sf (U)S 152 680 :M f3_12 sf 1.001 .1( is an ancestor of either )J 276 680 :M f4_12 sf .892(A)A f3_12 sf .846 .085( or )J 302 680 :M f4_12 sf .548(B)A f3_12 sf .867 .087( or a collider on )J f4_12 sf (U)S 404 680 :M f3_12 sf .925 .092(, and hence every)J endp %%Page: 36 36 %%BeginPageSetup initializepage (peter; page: 36 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (36)S gR gS 0 0 552 730 rC 59 56 :M f3_12 sf 3.359 .336(vertex on )J 117 56 :M f4_12 sf (U)S 126 56 :M f3_12 sf 3.87 .387( is an ancestor of )J f4_12 sf 2.322(A)A f3_12 sf 2.203 .22( or )J 267 56 :M f4_12 sf 3.061(B)A f3_12 sf 2.904 .29( or )J 300 56 :M f0_12 sf (S)S 307 56 :M f3_12 sf 4.058 .406(. If )J 334 56 :M f4_12 sf (U)S 343 56 :M f3_12 sf 3.512 .351( d-connects )J f4_12 sf 1.421(A)A f3_12 sf 1.808 .181( and )J f4_12 sf 1.421(B)A f3_12 sf 3.833 .383( given)J 59 74 :M <2828>S 67 74 :M f0_12 sf (Ancestors)S 118 74 :M f3_12 sf <28>S 122 74 :M f4_12 sf (G)S 131 74 :M f3_12 sf (,{)S 140 74 :M f4_12 sf 1.912(A)A f3_12 sf .783(,)A f4_12 sf 1.912(B)A f3_12 sf 1.904 .19(} )J f1_12 sf 2.404A f3_12 sf .783A f0_12 sf (S)S 196 74 :M f3_12 sf <29>S 200 74 :M f1_12 sf 15.814 1.581<20C7>J f3_12 sf 4.661 .466( )J 239 74 :M f0_12 sf 6.757(O)A f3_12 sf 4.602 .46(\) )J 267 74 :M f1_12 sf 8.916A f3_12 sf 2.903 .29( )J 291 74 :M f0_12 sf (S)S 298 74 :M f3_12 sf 1.392(\)\\{)A f4_12 sf 2.34(A)A f3_12 sf .958(,)A f4_12 sf 2.34(B)A f3_12 sf 6.042 .604(}, then every member of)J 59 92 :M <2828>S 67 92 :M f0_12 sf (Ancestors)S 118 92 :M f3_12 sf <28>S 122 92 :M f4_12 sf (G)S 131 92 :M f3_12 sf (,{)S 140 92 :M f4_12 sf .258(A)A f3_12 sf .106(,)A f4_12 sf .258(B)A f3_12 sf .28 .028(} )J 167 92 :M f1_12 sf S f3_12 sf S f0_12 sf (S)S 186 92 :M f3_12 sf <29>S 190 92 :M f1_12 sf 1.244 .124<20C7>J f3_12 sf .367 .037( )J 207 92 :M f0_12 sf .532(O)A f3_12 sf .362 .036(\) )J 224 92 :M f1_12 sf .701A f3_12 sf .228 .023( )J 237 92 :M f0_12 sf (S)S 244 92 :M f3_12 sf .17(\)\\{)A f4_12 sf .286(A)A f3_12 sf .117(,)A f4_12 sf .286(B)A f3_12 sf .411 .041(} that is on )J f4_12 sf (U)S 342 92 :M f3_12 sf .613 .061(, except for the endpoints, is a)J 59 110 :M .22 .022(collider. Since every vertex on )J 211 110 :M f4_12 sf (U)S 220 110 :M f3_12 sf .099 .01( is in )J f0_12 sf .088(Ancestors)A 298 110 :M f3_12 sf <28>S 302 110 :M f4_12 sf (G)S 311 110 :M f3_12 sf (,{)S 320 110 :M f4_12 sf .058(A)A f3_12 sf (,)S f4_12 sf .058(B)A f3_12 sf .057 .006(} )J f1_12 sf .073A f3_12 sf S f0_12 sf (S)S 365 110 :M f3_12 sf .246 .025(\), every member of )J 462 110 :M f0_12 sf .11(O)A f3_12 sf .176 .018( that)J 59 128 :M 1.11 .111(is on )J 88 128 :M f4_12 sf (U)S 97 128 :M f3_12 sf .909 .091(, except for the endpoints, is a collider. Every member of )J 389 128 :M f0_12 sf (S)S 396 128 :M f3_12 sf 1.206 .121( on )J 417 128 :M f4_12 sf (U)S 426 128 :M f3_12 sf .946 .095( is a collider.)J 59 146 :M (Hence )S 93 146 :M f4_12 sf (U)S 102 146 :M f3_12 sf ( is an inducing path between )S 242 146 :M f4_12 sf (A)S f3_12 sf ( and )S f4_12 sf (B)S f3_12 sf (. )S 285 137 9 9 rC gS 1.286 1 scale 221.668 146 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 77 170 :M f0_12 sf .654 .065(Lemma 9: )J 134 170 :M f4_12 sf (G)S 143 170 :M f3_12 sf <28>S 147 170 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 166 170 :M f3_12 sf .112(,)A f0_12 sf .299(L)A f3_12 sf .559 .056(\)\) entails that for all subsets )J 319 170 :M f0_12 sf (X)S 328 170 :M f3_12 sf .797 .08( of )J 346 170 :M f0_12 sf .411(O)A f3_12 sf .22 .022(, )J f4_12 sf .323(A)A f3_12 sf .64 .064( is dependent on )J 453 170 :M f4_12 sf .182(B)A f3_12 sf .49 .049( given)J 59 188 :M <28>S 63 188 :M f0_12 sf (X)S 72 188 :M f3_12 sf ( )S f1_12 sf S f3_12 sf ( )S f0_12 sf (S)S 94 188 :M f3_12 sf (\)\\{)S f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf (} if and only if there is an inducing path between )S f4_12 sf (A)S f3_12 sf ( and )S f4_12 sf (B)S f3_12 sf (.)S 77 212 :M (Proof. This follows from Lemma 4)S 245 212 :M (, Lemma 5)S 297 212 :M (, Lemma 7)S 349 212 :M (, and Lemma 8)S 421 212 :M ( )S 424 203 9 9 rC gS 1.286 1 scale 329.78 212 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 77 236 :M f3_12 sf 1.145 .114(If )J f4_12 sf (G)S 99 236 :M f3_12 sf <28>S 103 236 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 122 236 :M f3_12 sf .351(,)A f0_12 sf .937(L)A f3_12 sf 1.71 .171(\) is a directed acyclic graph, and in )J 326 236 :M f4_12 sf (G)S 335 236 :M f3_12 sf <28>S 339 236 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 358 236 :M f3_12 sf .329(,)A f0_12 sf .878(L)A f3_12 sf 1.658 .166(\) there is a sequence of)J 59 254 :M .126 .013(vertices )J f4_12 sf (M)S 109 254 :M f3_12 sf .222 .022( \(each of which is in )J 212 254 :M f0_12 sf .131(O)A f3_12 sf .215 .021(\) starting with )J f4_12 sf .103(A)A f3_12 sf .205 .021( and ending with )J f4_12 sf .112(C)A f3_12 sf .16 .016(, and a set of paths )J f4_12 sf (F)S 59 272 :M f3_12 sf .229 .023(such that for every pair of vertices )J 230 272 :M f4_12 sf (I)S 234 272 :M f3_12 sf .167 .017( and )J f4_12 sf .095(J)A f3_12 sf .243 .024( adjacent in )J f4_12 sf (M)S 332 272 :M f3_12 sf .233 .023( in that order there is exactly one)J 59 290 :M .097 .01(inducing path )J 128 290 :M f4_12 sf .062 .006(W )J f3_12 sf .145 .014(between )J 185 290 :M f4_12 sf (I)S 189 290 :M f3_12 sf .119 .012( and )J f4_12 sf .068(J)A f3_12 sf .081 .008( in )J f4_12 sf .093(F)A f3_12 sf .123 .012(, and if )J 277 290 :M f4_12 sf .057(J)A f3_12 sf ( )S f1_12 sf S 292 290 :M f3_12 sf ( )S f4_12 sf .102(C)A f3_12 sf .136 .014( then )J 330 290 :M f4_12 sf .145 .015(W )J 344 290 :M f3_12 sf .118 .012(is into )J 377 290 :M f4_12 sf .056(J)A f3_12 sf .098 .01(, and if )J f4_12 sf (I)S 423 290 :M f3_12 sf .16 .016( )J 427 290 :M f1_12 sf S 434 290 :M f3_12 sf S f4_12 sf .063(A)A f3_12 sf .092 .009( then )J 471 290 :M f4_12 sf .083 .008(W )J f3_12 sf .061(is)A 59 308 :M .15 .015(into )J 81 308 :M f4_12 sf (I)S 85 308 :M f3_12 sf .168 .017(, and )J 112 308 :M f4_12 sf (I)S 116 308 :M f3_12 sf .102 .01( and )J f4_12 sf .058(J)A f3_12 sf .164 .016( are ancestors of {)J 232 308 :M f4_12 sf .057(A)A f3_12 sf (,)S f4_12 sf .062(C)A f3_12 sf .062 .006(} )J 259 308 :M f1_12 sf .103A f3_12 sf ( )S f0_12 sf (S)S 278 308 :M f3_12 sf .162 .016(, then )J 309 308 :M f4_12 sf (F)S f3_12 sf .05 .005( is an )J f0_12 sf .235 .023(inducing sequence)J 439 308 :M f3_12 sf .125 .013( between )J f4_12 sf (A)S 59 326 :M f3_12 sf (and )S f4_12 sf (C)S f3_12 sf (.)S 77 350 :M f0_12 sf .113 .011(Lemma 10:)J 135 350 :M f3_12 sf .177 .018( If )J 150 350 :M f4_12 sf (G)S 159 350 :M f3_12 sf <28>S 163 350 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 182 350 :M f3_12 sf (,)S f0_12 sf .058(L)A f3_12 sf .126 .013(\) is a directed acyclic graph and there is an inducing sequence)J 59 368 :M f4_12 sf .199(F)A f3_12 sf .446 .045( between )J 114 368 :M f4_12 sf .398(A)A f3_12 sf .507 .051( and )J f4_12 sf .398(B)A f3_12 sf .362 .036( in )J 169 368 :M f4_12 sf (G)S 178 368 :M f3_12 sf <28>S 182 368 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 201 368 :M f3_12 sf .099(,)A f0_12 sf .265(L)A f3_12 sf .371 .037(\), then in )J f4_12 sf (G)S 269 368 :M f3_12 sf <28>S 273 368 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 292 368 :M f3_12 sf .078(,)A f0_12 sf .208(L)A f3_12 sf .443 .044(\) a subpath of the concatenation of the)J 59 386 :M .133 .013(paths in )J f4_12 sf .07(F)A f3_12 sf .13 .013( is an inducing path )J f4_12 sf (T)S 212 386 :M f3_12 sf .147 .015( between )J 259 386 :M f4_12 sf .081(A)A f3_12 sf .103 .01( and )J f4_12 sf .089(C)A f3_12 sf .13 .013( such that if the path in )J f4_12 sf .081(F)A f3_12 sf .182 .018( between )J 465 386 :M f4_12 sf .055(A)A f3_12 sf .116 .012( and)J 59 404 :M 1.201 .12(its successor in )J 141 404 :M f4_12 sf (M)S 151 404 :M f3_12 sf 1.208 .121( is into )J f4_12 sf .969(A)A f3_12 sf 1.409 .141( then )J 230 404 :M f4_12 sf (U)S 239 404 :M f3_12 sf 1.453 .145( is into )J 281 404 :M f4_12 sf .697(A)A f3_12 sf 1.06 .106(, and if the path in )J f4_12 sf .697(F)A f3_12 sf 1.508 .151( between )J f4_12 sf .761(C)A f3_12 sf 1.221 .122( and its)J 59 422 :M (predecessor in )S 131 422 :M f4_12 sf (M)S 141 422 :M f3_12 sf ( is into )S 177 422 :M f4_12 sf (C)S f3_12 sf ( then )S 212 422 :M f4_12 sf (U)S 221 422 :M f3_12 sf ( is into )S 257 422 :M f4_12 sf (C)S f3_12 sf (.)S 77 446 :M (Proof)S 104 446 :M f0_12 sf (.)S f3_12 sf .038 .004( Suppose that in )J f4_12 sf (G)S 196 446 :M f3_12 sf <28>S 200 446 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 219 446 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf .04 .004(\) there is a sequence )J 331 446 :M f4_12 sf (M)S 341 446 :M f3_12 sf .045 .005( of vertices in )J 410 446 :M f0_12 sf (O)S f3_12 sf .033 .003( starting with )J f4_12 sf (A)S 59 464 :M f3_12 sf 1.874 .187(and ending with )J f4_12 sf .859(C)A f3_12 sf 1.223 .122(, and a set of paths )J f4_12 sf .787(F)A f3_12 sf 1.564 .156( such that for every pair of vertices )J 455 464 :M f4_12 sf (I)S 459 464 :M f3_12 sf 1.886 .189( and )J 487 464 :M f4_12 sf (J)S 59 482 :M f3_12 sf .025 .003(adjacent in )J f4_12 sf (M)S 124 482 :M f3_12 sf .032 .003( there is exactly one inducing path )J 292 482 :M f4_12 sf (W )S f3_12 sf .043 .004(between )J 349 482 :M f4_12 sf (I)S 353 482 :M f3_12 sf .034 .003( and )J f4_12 sf (J)S f3_12 sf .035 .004(, and if )J 418 482 :M f4_12 sf (J )S f1_12 sf S 433 482 :M f3_12 sf ( )S f4_12 sf (C)S f3_12 sf .04 .004( then )J 471 482 :M f4_12 sf (W )S f3_12 sf (is)S 59 500 :M .143 .014(into )J 81 500 :M f4_12 sf .076(J)A f3_12 sf .138 .014(, and if )J 124 500 :M f4_12 sf (I)S 128 500 :M f3_12 sf .057 .006( )J f1_12 sf S 138 500 :M f3_12 sf S f4_12 sf .088(A)A f3_12 sf .123 .012( then )J f4_12 sf .142 .014(W )J 189 500 :M f3_12 sf .15 .015(is into )J f4_12 sf .094(A)A f3_12 sf .199 .02(, and)J f4_12 sf .075 .007( I)J f3_12 sf .125 .012( and )J 283 500 :M f4_12 sf (J)S f3_12 sf .129 .013( are ancestors of either )J 401 500 :M f4_12 sf .124(A)A f3_12 sf .113 .011( or )J f4_12 sf .135(C)A f3_12 sf .117 .012( or )J 449 500 :M f0_12 sf (S)S 456 500 :M f3_12 sf .108 .011(. Let )J f4_12 sf .12<54D5>A 59 518 :M f3_12 sf -.005(be the concatenation of the paths in )A 232 518 :M f4_12 sf (F)S f3_12 sf (. )S 245 518 :M f4_12 sf <54D5>S 256 518 :M f3_12 sf -.005( may not be an acyclic undirected path because it)A 59 536 :M .332 .033(might contain undirected cycles. Let )J 240 536 :M f4_12 sf (T)S 247 536 :M f3_12 sf .377 .038( be an acyclic undirected subpath of )J 427 536 :M f4_12 sf <54D5>S 438 536 :M f3_12 sf .34 .034( between )J f4_12 sf (A)S 59 554 :M f3_12 sf .392 .039(and )J f4_12 sf .216(C)A f3_12 sf .439 .044(. We will now show that except for the endpoints, every vertex in )J 413 554 :M f0_12 sf .504(O)A f3_12 sf .162 .016( )J 426 554 :M f1_12 sf .325A f3_12 sf .096 .01( )J f0_12 sf (S)S 445 554 :M f3_12 sf .579 .058( on )J 465 554 :M f4_12 sf (T)S 472 554 :M f3_12 sf .555 .056( is a)J 59 572 :M (collider, and every collider on )S f4_12 sf (T)S 213 572 :M f3_12 sf ( is an ancestor of {)S f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (C)S f3_12 sf (} )S 330 572 :M f1_12 sf S f3_12 sf ( )S f0_12 sf (S)S 349 572 :M f3_12 sf (.)S 77 596 :M .844 .084(If )J 89 596 :M f4_12 sf .495(V)A f3_12 sf .703 .07( is a vertex in )J 169 596 :M f0_12 sf .605(O)A f3_12 sf .177 .018( )J f1_12 sf .598A f3_12 sf .177 .018( )J f0_12 sf (S)S 202 596 :M f3_12 sf .687 .069( that is on )J f4_12 sf (T)S 263 596 :M f3_12 sf .657 .066( but that is not equal to )J f4_12 sf .409(A)A f3_12 sf .372 .037( or )J f4_12 sf .447(C)A f3_12 sf .93 .093(, every edge on)J 59 614 :M .757 .076(every path in )J 129 614 :M f4_12 sf .555(F)A f3_12 sf .711 .071( is into )J 175 614 :M f4_12 sf .368(V)A f3_12 sf .776 .078(. Hence, every edge on )J f4_12 sf .441 .044(T )J 311 614 :M f3_12 sf .658 .066(that contains )J 378 614 :M f4_12 sf .555(V)A f3_12 sf .711 .071( is into )J 424 614 :M f4_12 sf .241(V)A f3_12 sf .623 .062( because the)J 59 632 :M (edges on )S f4_12 sf (T)S 111 632 :M f3_12 sf ( are a subset of the edges on inducing paths in )S 335 632 :M f4_12 sf (F)S f3_12 sf (.)S 77 656 :M (Let )S 96 656 :M f4_12 sf (R)S f3_12 sf ( and )S f4_12 sf (H)S 135 656 :M f3_12 sf -.005( be the endpoints of a path )A 265 656 :M f4_12 sf (W)S 275 656 :M f3_12 sf ( in)S f4_12 sf ( )S 290 656 :M f3_12 sf -.004(F. We will now show that every vertex on)A 59 674 :M f4_12 sf .111 .011(W )J f3_12 sf .155 .015(is an ancestor of {)J 160 674 :M f4_12 sf .056(A)A f3_12 sf (,)S f4_12 sf .061(C)A f3_12 sf .06 .006(} )J 187 674 :M f1_12 sf .101A f3_12 sf ( )S f0_12 sf (S)S 206 674 :M f3_12 sf .134 .013(. By hypothesis, )J 287 674 :M f4_12 sf .073(R)A f3_12 sf .128 .013( is an ancestor of {)J f4_12 sf .073(A)A f3_12 sf (,)S f4_12 sf .08(C)A f3_12 sf .079 .008(} )J 412 674 :M f1_12 sf .101A f3_12 sf ( )S f0_12 sf (S)S 431 674 :M f3_12 sf .165 .016(, and )J 458 674 :M f4_12 sf (H)S 467 674 :M f3_12 sf .165 .016( is an)J endp %%Page: 37 37 %%BeginPageSetup initializepage (peter; page: 37 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (37)S gR gS 0 0 552 730 rC 59 56 :M f3_12 sf .187 .019(ancestor of {)J f4_12 sf .073(A)A f3_12 sf (,)S f4_12 sf .08(C)A f3_12 sf .073 .007(} )J f1_12 sf .092A f3_12 sf ( )S 161 56 :M f0_12 sf (S)S 168 56 :M f3_12 sf .167 .017(. Because )J f4_12 sf .111 .011(W )J f3_12 sf .17 .017(is an inducing path, every collider on )J f4_12 sf .111 .011(W )J f3_12 sf .202 .02(is an ancestor)J 59 74 :M .326 .033(of {)J 79 74 :M f4_12 sf (R)S f3_12 sf (,)S f4_12 sf (H)S 98 74 :M f3_12 sf .295 .03(} )J f1_12 sf .373A f3_12 sf .121 .012( )J 120 74 :M f0_12 sf (S)S 127 74 :M f3_12 sf .281 .028(, and hence an ancestor of {)J f4_12 sf .136(A)A f3_12 sf .056(,)A f4_12 sf .148(C)A f3_12 sf .147 .015(} )J 290 74 :M f1_12 sf .207A f3_12 sf .061 .006( )J f0_12 sf (S)S 309 74 :M f3_12 sf .269 .027(. Every non-collider on )J 425 74 :M f4_12 sf .21 .021(W )J f3_12 sf .329 .033(is either an)J 59 92 :M 1.311 .131(ancestor of )J f4_12 sf .55(R)A f3_12 sf .5 .05( or )J f4_12 sf (H)S 153 92 :M f3_12 sf 1.086 .109(, or an ancestor of a collider on )J f4_12 sf (W)S 328 92 :M f3_12 sf 1.129 .113(. Hence every vertex on )J 454 92 :M f4_12 sf 1.011 .101(W )J f3_12 sf 1.489 .149(is an)J 59 110 :M .034 .003(ancestor of either {)J f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (C)S f3_12 sf (} )S 178 110 :M f1_12 sf S f3_12 sf ( )S f0_12 sf (S)S 197 110 :M f3_12 sf .037 .004(. It follows that every collider on )J 359 110 :M f4_12 sf (T)S 366 110 :M f3_12 sf .036 .004( is an ancestor of {)J f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (C)S f3_12 sf (} )S 483 110 :M f1_12 sf S 59 128 :M f0_12 sf (S)S 66 128 :M f3_12 sf (, because the vertices on )S 186 128 :M f4_12 sf (T)S 193 128 :M f3_12 sf ( are a subset of the vertices on paths in )S 382 128 :M f4_12 sf <54D5>S 393 128 :M f3_12 sf (.)S 77 152 :M .256 .026(By definition, )J 148 152 :M f4_12 sf (T)S 155 152 :M f3_12 sf .297 .03( is an inducing path between )J 298 152 :M f4_12 sf .208(A)A f3_12 sf .276 .028( and )J 328 152 :M f4_12 sf .157(C)A f3_12 sf .285 .028(. Suppose the path in )J 442 152 :M f4_12 sf .064(F)A f3_12 sf .219 .022( between)J 59 170 :M f4_12 sf (A)S f3_12 sf .076 .008( and its successor is into )J f4_12 sf (A)S f3_12 sf .063 .006(. If the edge on )J 268 170 :M f4_12 sf (T)S 275 170 :M f3_12 sf .094 .009( with endpoint )J f4_12 sf (A)S f3_12 sf .062 .006( is on the path in )J 438 170 :M f4_12 sf (F)S f3_12 sf .065 .007( on which)J 59 188 :M f4_12 sf (A)S f3_12 sf .022 .002( is an endpoint, then )J 167 188 :M f4_12 sf (T)S 174 188 :M f3_12 sf .027 .003( is into )J 210 188 :M f4_12 sf (A)S f3_12 sf .021 .002( because by hypothesis that inducing path is into )J 453 188 :M f4_12 sf (A)S f3_12 sf .02 .002(. If the)J 59 206 :M .753 .075(edge on )J f4_12 sf (T)S 109 206 :M f3_12 sf .918 .092( with endpoint )J f4_12 sf .419(A)A f3_12 sf .799 .08( is on an inducing path in which )J 359 206 :M f4_12 sf .426(A)A f3_12 sf .816 .082( is not an endpoint of the)J 59 224 :M .476 .048(path, then )J 111 224 :M f4_12 sf (T)S 118 224 :M f3_12 sf .564 .056( is into )J 156 224 :M f4_12 sf .308(A)A f3_12 sf .637 .064( because )J f4_12 sf .308(A)A f3_12 sf .307 .031( is in )J 244 224 :M f0_12 sf .248(O)A f3_12 sf .448 .045(, and hence a collider on every inducing path for)J 59 242 :M 1.12 .112(which it is not an endpoint. Similarly, )J f4_12 sf (T)S 263 242 :M f3_12 sf 1.44 .144( is into )J 305 242 :M f4_12 sf .769(C)A f3_12 sf .666 .067( if in )J f4_12 sf .704(F)A f3_12 sf 1.445 .145( the path between )J f4_12 sf .769(C)A f3_12 sf 1.233 .123( and its)J 59 260 :M (predecessor is into )S f4_12 sf (A)S f3_12 sf (. )S 164 251 9 9 rC gS 1.286 1 scale 127.556 260 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 77 284 :M f0_12 sf .416 .042(Lemma )J 120 284 :M .565 .057(11: )J f3_12 sf .381 .038(If )J 151 284 :M f4_12 sf (G)S 160 284 :M f3_12 sf <28>S 164 284 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 183 284 :M f3_12 sf .075(,)A f0_12 sf .201(L)A f3_12 sf .385 .039(\) is a directed acyclic graph, )J 336 284 :M f4_12 sf .323(A)A f3_12 sf .41 .041( and )J f4_12 sf .323(B)A f3_12 sf .392 .039( are in )J 409 284 :M f0_12 sf .263(O)A f3_12 sf .285 .029(, and )J f4_12 sf (G)S 454 284 :M f3_12 sf <28>S 458 284 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 477 284 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf <29>S 59 302 :M .371 .037(contains an inducing path )J 188 302 :M f4_12 sf (U)S 197 302 :M f3_12 sf .566 .057( between )J f4_12 sf .262(A)A f3_12 sf .333 .033( and )J f4_12 sf .262(B)A f3_12 sf .359 .036( that is out of )J f4_12 sf .262(A)A f3_12 sf .412 .041( and into )J 405 302 :M f4_12 sf .261(B)A f3_12 sf .361 .036(, and )J f4_12 sf .261(A)A f3_12 sf .399 .04( is not an)J 59 320 :M (ancestor of )S 115 320 :M f0_12 sf (S)S 122 320 :M f3_12 sf (, then there is a directed path from )S 290 320 :M f4_12 sf (A)S f3_12 sf ( to )S f4_12 sf (B)S f3_12 sf ( in )S f4_12 sf (G)S 343 320 :M f3_12 sf <28>S 347 320 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 366 320 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (\).)S 77 344 :M .944 .094(Proof. Suppose that )J f4_12 sf .356(A)A f3_12 sf .607 .061( is not an ancestor of )J 294 344 :M f0_12 sf (S)S 301 344 :M f3_12 sf .647 .065(, and )J f4_12 sf (U)S 338 344 :M f3_12 sf .762 .076( is an inducing path between )J 485 344 :M f4_12 sf (A)S 59 362 :M f3_12 sf .068 .007(and )J f4_12 sf (B)S f3_12 sf .048 .005( that is out of )J 153 362 :M f4_12 sf (A)S f3_12 sf .049 .005( and into )J f4_12 sf (B)S f3_12 sf .029 .003(. If )J f4_12 sf (U)S 238 362 :M f3_12 sf .05 .005( does not contain a collider, then )J 398 362 :M f4_12 sf (U)S 407 362 :M f3_12 sf .051 .005( is a directed path)J 59 380 :M .438 .044(from )J 86 380 :M f4_12 sf .323(A)A f3_12 sf .282 .028( to )J f4_12 sf .323(B)A f3_12 sf .3 .03(. If )J f4_12 sf (U)S 143 380 :M f3_12 sf .414 .041( does contain a collider, let )J 279 380 :M f4_12 sf (D)S 288 380 :M f3_12 sf .418 .042( be the first collider after )J 414 380 :M f4_12 sf .117(A)A f3_12 sf .327 .033(. By definition)J 59 398 :M .258 .026(of inducing path, there is either a directed path from )J 316 398 :M f4_12 sf (D)S 325 398 :M f3_12 sf .345 .035( to )J 340 398 :M f4_12 sf .218(B)A f3_12 sf .162 .016(, )J 354 398 :M f4_12 sf (D)S 363 398 :M f3_12 sf .305 .031( to a member of )J 444 398 :M f0_12 sf (S)S 451 398 :M f3_12 sf .331 .033(, or )J 471 398 :M f4_12 sf (D)S 480 398 :M f3_12 sf .331 .033( to)J 59 416 :M f4_12 sf .701(A)A f3_12 sf 1.285 .129(. There is no path from )J f4_12 sf (D)S 200 416 :M f3_12 sf .737 .074( to )J f4_12 sf .845(A)A f3_12 sf 1.603 .16( because there is no cycle in )J 379 416 :M f4_12 sf (G)S 388 416 :M f3_12 sf <28>S 392 416 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 411 416 :M f3_12 sf .247(,)A f0_12 sf .658(L)A f3_12 sf 1.223 .122(\). There is no)J 59 434 :M .065 .006(directed path from )J 151 434 :M f4_12 sf (D)S 160 434 :M f3_12 sf .08 .008( to a member of )J 240 434 :M f0_12 sf (S)S 247 434 :M f3_12 sf .082 .008(, because )J f4_12 sf (A)S f3_12 sf .065 .006( is an ancestor of )J 386 434 :M f4_12 sf (D)S 395 434 :M f3_12 sf .069 .007(, but not an ancestor)J 59 452 :M .233 .023(of a member of )J f0_12 sf (S)S 144 452 :M f3_12 sf .238 .024(. Hence there is a directed path from )J 325 452 :M f4_12 sf (D)S 334 452 :M f3_12 sf .111 .011( to )J f4_12 sf .127(B)A f3_12 sf .284 .028(. Because )J f4_12 sf (U)S 415 452 :M f3_12 sf .279 .028( is out of )J 462 452 :M f4_12 sf .083(A)A f3_12 sf .19 .019(, and)J 59 470 :M f4_12 sf (D)S 68 470 :M f3_12 sf 1.434 .143( is the first collider after )J 200 470 :M f4_12 sf .797(A)A f3_12 sf 1.535 .154(, there is a directed path from )J f4_12 sf .797(A)A f3_12 sf .725 .073( to )J 392 470 :M f4_12 sf (D)S 401 470 :M f3_12 sf 1.476 .148(. Hence there is a)J 59 488 :M (directed path from )S 151 488 :M f4_12 sf (A)S f3_12 sf ( to )S f4_12 sf (B)S f3_12 sf (. )S 186 479 9 9 rC gS 1.286 1 scale 144.668 488 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 77 511 :M f4_12 sf .141(V)A f3_12 sf .058 .006( )J 88 511 :M f1_12 sf S 97 511 :M f2_12 sf ( )S f0_12 sf .046(D-SEP)A 135 511 :M f3_12 sf <28>S 139 511 :M f4_12 sf .059(A)A f3_12 sf (,)S f4_12 sf .059(B)A f3_12 sf .099 .01(\) in DAG )J f4_12 sf (G)S 213 511 :M f3_12 sf <28>S 217 511 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 236 511 :M f3_12 sf (,)S f0_12 sf .076(L)A f3_12 sf .132 .013(\) if and only if there is a sequence of vertices )J 469 511 :M f4_12 sf (U)S 478 511 :M f3_12 sf .086 .009( )J f1_12 sf S 59 529 :M f3_12 sf (<)S 66 529 :M f4_12 sf (A)S f3_12 sf S f4_12 sf (V)S f3_12 sf (> in )S f0_12 sf (O )S f1_12 sf S f0_12 sf (Ancestors)S 189 529 :M f3_12 sf (\({)S 199 529 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf (} )S 225 529 :M f1_12 sf S f0_12 sf (S)S 244 529 :M f3_12 sf (\) such that)S 77 554 :M (1)S 83 554 :M (.)S 86 554 :M ( )S 95 554 :M (there is an inducing path between every consecutive pair of vertices on )S f4_12 sf (U)S 77 572 :M f3_12 sf (2)S 83 572 :M (.)S 86 572 :M 5 .5( )J 95 572 :M .863 .086(with the exception of the endpoints every vertex on )J 357 572 :M f4_12 sf (U)S 366 572 :M f3_12 sf .955 .095( is not an ancestor of the)J 95 584 :M (vertices preceding or succeeding it in the sequence )S 342 584 :M f4_12 sf (U)S 351 584 :M f3_12 sf ( nor an ancestor of )S f0_12 sf (S)S 450 584 :M f3_12 sf (.)S 77 607 :M f0_12 sf .449 .045(Lemma 12:)J 136 607 :M f3_12 sf .588 .059( If there is some subset )J f0_12 sf .602 .06(W )J 270 607 :M f1_12 sf S 279 607 :M f3_12 sf .094 .009( )J f0_12 sf .322(O)A f3_12 sf .157(\\{)A f4_12 sf .253(A)A f3_12 sf .103(,)A f4_12 sf .253(B)A f3_12 sf .46 .046(} such that )J 375 607 :M f4_12 sf .208(A)A f3_12 sf .265 .026( and )J f4_12 sf .208(B)A f3_12 sf .616 .062( are d-separated)J 59 625 :M (by )S f0_12 sf (W )S f1_12 sf S f0_12 sf (S)S 108 625 :M f3_12 sf ( then )S 135 625 :M f4_12 sf (A)S f3_12 sf ( and )S f4_12 sf (B)S f3_12 sf ( are d)S 199 625 :M (-separated given )S 280 625 :M f0_12 sf (D-SEP)S 315 625 :M f3_12 sf <28>S 319 625 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf (\) )S 343 625 :M f1_12 sf S f0_12 sf (S)S 362 625 :M f3_12 sf (.)S 77 649 :M .334 .033(Proof. Suppose there is some subset )J 256 649 :M f0_12 sf .481 .048(W )J 272 649 :M f1_12 sf S 281 649 :M f3_12 sf .061 .006( )J f0_12 sf .211(O)A f3_12 sf .103(\\{)A f4_12 sf .165(A)A f3_12 sf .068(,)A f4_12 sf .165(B)A f3_12 sf .301 .03(} such that )J 376 649 :M f4_12 sf .136(A)A f3_12 sf .173 .017( and )J f4_12 sf .136(B)A f3_12 sf .403 .04( are d-separated)J 59 667 :M .724 .072(by )J f0_12 sf .784 .078(W )J f1_12 sf .638 .064J f0_12 sf (S)S 111 667 :M f3_12 sf .645 .065(, but )J f4_12 sf .505(A)A f3_12 sf .643 .064( and )J f4_12 sf .505(B)A f3_12 sf .735 .073( are d)J 205 667 :M .393 .039(-connected given )J f0_12 sf .148(D-SEP)A 326 667 :M f3_12 sf <28>S 330 667 :M f4_12 sf .344(A)A f3_12 sf .141(,)A f4_12 sf .344(B)A f3_12 sf .274 .027(\) )J f1_12 sf .478 .048J f0_12 sf (S)S 375 667 :M f3_12 sf .875 .088(. Let )J f4_12 sf .784 .078(P )J f1_12 sf .793 .079J 424 667 :M f3_12 sf (<)S 431 667 :M f4_12 sf .254(A)A f3_12 sf .26<2CC9>A f4_12 sf .254(B)A f3_12 sf .407 .041(> be a)J 59 685 :M (path d-connecting )S f4_12 sf (A)S f3_12 sf ( and )S f4_12 sf (B)S f3_12 sf ( given )S 218 685 :M f0_12 sf (D-SEP)S 253 685 :M f3_12 sf <28>S 257 685 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf (\) )S 281 685 :M f1_12 sf S f0_12 sf (S)S 300 685 :M f3_12 sf (.)S endp %%Page: 38 38 %%BeginPageSetup initializepage (peter; page: 38 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (38)S gR gS 0 0 552 730 rC 77 50 :M f3_12 sf .7 .07(If every vertex in )J f0_12 sf .449(O)A f3_12 sf .376 .038( on )J 195 50 :M f4_12 sf .263(P)A f3_12 sf .583 .058( occurs as a collider then every observed vertex on )J 457 50 :M f4_12 sf .407(P)A f3_12 sf .563 .056( is an)J 59 67 :M .223 .022(ancestor of )J f4_12 sf .094(A)A f3_12 sf .089 .009( or )J 139 67 :M f4_12 sf .123(B)A f3_12 sf .112 .011( or )J f0_12 sf (S)S 169 67 :M f3_12 sf .187 .019( \(since )J 205 67 :M f0_12 sf (D-SEP)S 240 67 :M f3_12 sf <28>S 244 67 :M f4_12 sf .119(A)A f3_12 sf (,)S f4_12 sf .119(B)A f3_12 sf .095 .009(\) )J f1_12 sf .166 .017J f0_12 sf .143 .014(S )J 290 67 :M f3_12 sf <29>S 294 67 :M f1_12 sf .23 .023J 307 67 :M f0_12 sf (Ancestors)S 358 67 :M f3_12 sf (\({)S 368 67 :M f4_12 sf .082(A)A f3_12 sf (,)S f4_12 sf .082(B)A f3_12 sf .081 .008(} )J f1_12 sf .114 .011J f0_12 sf (S)S 413 67 :M f3_12 sf .165 .016(\).\) Hence in this)J 59 85 :M 1.302 .13(case )J 85 85 :M f4_12 sf .566(P)A f3_12 sf 1.443 .144( constitutes an inducing path between )J f4_12 sf .566(A)A f3_12 sf .72 .072( and )J f4_12 sf .566(B)A f3_12 sf 1.002 .1(, and so there is no subset )J 465 85 :M f0_12 sf 1.658 .166(W )J 482 85 :M f1_12 sf S 59 103 :M f0_12 sf (O)S f3_12 sf (\\{)S f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf (} such that )S 149 103 :M f4_12 sf (A)S f3_12 sf ( and )S f4_12 sf (B)S f3_12 sf ( are d)S 213 103 :M (-separated by some )S 308 103 :M f0_12 sf (W )S f1_12 sf S f0_12 sf (S)S 342 103 :M f3_12 sf (.)S 77 127 :M 1.384 .138(Hence there is some vertex )J 221 127 :M f4_12 sf (O)S 230 127 :M f3_12 sf .788 .079( )J f1_12 sf 3.037 .304J 249 127 :M f0_12 sf .842(O)A f3_12 sf 1.116 .112(, such that )J f4_12 sf (O)S 325 127 :M f3_12 sf 1.482 .148( is a non-collider on )J 435 127 :M f4_12 sf .286(P)A f3_12 sf 1.01 .101(. Suppose)J 59 145 :M .106 .011(without loss that )J 142 145 :M f4_12 sf (O)S 151 145 :M f3_12 sf .112 .011( is the first such vertex on )J f4_12 sf .064(P)A f3_12 sf .125 .013(. We will now show that )J f4_12 sf .093 .009(O )J 419 145 :M f1_12 sf .151 .015J 431 145 :M f0_12 sf (D-SEP)S 466 145 :M f3_12 sf <28>S 470 145 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf <29>S 59 163 :M f1_12 sf S f0_12 sf (S)S 78 163 :M f3_12 sf (.)S 77 187 :M .603 .06(Consider the subpath )J f4_12 sf .216(P)A f3_12 sf <28>S 195 187 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (O)S 214 187 :M f3_12 sf .602 .06(\). Let <)J f4_12 sf .377(C)A f3_9 sf 0 2 rm (1)S 0 -2 rm 265 187 :M f3_12 sf .193<2CC9>A f4_12 sf .206(C)A f3_9 sf 0 2 rm .181(m)A 0 -2 rm f3_12 sf .564 .056(> denote the \(possibly empty\) sequence)J 59 206 :M .078 .008(of colliders on )J 132 206 :M f4_12 sf (P)S f3_12 sf <28>S 143 206 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (O)S 162 206 :M f3_12 sf .06 .006(\), which are ancestors of )J f0_12 sf .033(D-SEP)A 318 206 :M f3_12 sf <28>S 322 206 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf .069 .007(\), but not ancestors of )J 447 206 :M f0_12 sf (S)S 454 206 :M f3_12 sf .073 .007(. Hence)J 59 223 :M .089 .009(there is a directed path \(possibly of length 0\) from )J f4_12 sf (C)S f4_10 sf 0 2 rm (i)S 0 -2 rm 315 223 :M f1_12 sf S 327 223 :M f3_12 sf S f1_12 sf S 351 223 :M f4_12 sf (D)S 360 225 :M f4_10 sf (i)S 363 223 :M f3_12 sf .105 .011(, where )J 402 223 :M f4_12 sf (D)S 411 225 :M f4_10 sf .074 .007(i )J f1_12 sf 0 -2 rm .178 .018J 0 2 rm 428 223 :M f0_12 sf (D-SEP)S 463 223 :M f3_12 sf <28>S 467 223 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf (\).)S 59 241 :M .18 .018(Let )J f4_12 sf .098(F)A f3_10 sf 0 2 rm (i)S 0 -2 rm 88 241 :M f3_12 sf .266 .027( be the first measured vertex on the path )J 288 241 :M f4_12 sf (C)S f4_10 sf 0 2 rm (i)S 0 -2 rm 299 243 :M f3_10 sf .387 .039( )J 302 241 :M f1_12 sf S 314 241 :M f3_12 sf .173A f1_12 sf .195 .019J 342 241 :M f4_12 sf (D)S 351 243 :M f4_10 sf (i)S 354 241 :M f3_12 sf .298 .03(; such an )J 401 241 :M f4_12 sf (F)S f4_10 sf 0 2 rm (i)S 0 -2 rm 411 241 :M f3_12 sf .252 .025( is guaranteed to)J 59 259 :M (exist since )S f4_12 sf (D)S 121 261 :M f4_10 sf (i )S f1_12 sf 0 -2 rm S 0 2 rm 138 259 :M f0_12 sf (O)S f3_12 sf (.)S 77 283 :M .372 .037(We will now show that there is an inducing path between )J f4_12 sf .163(F)A f4_10 sf 0 2 rm (i)S 0 -2 rm 371 283 :M f3_12 sf .325 .032( and )J f4_12 sf .255(F)A f4_10 sf 0 2 rm (i)S 0 -2 rm 405 285 :M f3_10 sf (+)S 411 285 :M f3_9 sf (1)S 416 283 :M f3_12 sf .379 .038(. It follows that)J 59 301 :M .277 .028(no )J 75 301 :M f4_12 sf (F)S f4_7 sf 0 3 rm (i)S 0 -3 rm 84 301 :M f3_12 sf .223 .022( is an ancestor of )J f0_12 sf (S)S 176 301 :M f3_12 sf .237 .024(, because no )J 239 301 :M f4_12 sf (C)S f4_7 sf 0 3 rm (i)S 0 -3 rm 249 301 :M f3_12 sf .244 .024( is an ancestor of )J 335 301 :M f0_12 sf (S)S 342 301 :M f3_12 sf .225 .023(. Consider the path )J 437 301 :M f4_12 sf (Q)S 446 303 :M f4_10 sf (i)S 449 301 :M f3_12 sf .246 .025( formed)J 59 319 :M .12 .012(by concatenating the directed path )J 228 319 :M f4_12 sf (F)S f4_10 sf 0 2 rm (i)S 0 -2 rm 238 321 :M f3_10 sf .201 .02( )J 242 319 :M f1_12 sf S 254 319 :M f3_12 sf .064A f1_12 sf .066 .007J f4_12 sf (C)S f4_10 sf 0 2 rm (i)S 0 -2 rm 292 319 :M f3_12 sf .117 .012(, the subpath )J f4_12 sf .059(P)A f3_12 sf <28>S 367 319 :M f4_12 sf (C)S f4_10 sf 0 2 rm (i)S 0 -2 rm 378 319 :M f3_12 sf (,)S f4_12 sf (C)S f4_10 sf 0 2 rm (i)S 0 -2 rm 392 321 :M f3_10 sf (+)S 398 321 :M f3_9 sf (1)S 403 319 :M f3_12 sf .126 .013(\), and the directed)J 59 337 :M .632 .063(path )J f4_12 sf .321(C)A f4_10 sf 0 2 rm (i)S 0 -2 rm 95 339 :M f3_10 sf (+)S 101 339 :M f3_9 sf .418 .042(1 )J f1_12 sf 0 -2 rm S 0 2 rm 121 337 :M f3_12 sf S f4_12 sf (F)S f4_10 sf 0 2 rm (i)S 0 -2 rm 143 339 :M f3_10 sf (+)S 149 339 :M f3_9 sf (1)S 154 337 :M f3_12 sf .883 .088(. It follows from the construction, that with the exception of )J 460 337 :M f4_12 sf (F)S f4_10 sf 0 2 rm (i)S 0 -2 rm 470 337 :M f3_12 sf 1.062 .106( and)J 59 355 :M f4_12 sf (F)S f4_10 sf 0 2 rm (i)S 0 -2 rm 69 357 :M f3_10 sf (+)S 75 357 :M f3_9 sf (1)S 80 355 :M f3_12 sf .506 .051( the only measured vertices on )J f4_12 sf (Q)S 243 357 :M f4_10 sf (i)S 246 355 :M f3_12 sf .676 .068( are on )J 288 355 :M f4_12 sf (P)S f3_12 sf <28>S 299 355 :M f4_12 sf (C)S f4_10 sf 0 2 rm (i)S 0 -2 rm 310 355 :M f3_12 sf (,)S f4_12 sf (C)S f4_10 sf 0 2 rm (i)S 0 -2 rm 324 357 :M f3_10 sf (+)S 330 357 :M f3_9 sf (1)S 335 355 :M f3_12 sf .507 .051(\). Moreover, since )J 429 355 :M f4_12 sf (O)S 438 355 :M f3_12 sf .58 .058( is the first)J 59 373 :M .442 .044(non-collider on )J 138 373 :M f4_12 sf .26(P)A f3_12 sf .322 .032( that is in )J f0_12 sf .331(O)A f3_12 sf .618 .062(, every measured vertex on )J 341 373 :M f4_12 sf (P)S f3_12 sf <28>S 352 373 :M f4_12 sf (C)S f4_10 sf 0 2 rm (i)S 0 -2 rm 363 373 :M f3_12 sf (,)S f4_12 sf (C)S f4_10 sf 0 2 rm (i)S 0 -2 rm 377 375 :M f3_10 sf (+)S 383 375 :M f3_9 sf (1)S 388 373 :M f3_12 sf .508 .051(\) is a collider. Hence)J 59 391 :M .621 .062(by construction of the sequence <)J f4_12 sf .255(C)A f3_9 sf 0 2 rm (1)S 0 -2 rm 239 391 :M f3_12 sf .247<2CC9>A f4_12 sf .263(C)A f4_9 sf 0 2 rm .214(m)A 0 -2 rm f3_12 sf .595 .059(>, every measured vertex on )J 413 391 :M f4_12 sf (P)S f3_12 sf <28>S 424 391 :M f4_12 sf (C)S f4_10 sf 0 2 rm (i)S 0 -2 rm 435 391 :M f3_12 sf (,)S f4_12 sf (C)S f4_10 sf 0 2 rm (i)S 0 -2 rm 449 393 :M f3_10 sf (+)S 455 393 :M f3_9 sf (1)S 460 391 :M f3_12 sf .854 .085(\) is an)J 59 409 :M (ancestor of )S 115 409 :M f0_12 sf (S)S 122 409 :M f3_12 sf (. Hence )S 162 409 :M f4_12 sf (Q)S 171 411 :M f4_10 sf (i)S 174 409 :M f3_12 sf ( is an inducing path.)S 77 433 :M 1.26 .126(Similarly, by concatenating the path )J f4_12 sf .455(P)A f3_12 sf <28>S 276 433 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (C)S f3_9 sf 0 2 rm (1)S 0 -2 rm 299 433 :M f3_12 sf 1.728 .173(\) and the path )J 378 433 :M f4_12 sf .702(C)A f3_9 sf 0 2 rm .493 .049(1 )J 0 -2 rm f1_12 sf S 407 433 :M f3_12 sf .724A f1_12 sf .746 .075J f4_12 sf .442(F)A f3_9 sf 0 2 rm (1)S 0 -2 rm 448 433 :M f3_12 sf 1.728 .173(, and by)J 59 451 :M .228 .023(concatenating the path )J f4_12 sf .08(F)A f4_10 sf 0 2 rm .088 .009(m )J 0 -2 rm f1_12 sf S 200 451 :M f3_12 sf .17A f1_12 sf .191 .019J 228 451 :M f4_12 sf .158(C)A f4_10 sf 0 2 rm .143(m)A 0 -2 rm f3_12 sf .192 .019( and )J 266 451 :M f4_12 sf (P)S f3_12 sf <28>S 277 451 :M f4_12 sf (C)S f4_10 sf 0 2 rm (m)S 0 -2 rm f3_12 sf (,)S f4_12 sf (O)S 304 451 :M f3_12 sf .256 .026(\), we may form inducing paths )J 457 451 :M f4_12 sf (Q)S 466 453 :M f3_9 sf (0)S 471 451 :M f3_12 sf .293 .029( and)J 59 469 :M f4_12 sf (Q)S 68 471 :M f4_10 sf (m)S f3_12 sf 0 -2 rm ( between )S 0 2 rm 121 469 :M f4_12 sf (A)S f3_12 sf ( and )S f4_12 sf (F)S f3_9 sf 0 2 rm (1)S 0 -2 rm 163 469 :M f3_12 sf (, and between )S f4_12 sf (F)S f4_10 sf 0 2 rm (m)S 0 -2 rm f3_12 sf ( and )S f4_12 sf (O)S 278 469 :M f3_12 sf ( respectively.)S 77 494 :M .181 .018(Note that all of the inducing path )J 241 494 :M f4_12 sf (Q)S 250 496 :M f4_10 sf (k)S f3_12 sf 0 -2 rm .167 .017( that we have constructed are into the )J 0 2 rm f4_12 sf 0 -2 rm .08(F)A 0 2 rm f4_10 sf (i)S 448 494 :M f3_12 sf .141 .014( vertices.)J 59 512 :M .197 .02(The sequence )J f4_12 sf .081 .008(R )J f1_12 sf .082 .008J 148 512 :M f3_12 sf (<)S 155 512 :M f4_12 sf .19 .019(A )J 166 512 :M f1_12 sf .19 .019J 176 512 :M f4_12 sf (F)S f3_9 sf 0 2 rm (0)S 0 -2 rm 188 512 :M f3_12 sf (,)S f4_12 sf (F)S f3_9 sf 0 2 rm (1)S 0 -2 rm 203 512 :M f3_12 sf .046<2CC9>A f4_12 sf (F)S f4_10 sf 0 2 rm (m)S 0 -2 rm f3_12 sf (,)S f4_12 sf (F)S f4_10 sf 0 2 rm (m)S 0 -2 rm f3_9 sf 0 2 rm .056 .006(+1 )J 0 -2 rm f1_12 sf .053 .005J 271 512 :M f4_12 sf (O)S 280 512 :M f3_12 sf .137 .014(> thus consists of a sequence each of which)J 59 530 :M 1.128 .113(is an ancestor of )J 147 530 :M f4_12 sf .884(A)A f3_12 sf .804 .08( or )J f4_12 sf .884(B)A f3_12 sf .804 .08( or )J f0_12 sf (S)S 206 530 :M f3_12 sf 1.066 .107(, and such that each consecutive pair in the sequence is)J 59 548 :M .103 .01(connected by an inducing path. Hence this sequence satisfies the first condition necessary)J 59 566 :M (to show that )S 121 566 :M f4_12 sf (O )S 133 566 :M f1_12 sf S 145 566 :M f0_12 sf (D-SEP)S 180 566 :M f3_12 sf <28>S 184 566 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf (\).)S 77 589 :M 1.764 .176(We now construct a subsequence of )J f4_12 sf (T)S 276 589 :M f3_12 sf 2.868 .287( )J 282 589 :M f1_12 sf S 289 589 :M f3_12 sf .754 .075(J f4_12 sf .39(F)A f1_10 sf 0 2 rm .335(a)A 0 -2 rm f3_9 sf 0 2 rm .186(\(0\))A 0 -2 rm f3_12 sf .159(,)A f4_12 sf .39(F)A f1_10 sf 0 2 rm .335(a)A 0 -2 rm f3_9 sf 0 2 rm .186(\(1\))A 0 -2 rm f3_12 sf .638A f4_12 sf .39(F)A f1_10 sf 0 2 rm .335(a)A 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 399 591 :M f3_10 sf (t)S 402 591 :M f3_9 sf 1.691 .169(\) )J f1_12 sf 0 -2 rm 3.371 .337J 0 2 rm 423 589 :M f4_12 sf (O)S 432 589 :M f3_12 sf 1.366 .137(> satisfying)J 59 608 :M (property \(ii\) as follows:)S 77 632 :M (1)S 83 632 :M (.)S 86 632 :M ( )S 95 632 :M (Let )S 114 632 :M f5_12 sf (a)S 122 632 :M f3_12 sf (\(0\) = 0, so )S f4_12 sf (F)S f5_10 sf 0 2 rm (a)S 0 -2 rm f3_9 sf 0 2 rm (\(0\))S 0 -2 rm f3_12 sf ( = )S 210 632 :M f4_12 sf (F)S f3_9 sf 0 2 rm (0 )S 0 -2 rm 224 632 :M f1_12 sf S 234 632 :M f3_12 sf (A.)S 77 656 :M (2)S 83 656 :M (.)S 86 656 :M 6 .6( )J 95 656 :M .159 .016(Let )J f5_12 sf (a)S 122 656 :M f3_12 sf .215 .022(\(1\) be the largest )J 208 656 :M f5_12 sf .086(h)A f3_12 sf .187 .019( such that there is an inducing path between )J f4_12 sf .087(F)A f1_10 sf 0 2 rm .075(a)A 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 446 658 :M f3_10 sf (0)S f3_9 sf <29>S 454 656 :M f3_12 sf .26 .026( and )J 478 656 :M f4_12 sf (F)S f5_10 sf 0 2 rm (h)S 0 -2 rm 95 674 :M f3_12 sf (which is into )S 160 674 :M f4_12 sf (F)S f5_10 sf 0 2 rm (h)S 0 -2 rm f3_5 sf 0 4 rm ( )S 0 -4 rm f3_12 sf ( if there is such an inducing path, else let )S 376 674 :M f4_12 sf (F)S f5_10 sf 0 2 rm (a)S 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 392 676 :M f3_10 sf (1)S f3_9 sf (\) )S f1_12 sf 0 -2 rm S 0 2 rm 412 674 :M f4_12 sf (O)S 421 674 :M f3_12 sf (.)S endp %%Page: 39 39 %%BeginPageSetup initializepage (peter; page: 39 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (39)S gR gS 0 0 552 730 rC 77 56 :M f3_12 sf (3)S 83 56 :M (.)S 86 56 :M 6 .6( )J 95 56 :M .24 .024(Let )J f5_12 sf (a)S 122 56 :M f3_12 sf <28>S 126 56 :M f4_12 sf .109(k)A f3_12 sf .306 .031(+1\) be the largest )J 220 56 :M f5_12 sf .444(h)A f3_12 sf .293 .029( > )J 245 56 :M f4_12 sf .096(k)A f3_12 sf .284 .028( such that there is an inducing path between )J f4_12 sf .132(F)A f5_10 sf 0 2 rm .114(a)A 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 483 58 :M f3_10 sf (k)S f3_9 sf <29>S 95 74 :M f3_12 sf (and )S f4_12 sf (F)S f5_10 sf 0 2 rm (h)S 0 -2 rm f3_12 sf ( which is into )S 196 74 :M f4_12 sf (F)S f5_10 sf 0 2 rm (a)S 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 212 76 :M f3_10 sf (k)S f3_9 sf <29>S 220 74 :M f3_12 sf ( and, if )S 257 74 :M f5_12 sf (h )S f3_12 sf (< )S 277 74 :M f4_12 sf (m)S 286 74 :M f3_12 sf (+1, into )S f4_12 sf (F)S f5_10 sf 0 2 rm (h)S 0 -2 rm f3_12 sf (.)S 77 98 :M (4)S 83 98 :M (.)S 86 98 :M ( )S 95 98 :M (If )S 106 98 :M f5_12 sf (a)S 114 98 :M f3_12 sf <28>S 118 98 :M f4_12 sf (k)S f3_12 sf (\) = )S 140 98 :M f4_12 sf (m)S 149 98 :M f3_12 sf (+1 then stop.)S 77 121 :M .015 .002(\(Note that at each stage in the construction, so long as )J f1_12 sf (a)S 347 121 :M f3_12 sf <28>S 351 121 :M f4_12 sf (k)S f3_12 sf (\) < )S 373 121 :M f4_12 sf (m)S 382 121 :M f3_12 sf .014 .001(+1, there is guaranteed)J 59 139 :M .228 .023(to be some )J f5_12 sf .123(h)A f3_12 sf .269 .027( such that there is an inducing path between )J 339 139 :M f4_12 sf (F)S f5_10 sf 0 2 rm (a)S 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 355 141 :M f4_10 sf (k)S f3_9 sf <29>S 362 139 :M f3_12 sf .324 .032( and )J 385 139 :M f4_12 sf .14(F)A f5_10 sf 0 2 rm .115(h)A 0 -2 rm f3_12 sf .241 .024( which is into )J 468 139 :M f4_12 sf (F)S f5_10 sf 0 2 rm (a)S 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 484 141 :M f4_10 sf (k)S f3_9 sf <29>S 59 157 :M f3_12 sf .459 .046(and, if )J 94 157 :M f5_12 sf .689(h)A f1_12 sf .26 .026( )J f3_12 sf .553 .055(< )J 123 157 :M f4_12 sf (m)S 132 157 :M f3_12 sf .402 .04(+1, into )J f4_12 sf .211(F)A f5_10 sf 0 2 rm .174(h)A 0 -2 rm f3_12 sf .368 .037(, since, for )J 242 157 :M f4_12 sf .089(i)A f3_12 sf .416 .042( >0 there is an inducing path between )J 433 157 :M f4_12 sf (F)S f3_10 sf 0 2 rm (i)S 0 -2 rm 443 157 :M f3_12 sf .345 .035( and )J f4_12 sf .271(F)A f4_10 sf 0 2 rm (i)S 0 -2 rm 477 159 :M f3_10 sf (+)S 483 159 :M f3_9 sf (1)S 488 157 :M f3_12 sf (,)S 59 175 :M (which is into )S 124 175 :M f4_12 sf (F)S f4_10 sf 0 2 rm (i)S 0 -2 rm 134 175 :M f3_12 sf (, and for )S f4_12 sf (i)S f3_12 sf (+1 < )S 209 175 :M f4_12 sf (m)S 218 175 :M f3_12 sf (+1, into )S f4_12 sf (F)S f4_10 sf 0 2 rm (i)S 0 -2 rm 268 177 :M f3_10 sf (+)S 274 177 :M f3_9 sf (1)S 279 175 :M f3_12 sf (.\))S 77 199 :M .994 .099(We will now show that for )J f4_12 sf .217(i)A f3_12 sf .629 .063( )J cF f1_12 sf .063A sf .629 .063( 0, and i )J cF f1_12 sf .063A sf .629 .063J 285 199 :M f4_12 sf (m)S 294 199 :M f3_12 sf .89 .089(+1, )J 314 199 :M f4_12 sf (F)S f5_10 sf 0 2 rm (a)S 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 330 201 :M f4_10 sf (i)S 333 201 :M f3_9 sf <29>S 336 199 :M f3_12 sf .815 .081( is not an ancestor of)J 442 199 :M f4_12 sf .508 .051( F)J f5_10 sf 0 2 rm .372(a)A 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 462 201 :M f4_10 sf (i)S 465 201 :M f3_9 sf S 477 199 :M f3_12 sf .965 .096( or)J 59 217 :M f4_12 sf (F)S f5_10 sf 0 2 rm (a)S 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 75 219 :M f4_10 sf (i)S 78 219 :M f3_9 sf (+1\))S 91 217 :M f3_12 sf .392 .039(. Suppose that )J 164 217 :M f4_12 sf (F)S f5_10 sf 0 2 rm (a)S 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 180 219 :M f4_10 sf (i)S 183 219 :M f3_9 sf <29>S 186 217 :M f3_12 sf .348 .035( is an ancestor of )J f4_12 sf .209(F)A f5_10 sf 0 2 rm .18(a)A 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 289 219 :M f4_10 sf (i)S 292 219 :M f3_9 sf S 304 217 :M f3_12 sf .235 .023( or )J f4_12 sf .258(F)A f5_10 sf 0 2 rm .222(a)A 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 337 219 :M f4_10 sf (i)S 340 219 :M f3_9 sf (+1\))S 353 217 :M f3_12 sf .369 .037(. By construction there is an)J 59 235 :M .297 .03(inducing path between )J f4_12 sf .101(F)A f5_10 sf 0 2 rm .087(a)A 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 188 237 :M f4_10 sf (i)S 191 237 :M f3_9 sf <29>S 194 235 :M f3_12 sf .493 .049( and )J 219 235 :M f4_12 sf (F)S f5_10 sf 0 2 rm (a)S 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 235 237 :M f4_10 sf (i)S 238 237 :M f3_9 sf S 253 235 :M f3_12 sf .346 .035( which is into )J f4_12 sf .205(F)A f5_10 sf 0 2 rm .176(a)A 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 339 237 :M f4_10 sf (i)S 342 237 :M f3_9 sf <29>S 345 235 :M f3_12 sf .37 .037(, likewise there is an inducing)J 59 253 :M .333 .033(path between )J f4_12 sf .121(F)A f5_10 sf 0 2 rm .104(a)A 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 143 255 :M f4_10 sf (i)S 146 255 :M f3_9 sf <29>S 149 253 :M f3_12 sf .55 .055( and F)J 181 255 :M f1_10 sf (a)S f3_9 sf <28>S 190 255 :M f3_10 sf (i)S 193 255 :M f3_9 sf (+1\))S 206 253 :M f3_12 sf .402 .04( which is into )J f4_12 sf .238(F)A f5_10 sf 0 2 rm .205(a)A 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 292 255 :M f4_10 sf (i)S 295 255 :M f3_9 sf <29>S 298 253 :M f3_12 sf .516 .052(. Hence if )J 351 253 :M f4_12 sf (F)S f5_10 sf 0 2 rm (a)S 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 367 255 :M f4_10 sf (i)S 370 255 :M f3_9 sf <29>S 373 253 :M f3_12 sf .416 .042( is an ancestor of )J f4_12 sf .25(F)A f5_10 sf 0 2 rm .215(a)A 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 476 255 :M f4_10 sf (i)S 479 255 :M f3_9 sf S 59 271 :M f3_12 sf .48 .048(or )J f4_12 sf .352(F)A f5_10 sf 0 2 rm .303(a)A 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 89 273 :M f4_10 sf (i)S 92 273 :M f3_9 sf (+1\))S 105 271 :M f3_12 sf .956 .096( then there is an inducing path between )J 307 271 :M f4_12 sf (F)S f5_10 sf 0 2 rm (a)S 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 323 273 :M f4_10 sf (i)S 326 273 :M f3_9 sf S 338 271 :M f3_12 sf .664 .066( and )J f4_12 sf .522(F)A f5_10 sf 0 2 rm .449(a)A 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 380 273 :M f4_10 sf (i)S 383 273 :M f3_9 sf (+1\))S 396 271 :M f3_12 sf .98 .098( which is into both)J 59 289 :M f4_12 sf (F)S f5_10 sf 0 2 rm (a)S 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 75 291 :M f4_10 sf (i)S 78 291 :M f3_9 sf S 90 289 :M f3_12 sf .082 .008( and )J f4_12 sf .065(F)A f5_10 sf 0 2 rm .056(a)A 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 129 291 :M f4_10 sf (i)S 132 291 :M f3_9 sf (+1\))S 145 289 :M f3_12 sf .125 .013( \(unless )J 186 289 :M f4_12 sf (F)S f5_10 sf 0 2 rm (a)S 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 202 291 :M f4_10 sf (i)S 205 291 :M f3_9 sf S 217 289 :M f3_12 sf .072 .007( or )J f4_12 sf .079(F)A f5_10 sf 0 2 rm .068(a)A 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 249 291 :M f4_10 sf (i)S 252 291 :M f3_9 sf (+1\))S 265 289 :M f3_12 sf .117 .012( is an endpoint\). But in that case )J f5_12 sf (a)S 432 289 :M f3_12 sf <28>S 436 289 :M f4_12 sf (i)S f3_12 sf .121 .012(\) is not the)J 59 307 :M .404 .04(largest )J 95 307 :M f5_12 sf .189(h)A f3_12 sf .412 .041( such that there is an inducing path between )J f4_12 sf .191(F)A f5_10 sf 0 2 rm .165(a)A 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 337 309 :M f4_10 sf (i)S 340 309 :M f3_9 sf S 352 307 :M f3_12 sf .368 .037( and )J f4_12 sf .289(F)A f5_10 sf 0 2 rm .238(h)A 0 -2 rm f3_12 sf .496 .05( which is into )J 460 307 :M f4_12 sf (F)S f5_10 sf 0 2 rm (a)S 0 -2 rm f3_9 sf 0 2 rm <28>S 0 -2 rm 476 309 :M f4_10 sf (i)S 479 309 :M f3_9 sf S 59 325 :M f3_12 sf (and, if )S 93 325 :M f5_12 sf (h)S f1_12 sf ( )S f3_12 sf (< )S 113 325 :M f4_12 sf (m)S 122 325 :M f3_12 sf (+1, into )S f4_12 sf (F)S f5_10 sf 0 2 rm (h)S 0 -2 rm f3_12 sf (. This is a contradiction.)S 291 325 :M f1_12 sf ( \\)S 77 350 :M f0_12 sf 1.19 .119(Lemma 13: )J 142 350 :M f3_12 sf .83 .083(If )J f5_12 sf (p)S 161 353 :M f1_7 sf (0)S 165 350 :M f4_12 sf 1.726 .173( )J 170 350 :M f3_12 sf 1.085 .108(is the partially oriented graph constructed in step C\) of the Fast)J 59 368 :M .127 .013(Causal Inference Algorithm for )J 214 368 :M f4_12 sf (G)S 223 368 :M f3_12 sf <28>S 227 368 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 246 368 :M f3_12 sf (,)S f0_12 sf .103(L)A f3_12 sf .099 .01(\), )J f4_12 sf .095(A)A f3_12 sf .125 .013( and )J 298 368 :M f4_12 sf .107(B)A f3_12 sf .127 .013( are in )J f0_12 sf .136(O)A f3_12 sf .154 .015(, and )J 374 368 :M f4_12 sf .089(A)A f3_12 sf .152 .015( is not an ancestor of )J 485 368 :M f4_12 sf (B)S 59 386 :M f3_12 sf (in )S f4_12 sf <47D5>S 84 386 :M f3_12 sf (, then every vertex in )S 189 386 :M f0_12 sf (D-SEP)S 224 386 :M f3_12 sf <28>S 228 386 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf (\) in )S f4_12 sf (G)S 273 386 :M f3_12 sf <28>S 277 386 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 296 386 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (\) is in )S f0_12 sf (Possible-D-SEP)S 417 386 :M f3_12 sf <28>S 421 386 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf (,)S f5_12 sf ( p)S 451 389 :M f1_7 sf (0)S 455 386 :M f3_12 sf <29>S 459 386 :M f4_12 sf (.)S 77 410 :M f3_12 sf (Proof)S 104 410 :M f0_12 sf .199(.)A f3_12 sf 1.044 .104( Suppose that )J 180 410 :M f4_12 sf .625(A)A f3_12 sf 1.067 .107( is not an ancestor of )J 299 410 :M f4_12 sf .618(B)A f3_12 sf .573 .057(. If )J f4_12 sf .618(V)A f3_12 sf .6 .06( is in )J f0_12 sf .731(D-SEP)A 399 410 :M f3_12 sf <28>S 403 410 :M f4_12 sf .471(A)A f3_12 sf .193(,)A f4_12 sf .471(B)A f3_12 sf .551 .055(\), in )J f4_12 sf (G)S 454 410 :M f3_12 sf <28>S 458 410 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 477 410 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf <29>S 59 428 :M .374 .037(there is a sequence of vertices )J f4_12 sf (U)S 218 428 :M f3_12 sf .578 .058( )J 226 428 :M f1_12 sf .296 .03J f3_12 sf (<)S 243 428 :M f4_12 sf .218(A)A f3_12 sf .357A f4_12 sf .218(V)A f3_12 sf .263 .026(> in )J f0_12 sf .334 .033(O )J 305 428 :M f1_12 sf .551 .055J 318 428 :M f0_12 sf (Ancestors)S 369 428 :M f3_12 sf (\({)S 379 428 :M f4_12 sf .23(A)A f3_12 sf .094(,)A f4_12 sf .23(B)A f3_12 sf .229 .023(} )J f1_12 sf .349 .035J 418 428 :M f0_12 sf (S)S 425 428 :M f3_12 sf .368 .037(\), an inducing)J 59 446 :M .13 .013(path between every consecutive pair of vertices on )J 307 446 :M f4_12 sf (U)S 316 446 :M f3_12 sf .132 .013( with the exception of the endpoints,)J 59 464 :M .319 .032(and every vertex on )J f4_12 sf (U)S 168 464 :M f3_12 sf .342 .034( is not an ancestor of the vertices preceding or succeeding it in the)J 59 482 :M 1.111 .111(sequence )J f4_12 sf (U)S 117 482 :M f3_12 sf 1.736 .174(. The proof is by induction on the length of )J 352 482 :M f4_12 sf (U)S 361 482 :M f3_12 sf 1.838 .184(. If the length of )J 455 482 :M f4_12 sf (U)S 464 482 :M f3_12 sf 2 .2( is 1,)J 59 500 :M f0_12 sf (Possible-D-SEP)S 139 500 :M f3_12 sf <28>S 143 500 :M f4_12 sf .272(A)A f3_12 sf .111(,)A f4_12 sf .272(B)A f3_12 sf .111(,)A f5_12 sf .324 .032( p)J 174 503 :M f1_7 sf (0)S 178 500 :M f3_12 sf .428 .043(\) includes )J f0_12 sf .23(D-SEP)A 265 500 :M f3_12 sf <28>S 269 500 :M f4_12 sf .267(A)A f3_12 sf .109(,)A f4_12 sf .267(B)A f3_12 sf .645 .064(\) because it contains all edges adjacent to)J 59 518 :M .332 .033(A in )J f5_12 sf (p)S 91 521 :M f1_7 sf (0)S 95 518 :M f3_12 sf .348 .035(. Suppose that each vertex )J 227 518 :M f4_12 sf .174(V)A f3_12 sf .216 .022( that is in )J f0_12 sf .205(D-SEP)A 318 518 :M f3_12 sf <28>S 322 518 :M f4_12 sf .111(A)A f3_12 sf (,)S f4_12 sf .111(B)A f3_12 sf .273 .027(\)because of a sequence )J f4_12 sf (U)S 463 518 :M f3_12 sf .423 .042( of no)J 59 536 :M 1.31 .131(more than length )J f4_12 sf .438(n)A f3_12 sf .365 .037(, )J f4_12 sf .536(V)A f3_12 sf .534 .053( is in )J 198 536 :M f0_12 sf (Possible-D-SEP)S 278 536 :M f3_12 sf <28>S 282 536 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf (,)S f5_12 sf (p)S 309 539 :M f1_7 sf (0)S 313 536 :M f3_12 sf .747 .075(\). Let )J f4_12 sf (W)S 354 536 :M f3_12 sf 1.043 .104( be a vertex in )J 432 536 :M f0_12 sf (D-SEP)S 467 536 :M f3_12 sf <28>S 471 536 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf <29>S 59 554 :M .121 .012(because of a sequence )J f4_12 sf (U)S 178 554 :M f3_12 sf .159 .016( of length )J 231 554 :M f4_12 sf .095(n)A f3_12 sf .146 .015( + 1, and )J 283 554 :M f4_12 sf .061(X)A f3_12 sf .119 .012( is the predecessor of )J f4_12 sf (W)S 405 554 :M f3_12 sf .108 .011( in )J f4_12 sf (U)S 429 554 :M f3_12 sf .162 .016(, and )J 456 554 :M f4_12 sf (Y)S 463 554 :M f3_12 sf .156 .016( is the)J 59 572 :M 1.481 .148(predecessor of )J 137 572 :M f4_12 sf 1.179(X)A f3_12 sf 1.027 .103( in )J f4_12 sf (U)S 173 572 :M f3_12 sf 1.605 .16(. If there is an inducing path between )J f4_12 sf (W)S 384 572 :M f3_12 sf 1.567 .157( and )J f4_12 sf (Y)S 419 572 :M f3_12 sf 1.937 .194(, then )J 454 572 :M f4_12 sf (W)S 464 572 :M f3_12 sf 2.015 .201( is in)J 59 590 :M f0_12 sf (Possible-D-SEP)S 139 590 :M f3_12 sf <28>S 143 590 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf (,)S f5_12 sf (p)S 170 593 :M f1_7 sf (0)S 174 590 :M f3_12 sf 2.019 .202(\). Suppose that there is no inducing path between )J f4_12 sf (W)S 452 590 :M f3_12 sf 2.727 .273( and )J 482 590 :M f4_12 sf (Y)S 489 590 :M f3_12 sf (.)S 59 608 :M 1.495 .15(Because )J f4_12 sf .459(X)A f3_12 sf .784 .078( is not an ancestor of )J 222 608 :M f4_12 sf (Y)S 229 608 :M f3_12 sf .702 .07( or )J f4_12 sf (W)S 257 608 :M f3_12 sf .817 .082(, the inducing path between )J f4_12 sf (W)S 409 608 :M f3_12 sf .963 .096( and )J f4_12 sf .756(X)A f3_12 sf .968 .097( is into )J 482 608 :M f4_12 sf (X)S f3_12 sf (.)S 59 626 :M .417 .042(Similarly, the inducing path between )J f4_12 sf (Y)S 249 626 :M f3_12 sf .628 .063( and )J 274 626 :M f4_12 sf .385(X)A f3_12 sf .493 .049( is into )J 319 626 :M f4_12 sf .235(X)A f3_12 sf .471 .047(. Hence )J 368 626 :M f4_12 sf .323(X)A f3_12 sf .521 .052( is not in any set that d-)J 59 644 :M .451 .045(separates )J 107 644 :M f4_12 sf (Y)S 114 644 :M f3_12 sf .48 .048( and )J f4_12 sf (W)S 149 644 :M f3_12 sf .779 .078(. )J 156 644 :M f4_12 sf (Y)S 163 644 :M f3_12 sf .714 .071( and )J 187 644 :M f4_12 sf (W)S 197 644 :M f3_12 sf .577 .058( are d-separated given a subset of )J 366 644 :M f0_12 sf (O)S f3_12 sf (\\{)S f4_12 sf (W)S 394 644 :M f3_12 sf (,)S f4_12 sf (Y)S 404 644 :M f3_12 sf .547 .055(} because there is)J 59 662 :M 1.06 .106(no inducing path between them. Hence, step C\) of the FCI algorithm orients the edge)J 59 680 :M (between )S 102 680 :M f4_12 sf (W)S 112 680 :M f3_12 sf ( and )S f4_12 sf (X)S f3_12 sf ( into )S 167 680 :M f4_12 sf (X)S f3_12 sf (. It follows that )S f4_12 sf (W)S 260 680 :M f3_12 sf ( is in )S f0_12 sf (Possible-D-SEP)S 366 680 :M f3_12 sf <28>S 370 680 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf (,)S f5_12 sf (p)S 397 683 :M f1_7 sf (0)S 401 680 :M f3_12 sf (\). )S 411 668 9 12 rC gS 1.286 1 scale 319.669 679 :M f1_10 sf <5C>S gR endp %%Page: 40 40 %%BeginPageSetup initializepage (peter; page: 40 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (40)S gR gS 0 0 552 730 rC 77 56 :M f3_12 sf (Because )S 120 56 :M f5_12 sf (p)S 127 59 :M f3_7 sf (0)S 131 56 :M f3_12 sf .014 .001( contains more edges than the output PAG, the orientations in )J 430 56 :M f5_12 sf (p)S 437 56 :M f3_12 sf .016 .002( may not be)J 59 74 :M .871 .087(correct for the two following reasons. First, an edge that is in )J f5_12 sf (p)S 378 77 :M f3_7 sf (0)S 382 74 :M f3_12 sf .953 .095(, but not in the output)J 59 92 :M .074 .007(PAG, may hide some collider along a path. Second, a vertex may be oriented as a collider)J 59 110 :M .055 .006(in )J f5_12 sf (p)S 78 113 :M f3_7 sf (0)S 82 110 :M f3_12 sf .074 .007(, but not in the output PAG, because of a \322collision\323 involving an edge in )J 439 110 :M f5_12 sf (p)S 446 110 :M f3_12 sf .076 .008( that does)J 59 128 :M .113 .011(not occur in the output PAG. However, either of these mistakes in orientation in )J 449 128 :M f5_12 sf (p)S 456 128 :M f3_12 sf .111 .011( simply)J 59 146 :M (makes )S 93 146 :M f0_12 sf (Possible-D-SEP)S 173 146 :M f3_12 sf <28>S 177 146 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf (,)S f5_12 sf (p)S 204 146 :M f3_12 sf (\) larger, and so it still includes )S 353 146 :M f0_12 sf (D-SEP)S 388 146 :M f3_12 sf <28>S 392 146 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (B)S f3_12 sf (,)S f4_12 sf <47D5>S 425 146 :M f3_12 sf (\).)S 77 169 :M f0_12 sf (Lemma 14:)S 135 169 :M f3_12 sf .025 .002( Suppose that in a graph )J 255 169 :M f4_12 sf (G)S 264 169 :M f3_12 sf <28>S 268 169 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 287 169 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf .023 .002(\), X, Y, Z )J 348 169 :M f1_12 sf S 357 169 :M f3_12 sf ( )S f0_12 sf (O)S f3_12 sf .026 .003(, and Y is not an ancestor)J 59 187 :M 1.12 .112(of X or Z or )J 127 187 :M f0_12 sf (S)S 134 187 :M f3_12 sf 1.055 .105(. If there is a set )J 223 187 :M f0_12 sf 1.195 .12(Q )J f1_12 sf 1.221 .122J 250 187 :M f0_12 sf .51(O)A f3_12 sf .888 .089(, containing Y, such that for every subset )J f0_12 sf .501 .05(T )J f1_12 sf S 59 205 :M f0_12 sf .195(Q)A f3_12 sf .359 .036(\\{Y}, X and Z are d-connected given )J 253 205 :M f0_12 sf .448 .045(T )J f1_12 sf .543 .054J 277 205 :M f0_12 sf (S)S 284 205 :M f3_12 sf .409 .041(, then X and Z are d-connected given )J f0_12 sf .296 .03(Q )J 482 205 :M f1_12 sf S 59 224 :M f0_12 sf (S.)S 77 247 :M (Proof:)S 110 247 :M f3_12 sf 1.932 .193( Let )J f0_12 sf 1.758(T)A f3_10 sf 0 -3 rm 1.098(*)A 0 3 rm f3_12 sf 2.519 .252(= \()J 170 247 :M f0_12 sf .382(Ancenstors)A f3_12 sf 1.472 .147(\({X,Z} )J f1_12 sf .741 .074J 284 247 :M f0_12 sf (S)S 291 247 :M f3_12 sf 2.601 .26(\)\) )J f1_12 sf 3.418 .342J 321 247 :M f0_12 sf 1.749(Q)A f3_12 sf 2.33 .233(. Now )J f0_12 sf 1.5(T)A f3_10 sf 0 -3 rm 1.277 .128(* )J 0 3 rm 389 247 :M f1_12 sf S 398 247 :M f3_12 sf .491 .049( )J f0_12 sf 1.681(Q)A f3_12 sf 2.741 .274(, but since, by)J 59 265 :M .329 .033(hypothesis, Y)J 126 265 :M f1_12 sf S 135 265 :M f0_12 sf .362 .036( Ancenstors)J 195 265 :M f3_12 sf .453 .045(\({X,Z} )J 234 265 :M f1_12 sf .659 .066J 247 265 :M f0_12 sf (S)S 254 265 :M f3_12 sf .5 .05(\), it follows that )J 337 265 :M f0_12 sf .241(T)A f3_10 sf 0 -3 rm .188 .019(* )J 0 3 rm f1_12 sf S 362 265 :M f3_12 sf .69 .069( )J 370 265 :M f0_12 sf .17(Q)A f3_12 sf .393 .039(\\{Y}. Hence, again, by)J 59 283 :M .329 .033(hypothesis, there is a path )J 189 283 :M f4_12 sf .141 .014(P )J f3_12 sf .436 .044(d-connecting X and Z given )J f0_12 sf .195(T)A f3_10 sf 0 -3 rm .166 .017(* )J 0 3 rm 360 283 :M f1_12 sf .273 .027J f0_12 sf (S)S 379 283 :M f3_12 sf .347 .035(. By the definition of a)J 59 301 :M .207 .021(d-connecting path, every vertex on )J f4_12 sf .076(P)A f3_12 sf .126 .013( is either an ancestor of X or Z, or )J 405 301 :M f0_12 sf .122(T)A f3_10 sf 0 -3 rm .104 .01(* )J 0 3 rm 421 301 :M f1_12 sf .215 .022J 434 301 :M f0_12 sf (S)S 441 301 :M f3_12 sf .124 .012(. Since )J f0_12 sf .078(T)A f3_10 sf 0 -3 rm (*)S 0 3 rm 59 319 :M f1_12 sf S 68 319 :M f3_12 sf .644 .064( \()J f0_12 sf .633(Ancenstors)A f3_12 sf 2.589 .259(\({X,Z} )J 181 319 :M f1_12 sf 5.111 .511J 199 319 :M f0_12 sf (S)S 206 319 :M f3_12 sf 4.096 .41(\)\) )J f1_12 sf 5.382 .538J 240 319 :M f0_12 sf 2.428(Q)A f3_12 sf 3.833 .383(, it follows that every vertex on )J f4_12 sf 2.442 .244(P )J 457 319 :M f3_12 sf 4.498 .45( is in)J 59 337 :M f0_12 sf .166(Ancenstors)A f3_12 sf .678 .068(\({X,Z} )J 157 337 :M f1_12 sf 2.494 .249J 172 337 :M f0_12 sf (S)S 179 337 :M f3_12 sf 1.932 .193(\). Since no vertex in )J 293 337 :M f0_12 sf .698(Q)A f3_12 sf .249<5C>A f0_12 sf .598(T)A f3_10 sf 0 -3 rm .374(*)A 0 3 rm f3_12 sf .532 .053( is in )J f0_12 sf .428(Ancenstors)A f3_12 sf 1.752 .175(\({X,Z} )J 450 337 :M f1_12 sf 1.478 .148J f0_12 sf (S)S 471 337 :M f3_12 sf 1.959 .196(\), it)J 59 355 :M .897 .09(follows that no vertex in )J f0_12 sf .529(Q)A f3_12 sf .189<5C>A f0_12 sf .454(T)A f3_10 sf 0 -3 rm .283(*)A 0 3 rm f3_12 sf .562 .056( lies on )J 252 355 :M f4_12 sf .467(P)A f3_12 sf .838 .084(. But since )J 317 355 :M f0_12 sf .672(T)A f3_10 sf 0 -3 rm .572 .057(* )J 0 3 rm 334 355 :M f1_12 sf 1.183 .118J 348 355 :M f0_12 sf 1.182 .118(S )J 359 355 :M f1_12 sf 1.183 .118J 372 355 :M f0_12 sf 1.089 .109(Q )J f1_12 sf 1.177 .118J 399 355 :M f0_12 sf (S)S 406 355 :M f3_12 sf .929 .093(, the only way in)J 59 373 :M .77 .077(which )J f4_12 sf .279(P)A f3_12 sf .554 .055( could fail to d-connect X and Z given )J 292 373 :M f0_12 sf .567 .057(Q )J f1_12 sf .561 .056J f0_12 sf (S)S 325 373 :M f3_12 sf .602 .06( would be if some vertex in \()J f0_12 sf .443 .044(Q )J f1_12 sf S 59 391 :M f0_12 sf (S)S 66 391 :M f3_12 sf <295C28>S f0_12 sf (T )S f3_10 sf 0 -3 rm (*)S 0 3 rm f1_12 sf S f0_12 sf (S)S 112 391 :M f3_12 sf (\) = )S 129 391 :M f0_12 sf (Q)S f3_12 sf <5C>S f0_12 sf (T)S f3_10 sf 0 -3 rm (*)S 0 3 rm f3_12 sf ( lay on the path. Hence )S 268 391 :M f4_12 sf (P)S f3_12 sf ( still d-connects X and Z given )S 426 391 :M f0_12 sf (Q )S f1_12 sf S f0_12 sf (S)S 457 391 :M f3_12 sf (.)S f1_12 sf <5C>S 77 415 :M f3_12 sf .372 .037(In a graph )J f4_12 sf (G)S 139 415 :M f3_12 sf <28>S 143 415 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 162 415 :M f3_12 sf .081(,)A f0_12 sf .216(L)A f3_12 sf .39 .039(\) if X and Z are d-separated given )J 343 415 :M f0_12 sf (Q)S f1_12 sf S f0_12 sf (S)S 368 415 :M f3_12 sf .349 .035(, \({X,Z})J 410 415 :M f1_12 sf S f0_12 sf (Q)S f1_12 sf S 437 415 :M f0_12 sf .207(O)A f3_12 sf .338 .034(\), and for)J 59 433 :M 1.39 .139(any proper subset )J 154 433 :M f0_12 sf (T)S f1_12 sf S 171 433 :M f0_12 sf .747(Q)A f3_12 sf 1.234 .123(, X and Z are d-connected given )J f0_12 sf .64(T)A f1_12 sf .737A f0_12 sf (S)S 376 433 :M f3_12 sf 1.95 .195(, then )J f0_12 sf 1.657(Q)A f3_12 sf 1.133 .113( is a )J 448 433 :M f0_12 sf (minimal)S 59 452 :M (d)S 66 452 :M (-separating set)S 140 452 :M f3_12 sf ( for X and Z in )S 215 452 :M f4_12 sf (G)S 224 452 :M f3_12 sf <28>S 228 452 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 247 452 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (\).)S 77 476 :M f0_12 sf (Corollary:)S 131 476 :M f3_12 sf .568 .057( If Y is in a minimal d-separating set for X and Z in )J 393 476 :M f4_12 sf (G)S 402 476 :M f3_12 sf <28>S 406 476 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 425 476 :M f3_12 sf .11(,)A f0_12 sf .292(L)A f3_12 sf .475 .048(\), then Y is)J 59 494 :M (an ancestor of X or Z or )S f0_12 sf (S)S 184 494 :M f3_12 sf ( in )S f4_12 sf (G)S 208 494 :M f3_12 sf <28>S 212 494 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 231 494 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (\).)S 77 518 :M f0_12 sf (Proof:)S 110 518 :M f3_12 sf -.004( Suppose for a contradiction that there were a minimal d-separating set )A 453 518 :M f0_12 sf (Q)S f3_12 sf ( for X)S 59 535 :M .468 .047(and Z, which contained some vertex Y )J f1_12 sf .279 .028J 265 535 :M f0_12 sf .04(Ancenstors)A f3_12 sf .164 .016(\({X,Z} )J 361 535 :M f1_12 sf .358 .036J f0_12 sf (S)S 380 535 :M f3_12 sf .428 .043(\). It would then follow)J 59 554 :M 1.175 .118(from Lemma 14)J 141 554 :M 1.234 .123(, and the definition of a minimal d-separating set, that X and Z were)J 59 571 :M (d-connected given )S f0_12 sf (Q )S f1_12 sf S f0_12 sf (S)S 180 571 :M f3_12 sf (, which is a contradiction.)S 304 571 :M f1_12 sf <5C>S 77 596 :M f0_12 sf .443 .044(Theorem 5)J 134 596 :M (:)S 138 596 :M f3_12 sf .272 .027( If )J f4_12 sf .342(P)A f3_12 sf .654 .065(\(V\) is faithful to )J 245 596 :M f4_12 sf (G)S 254 599 :M f3_7 sf (1)S 258 596 :M f3_12 sf <28>S 262 596 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 281 599 :M f0_7 sf (1)S 285 596 :M f3_12 sf (,)S f0_12 sf (L)S f0_7 sf 0 3 rm (1)S 0 -3 rm 300 596 :M f3_12 sf .536 .054(\), and the input to the FCI algorithm is)J 59 614 :M (an oracle for )S 123 614 :M f4_12 sf (P)S f3_12 sf <28>S 134 614 :M f0_12 sf (V)S 143 614 :M f3_12 sf (\) over )S f0_12 sf (O)S f3_12 sf ( given )S 216 614 :M f0_12 sf (S)S 223 614 :M f3_12 sf ( = )S 236 614 :M f0_12 sf (1)S f3_12 sf (, the output is a PAG of )S 359 614 :M f4_12 sf (G)S 368 617 :M f3_7 sf (1)S 372 614 :M f3_12 sf <28>S 376 614 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 395 617 :M f0_7 sf (1)S 399 614 :M f3_12 sf (,)S f0_12 sf (L)S f0_7 sf 0 3 rm (1)S 0 -3 rm 414 614 :M f3_12 sf (\).)S 77 638 :M .455 .046(Proof. The adjacencies are correct by )J 263 638 :M .433 .043(Lemma 12)J 316 638 :M .526 .053( and Lemma 13)J 393 638 :M .517 .052(. We will now show)J 59 656 :M .371 .037(that the orientations are correct. The proof is by induction on the number of applications)J 59 674 :M .976 .098(of orientation rules in the repeat loop of the Fast Causal Inference Algorithm.. Let the)J endp %%Page: 41 41 %%BeginPageSetup initializepage (peter; page: 41 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (41)S gR gS 0 0 552 730 rC 59 56 :M f3_12 sf .459 .046(object constructed by the algorithm after the )J 280 56 :M f4_12 sf .197(n)A f3_9 sf 0 -3 rm .115(th)A 0 3 rm f3_12 sf .476 .048( iteration of the repeat loop be )J 446 56 :M f5_12 sf (p)S 453 56 :M f4_12 sf S 457 59 :M f3_7 sf (n)S 461 56 :M f3_12 sf .503 .05( \(Note)J 59 74 :M (that according to the notation of the algorithm, )S 286 74 :M f5_12 sf (p)S 293 74 :M f4_12 sf S 297 77 :M f3_7 sf (0)S 301 74 :M f3_12 sf ( = )S 314 74 :M f5_12 sf (p)S 321 77 :M f3_7 sf (2)S 325 74 :M f3_12 sf (.\) Note that each set )S 424 74 :M f0_12 sf (Sepset)S 457 74 :M f3_12 sf <28>S 461 74 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (C)S f3_12 sf (\) is)S 59 92 :M .387 .039(a minimal d-connecting set, because if there were any subset )J f0_12 sf (X)S 368 92 :M f3_12 sf .558 .056( of )J 386 92 :M f0_12 sf (Sepset)S 419 92 :M f3_12 sf <28>S 423 92 :M f4_12 sf .138(A)A f3_12 sf .056(,)A f4_12 sf .15(C)A f3_12 sf .313 .031(\) such that)J 59 110 :M f0_12 sf (X)S 68 110 :M f3_12 sf .297 .03( )J f1_12 sf 1.004A f3_12 sf .297 .03( )J f0_12 sf (S)S 92 110 :M f3_12 sf .963 .096( d-separated )J f4_12 sf .38(A)A f3_12 sf .504 .05( and )J 188 110 :M f4_12 sf .343(C)A f3_12 sf .748 .075(, then the algorithm would have found )J 391 110 :M f0_12 sf (X)S 400 110 :M f3_12 sf .808 .081( at an earlier stage)J 59 128 :M (and made )S f0_12 sf (Sepset)S 141 128 :M f3_12 sf <28>S 145 128 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (C)S f3_12 sf (\) equal to )S f0_12 sf (X)S 220 128 :M f3_12 sf (.)S 77 152 :M .948 .095(Base Case: Suppose that the only orientation rule that has been applied is that if )J 485 152 :M f4_12 sf (A)S 59 170 :M f3_12 sf .08(*)A f1_12 sf .16A f3_12 sf .1 .01(* )J f4_12 sf .098(B)A f3_12 sf .1 .01( *)J f1_12 sf .16A f3_12 sf .109 .011(* )J 124 170 :M f4_12 sf .153(C)A f3_12 sf .122 .012( in )J f5_12 sf (p)S 154 170 :M f4_12 sf S 158 173 :M f3_7 sf (0)S 162 170 :M f3_12 sf .229 .023(, but )J 187 170 :M f4_12 sf .111(A)A f3_12 sf .141 .014( and )J f4_12 sf .121(C)A f3_12 sf .203 .02( are not adjacent in )J 321 170 :M f5_12 sf (p)S 328 170 :M f4_12 sf S 332 173 :M f3_7 sf (0)S 336 170 :M f3_12 sf .099 .01(, )J f4_12 sf .146(A)A f3_12 sf .149 .015( *)J f1_12 sf .238A f3_12 sf .162 .016(* )J 380 170 :M f4_12 sf .094(B)A f3_12 sf .096 .01( *)J f1_12 sf .154A f3_12 sf .096 .01(* )J f4_12 sf .103(C)A f3_12 sf .193 .019( is oriented as)J 59 188 :M f4_12 sf .354(A)A f3_12 sf .363 .036( *)J f1_12 sf S 88 188 :M f3_12 sf .557 .056( )J f4_12 sf 1.498(B)A f3_12 sf .613 .061( )J 104 188 :M f1_12 sf S 116 188 :M f3_12 sf .553 .055(* )J f4_12 sf .59(C)A f3_12 sf .409 .041( if )J f4_12 sf .54(B)A f3_12 sf .884 .088( is not a member of )J f0_12 sf .481(Sepset)A 294 188 :M f3_12 sf <28>S 298 188 :M f4_12 sf .459(A)A f3_12 sf .188(,)A f4_12 sf .501(C)A f3_12 sf .647 .065(\) and as )J f4_12 sf .459(A)A f3_12 sf .47 .047( *)J f1_12 sf S 389 0 6 730 rC 389 188 :M f3_12 sf 12 f6_1 :p 4.362 :m 1.362 .136( )J 391 188 :M 8.724 :m 1.297 .13( )J gR gS 0 0 552 730 rC 389 188 :M f3_12 sf 12 f6_1 :p 10.362 :m 1.238 .124(* )J 395 0 5 730 rC 395 188 :M 4.362 :m 1.362 .136( )J 396 188 :M 8.724 :m 1.297 .13( )J gR gS 400 0 7 730 rC 400 188 :M f4_12 sf 12 f7_1 :p 4.362 :m 1.362 .136( )J 403 188 :M 8.724 :m 1.297 .13( )J gR gS 0 0 552 730 rC 400 188 :M f4_12 sf 12 f7_1 :p 7.33 :m (B)S 400 0 7 730 rC 400 188 :M 4.362 :m 1.362 .136( )J 403 188 :M 8.724 :m 1.297 .13( )J gR gS 407 0 4 730 rC 407 188 :M f3_12 sf 12 f6_1 :p 8.724 :m 1.297 .13( )J gR gS 0 0 552 730 rC 407 188 :M f3_12 sf 12 f6_1 :p 10.362 :m 1.238 .124( *)J 411 0 6 730 rC 411 188 :M 4.362 :m 1.362 .136( )J 413 188 :M 8.724 :m 1.297 .13( )J gR gS 0 0 552 730 rC 417 188 :M f1_12 sf .778A f3_12 sf .531 .053(* )J 440 188 :M f4_12 sf 1.022(C)A f3_12 sf .74 .074( if )J 463 188 :M f4_12 sf .749(B)A f3_12 sf .823 .082( is a)J 59 206 :M .438 .044(member of )J f0_12 sf .155(Sepset)A 149 206 :M f3_12 sf <28>S 153 206 :M f4_12 sf .266(A)A f3_12 sf .109(,)A f4_12 sf .29(C)A f3_12 sf .576 .058(\). Suppose first that )J 273 206 :M f4_12 sf .589(B)A f3_12 sf .718 .072( is not in )J 329 206 :M f0_12 sf (Sepset)S 362 206 :M f3_12 sf <28>S 366 206 :M f4_12 sf .251(A)A f3_12 sf .103(,)A f4_12 sf .274(C)A f3_12 sf .573 .057(\), and that contrary to)J 59 224 :M .191 .019(the hypothesis that )J 153 224 :M f4_12 sf .125(B)A f3_12 sf .211 .021( is an ancestor of )J 246 224 :M f4_12 sf .193(A)A f3_12 sf .176 .018( or )J f4_12 sf .211(C)A f3_12 sf .183 .018( or )J 294 224 :M f0_12 sf (S)S 301 224 :M f3_12 sf .165 .017(. If )J f4_12 sf (G)S 327 224 :M f3_12 sf <28>S 331 224 :M f0_12 sf .059(O,S,L)A f3_12 sf .166 .017(\) is an arbitrary member of)J 59 242 :M f0_12 sf (O)S f3_12 sf (-Equiv\()S 105 242 :M f8_12 sf (G)S 114 245 :M f0_7 sf (1)S 118 242 :M f3_12 sf <28>S 122 242 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 141 245 :M f0_7 sf (1)S 145 242 :M f3_12 sf (,)S f0_12 sf (L)S f0_7 sf 0 3 rm (1)S 0 -3 rm 160 242 :M f3_12 sf .782 .078(\)\), there is an inducing path between )J f4_12 sf .348(A)A f3_12 sf .461 .046( and )J 378 242 :M f4_12 sf .292(B)A f3_12 sf .685 .069( and an inducing path)J 59 260 :M 3.992 .399(between )J f4_12 sf 1.226(B)A f3_12 sf 1.625 .163( and )J 145 260 :M f4_12 sf 1.598(C)A f3_12 sf 2.592 .259(. Hence there are is a path )J f4_12 sf (U)S 317 263 :M f3_7 sf (1)S 321 260 :M f3_12 sf 2.676 .268( that d-connects )J 413 260 :M f4_12 sf 1.334(A)A f3_12 sf 1.698 .17( and )J f4_12 sf 1.334(B)A f3_12 sf 3.598 .36( given)J 59 278 :M f0_12 sf (Sepset)S 92 278 :M f3_12 sf <28>S 96 278 :M f4_12 sf .082(A)A f3_12 sf (,)S f4_12 sf .089(C)A f3_12 sf .049(\)\\{)A f4_12 sf .082(A)A f3_12 sf .081 .008(} )J f1_12 sf .103A f3_12 sf ( )S 156 278 :M f0_12 sf .152 .015(S )J f3_12 sf .241 .024(and a path )J f4_12 sf (U)S 228 281 :M f3_7 sf (2)S 232 278 :M f3_12 sf .217 .022( that d-connects )J 312 278 :M f4_12 sf .159(B)A f3_12 sf .211 .021( and )J 343 278 :M f4_12 sf .073(C)A f3_12 sf .11 .011( given )J f0_12 sf .059(Sepset)A 417 278 :M f3_12 sf <28>S 421 278 :M f4_12 sf .051(A)A f3_12 sf (,)S f4_12 sf .055(C)A f3_12 sf .03(\)\\{)A f4_12 sf .055(C)A f3_12 sf .055 .005(} )J 470 278 :M f1_12 sf .163A f3_12 sf .048 .005( )J f0_12 sf (S)S 489 278 :M f3_12 sf (.)S 59 296 :M .907 .091(If )J f4_12 sf (U)S 81 299 :M f0_7 sf (1)S 85 296 :M f3_12 sf 1.229 .123( and )J f4_12 sf (U)S 121 299 :M f3_7 sf (2)S 125 296 :M f3_12 sf 1.55 .155( do not collide at )J 220 296 :M f4_12 sf .891(B)A f3_12 sf 1.393 .139( then, by )J 278 296 :M 1.317 .132(Lemma 1)J 326 296 :M .862 .086(, )J f4_12 sf 1.264(A)A f3_12 sf 1.676 .168( and )J 369 296 :M f4_12 sf .433(C)A f3_12 sf 1.155 .115( are d-connected given)J 59 314 :M f0_12 sf (Sepset)S 92 314 :M f3_12 sf <28>S 96 314 :M f4_12 sf .568(A)A f3_12 sf .232(,)A f4_12 sf .62(C)A f3_12 sf .493 .049(\) )J 123 314 :M f1_12 sf 1.481A f3_12 sf .482 .048( )J 137 314 :M f0_12 sf (S)S 144 314 :M f3_12 sf 1.217 .122(, which is a contradiction. If )J f4_12 sf (U)S 302 317 :M f0_7 sf (1)S 306 314 :M f3_12 sf 1.145 .114( and )J f4_12 sf (U)S 342 317 :M f3_7 sf (2)S 346 314 :M f3_12 sf 1.416 .142( do collide at )J f4_12 sf .874(B)A f3_12 sf 1.208 .121(, and )J f4_12 sf .874(B)A f3_12 sf 1.208 .121( is an)J 59 332 :M .151 .015(ancestor of )J f0_12 sf (S)S 122 332 :M f3_12 sf .203 .02(, then, by )J 171 332 :M .169 .017(Lemma 1)J 217 332 :M .246 .025(, )J 224 332 :M f4_12 sf .065(A)A f3_12 sf .083 .008( and )J f4_12 sf .071(C)A f3_12 sf .157 .016( are d-connected given )J f0_12 sf .058(Sepset)A 408 332 :M f3_12 sf <28>S 412 332 :M f4_12 sf .105(A)A f3_12 sf (,)S f4_12 sf .115(C)A f3_12 sf .084 .008(\) )J f1_12 sf .133A f3_12 sf ( )S 450 332 :M f0_12 sf (S)S 457 332 :M f3_12 sf .169 .017(, which)J 59 350 :M .182 .018(is a contradiction. If )J 160 350 :M f4_12 sf .113(B)A f3_12 sf .191 .019( is an ancestor of )J 253 350 :M f4_12 sf .148(A)A f3_12 sf .134 .013( or )J f4_12 sf .161(C)A f3_12 sf .208 .021( but not of )J 338 350 :M f0_12 sf (S)S 345 350 :M f3_12 sf .184 .018(, then either there is a directed)J 59 368 :M .186 .019(path )J 83 368 :M f4_12 sf (D)S 92 368 :M f3_12 sf .209 .021( from )J 122 368 :M f4_12 sf .098(B)A f3_12 sf .085 .009( to )J f4_12 sf .107(C)A f3_12 sf .197 .02( that does not contain )J 260 368 :M f4_12 sf .128(A)A f3_12 sf .116 .012( or )J f0_12 sf (S)S 290 368 :M f3_12 sf .184 .018(, or there is a directed path )J 423 368 :M f4_12 sf (D)S 432 368 :M f3_12 sf .226 .023( from )J f4_12 sf .147(B)A f3_12 sf .134 .013( to )J 484 368 :M f4_12 sf (C)S 59 386 :M f3_12 sf .215 .021(that does not contain )J 164 386 :M f4_12 sf .16(A)A f3_12 sf .146 .015( or )J f0_12 sf (S)S 194 386 :M f3_12 sf .207 .021(. Suppose without loss of generality that the latter is the case.)J 59 404 :M 1.207 .121(It follows that )J f4_12 sf (D)S 144 404 :M f3_12 sf 1.747 .175( d-connects )J f4_12 sf .707(B)A f3_12 sf .9 .09( and )J f4_12 sf .772(C)A f3_12 sf 1.211 .121( given )J 285 404 :M f0_12 sf (Sepset)S 318 404 :M f3_12 sf <28>S 322 404 :M f4_12 sf .316(A)A f3_12 sf .129(,)A f4_12 sf .345(C)A f3_12 sf .188(\)\\{)A f4_12 sf .345(C)A f3_12 sf .343 .034(} )J 372 404 :M f1_12 sf 1.568A f3_12 sf .51 .051( )J 387 404 :M f0_12 sf (S)S 394 404 :M f3_12 sf 1.459 .146( and is out of )J f4_12 sf 1.059(B)A f3_12 sf 1.481 .148(. It)J 59 422 :M .944 .094(follows by )J 116 422 :M .885 .088(Lemma 1)J 164 422 :M .908 .091( that)J f4_12 sf .558 .056( A)J f3_12 sf .605 .061( and )J f4_12 sf .519(C)A f3_12 sf 1.171 .117( are d-connected given )J 349 422 :M f0_12 sf (Sepset)S 382 422 :M f3_12 sf <28>S 386 422 :M f4_12 sf .553(A)A f3_12 sf .226(,)A f4_12 sf .604(C)A f3_12 sf .439 .044(\) )J f1_12 sf .695A f3_12 sf .226 .023( )J 426 422 :M f0_12 sf (S)S 433 422 :M f3_12 sf 1.036 .104(, which is a)J 59 440 :M (contradiction.)S 77 464 :M .475 .048(Suppose next that )J f4_12 sf .189(B)A f3_12 sf .184 .018( is in )J f0_12 sf .169(Sepset)A 235 464 :M f3_12 sf <28>S 239 464 :M f4_12 sf .248(A)A f3_12 sf .101(,)A f4_12 sf .27(C)A f3_12 sf .396 .04(\), but that )J f4_12 sf .248(B)A f3_12 sf .423 .042( is not an ancestor of )J 422 464 :M f4_12 sf .362(A)A f3_12 sf .329 .033( or )J f4_12 sf .395(C)A f3_12 sf .329 .033( or )J f0_12 sf (S)S 478 464 :M f3_12 sf .513 .051(. It)J 59 482 :M .653 .065(follows from Lemma 11)J 179 482 :M .694 .069( that the inducing paths in )J f4_12 sf (G)S 321 482 :M f3_12 sf <28>S 325 482 :M f0_12 sf .259(O,S,L)A f3_12 sf .555 .056(\) are both into )J 431 482 :M f4_12 sf .267(B)A f3_12 sf .616 .062(. It follows)J 59 500 :M 2.412 .241(from )J 89 500 :M 2.412 .241(Lemma 1 that )J 169 500 :M f4_12 sf .81(A)A f3_12 sf 1.031 .103( and )J f4_12 sf .885(C)A f3_12 sf 1.964 .196( are d-connected given )J f0_12 sf .722(Sepset)A 372 500 :M f3_12 sf <28>S 376 500 :M f4_12 sf 1.319(A)A f3_12 sf .54(,)A f4_12 sf 1.44(C)A f3_12 sf 1.048 .105(\) )J f1_12 sf 1.657A f3_12 sf .54 .054( )J 420 500 :M f0_12 sf (S)S 427 500 :M f3_12 sf 2.471 .247(, which is a)J 59 518 :M (contradiction. Hence, )S 165 518 :M f4_12 sf (B)S f3_12 sf ( is an ancestor of )S f4_12 sf (A)S f3_12 sf ( or )S 279 518 :M f4_12 sf (C)S f3_12 sf ( or of )S 316 518 :M f0_12 sf (S)S 323 518 :M f3_12 sf (.)S 77 542 :M .688 .069(Induction Case: Suppose )J f5_12 sf (p)S 210 542 :M f4_12 sf S 214 544 :M f4_10 sf .288(n)A f3_12 sf 0 -2 rm .811 .081( is a partial ancestral graph of )J 0 2 rm f4_12 sf 0 -2 rm (G)S 0 2 rm 382 542 :M f3_12 sf <28>S 386 542 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 405 542 :M f3_12 sf .142(,)A f0_12 sf .378(L)A f3_12 sf .79 .079(\). We will now)J 59 560 :M (show that )S f5_12 sf (p)S 115 560 :M f4_12 sf S 119 562 :M f4_10 sf (n)S f3_10 sf (+1)S 135 560 :M f3_12 sf ( is a partial ancestral graph of )S 280 560 :M f4_12 sf (G)S 289 560 :M f3_12 sf <28>S 293 560 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 312 560 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (\).)S 77 584 :M .062 .006(Case 1: There is a directed path from )J 259 584 :M f4_12 sf (D)S 268 584 :M f3_12 sf .066 .007( from )J f4_12 sf (A)S f3_12 sf .038 .004( to )J f4_12 sf (B)S f3_12 sf .07 .007( and an edge )J f4_12 sf (A)S f3_12 sf .048 .005( *)J 406 584 :M f1_12 sf .065A f3_12 sf (* )S f4_12 sf (B)S f3_12 sf .035 .003( in )J f5_12 sf (p)S 456 584 :M f4_12 sf S 460 586 :M f4_10 sf (n)S f3_12 sf 0 -2 rm .06 .006(, so )J 0 2 rm 485 584 :M f4_12 sf (A)S 59 602 :M f3_12 sf .192(*)A f1_12 sf .383A f3_12 sf .239 .024(* )J f4_12 sf .234(B)A f3_12 sf .402 .04( is oriented as )J 165 602 :M f4_12 sf .297(A)A f3_12 sf .331 .033( *)J 182 602 :M f1_12 sf S 194 602 :M f3_12 sf .068 .007( )J f4_12 sf .183(B)A f3_12 sf .441 .044(. By the induction hypothesis, there is a directed path from)J 59 620 :M f4_12 sf (A)S f3_12 sf ( to )S f4_12 sf (B)S f3_12 sf ( in )S f4_12 sf (G)S 112 620 :M f3_12 sf <28>S 116 620 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 135 620 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (\). Hence )S 190 620 :M f4_12 sf (B)S f3_12 sf ( is not an ancestor of )S 300 620 :M f4_12 sf (A)S f3_12 sf ( in )S f4_12 sf (G)S 331 620 :M f3_12 sf <28>S 335 620 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 354 620 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (\).)S 77 644 :M .322 .032(Case 2: If )J f4_12 sf .195(B)A f3_12 sf .363 .036( is a collider along <)J f4_12 sf .195(A)A f3_12 sf .08(,)A f4_12 sf .195(B)A f3_12 sf .08(,)A f4_12 sf .213(C)A f3_12 sf .235 .023(> in )J f5_12 sf (p)S 294 647 :M f1_7 sf (2)S 298 644 :M f3_12 sf .514 .051(, )J 305 644 :M f4_12 sf .218(A)A f3_12 sf .378 .038( is not adjacent to )J 403 644 :M f4_12 sf .207(C)A f3_12 sf .129 .013(, )J f4_12 sf .189(B)A f3_12 sf .389 .039( is adjacent to)J 59 662 :M f4_12 sf (D)S 68 662 :M f3_12 sf .114 .011(, and )J f4_12 sf .119 .012(D )J 106 662 :M f3_12 sf .081 .008( is a non-collider along <)J f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (D)S 246 662 :M f3_12 sf (,)S f4_12 sf (C)S f3_12 sf .084 .008(>, then orient )J 325 662 :M f4_12 sf .053 .005(B )J f3_12 sf (*)S f1_12 sf .075A f3_12 sf .047 .005(* )J f4_12 sf (D)S 371 662 :M f3_12 sf .092 .009( as )J f4_12 sf .119 .012(B )J f3_12 sf ( )S f1_12 sf S 412 662 :M f3_12 sf .13 .013( * )J 425 662 :M f4_12 sf (D)S 434 662 :M f3_12 sf .081 .008(. Because )J f4_12 sf (D)S 59 680 :M f3_12 sf .085 .008(is a non-collider along <)J 177 680 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (D)S 196 680 :M f3_12 sf (,)S f4_12 sf (C)S f3_12 sf .061 .006(>, )J 220 680 :M f4_12 sf (D)S 229 680 :M f3_12 sf .103 .01( is an ancestor of )J f4_12 sf .069(A)A f3_12 sf .063 .006( or )J f4_12 sf .076(C)A f3_12 sf .063 .006( or )J f0_12 sf .091(S.)A 374 680 :M f3_12 sf .053 .005( If )J f4_12 sf .066(B)A f3_12 sf .112 .011( is an ancestor of )J 480 680 :M f4_12 sf (D)S 489 680 :M f3_12 sf (,)S endp %%Page: 42 42 %%BeginPageSetup initializepage (peter; page: 42 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (42)S gR gS 0 0 552 730 rC 59 56 :M f3_12 sf .502 .05(then )J 84 56 :M f4_12 sf .314(B)A f3_12 sf .523 .052( is an ancestor of )J f4_12 sf .314(A)A f3_12 sf .298 .03( or )J 203 56 :M f4_12 sf .367(C)A f3_12 sf .306 .031( or )J f0_12 sf (S)S 235 56 :M f3_12 sf .497 .05(. But because )J f4_12 sf .238(B)A f3_12 sf .451 .045( is a collider along <)J 413 56 :M f4_12 sf .2(A)A f3_12 sf .082(,)A f4_12 sf .2(B)A f3_12 sf .082(,)A f4_12 sf .218(C)A f3_12 sf .324 .032(>, it is not)J 59 74 :M (an ancestor of )S f4_12 sf (A)S f3_12 sf ( or )S 155 74 :M f4_12 sf (C)S f3_12 sf ( or )S 179 74 :M f0_12 sf (S)S 186 74 :M f3_12 sf (. Hence )S 226 74 :M f4_12 sf (B)S f3_12 sf ( is not an ancestor of )S 336 74 :M f4_12 sf (D)S 345 74 :M f3_12 sf (.)S 77 98 :M .625 .062(Case 3: If )J 130 98 :M f4_12 sf .374(P)A f3_12 sf .417 .042( *)J 147 98 :M f1_12 sf S 159 0 4 730 rC 159 98 :M f3_12 sf 12 f6_1 :p 3.833 :m .833 .083( )J 159 98 :M 7.666 :m .793 .079( )J gR gS 0 0 552 730 rC 159 98 :M f3_12 sf 12 f6_1 :p 3.833 :m .833 .083( )J 159 0 4 730 rC 159 98 :M 3.833 :m .833 .083( )J 159 98 :M 7.666 :m .793 .079( )J gR gS 163 0 10 730 rC 163 98 :M f4_12 sf 12 f7_1 :p 7.666 :m .793 .079( )J 169 98 :M 7.666 :m .793 .079( )J gR gS 0 0 552 730 rC 163 98 :M f4_12 sf 12 f7_1 :p 9.993 :m (M)S 163 0 10 730 rC 163 98 :M 7.666 :m .793 .079( )J 169 98 :M 7.666 :m .793 .079( )J gR gS 173 0 3 730 rC 172 98 :M f3_12 sf 12 f6_1 :p 7.666 :m .793 .079( )J gR gS 0 0 552 730 rC 173 98 :M f3_12 sf 12 f6_1 :p 9.833 :m .757 .076( *)J 176 0 6 730 rC 176 98 :M 3.833 :m .833 .083( )J 178 98 :M 7.666 :m .793 .079( )J gR gS 0 0 552 730 rC 182 98 :M f1_12 sf .476A f3_12 sf .325 .032(* )J 204 98 :M f4_12 sf .247(R)A f3_12 sf .532 .053( then the orientation is changed to )J 383 98 :M f4_12 sf .374(P)A f3_12 sf .417 .042( *)J 400 98 :M f1_12 sf S 412 98 :M f3_12 sf .833 .083( )J 416 98 :M f4_12 sf (M)S 426 98 :M f3_12 sf .833 .083( )J 430 98 :M f1_12 sf S 442 98 :M f3_12 sf .142 .014( )J f4_12 sf .382(R)A f3_12 sf .726 .073(. By the)J 59 116 :M .948 .095(induction hypothesis, if )J f4_12 sf .33(P)A f3_12 sf .368 .037( *)J 197 116 :M f1_12 sf S 209 0 4 730 rC 209 116 :M f3_12 sf 12 f6_1 :p 4.488 :m 1.488 .149( )J 209 116 :M 8.976 :m 1.417 .142( )J gR gS 0 0 552 730 rC 209 116 :M f3_12 sf 12 f6_1 :p 4.488 :m 1.488 .149( )J 209 0 4 730 rC 209 116 :M 4.488 :m 1.488 .149( )J 209 116 :M 8.976 :m 1.417 .142( )J gR gS 213 0 10 730 rC 213 116 :M f4_12 sf 12 f7_1 :p 8.976 :m 1.417 .142( )J 219 116 :M 8.976 :m 1.417 .142( )J gR gS 0 0 552 730 rC 213 116 :M f4_12 sf 12 f7_1 :p 9.993 :m (M)S 213 0 10 730 rC 213 116 :M 8.976 :m 1.417 .142( )J 219 116 :M 8.976 :m 1.417 .142( )J gR gS 223 0 5 730 rC 223 116 :M f3_12 sf 12 f6_1 :p 4.488 :m 1.488 .149( )J 224 116 :M 8.976 :m 1.417 .142( )J gR gS 0 0 552 730 rC 223 116 :M f3_12 sf 12 f6_1 :p 10.488 :m 1.353 .135( *)J 228 0 6 730 rC 228 116 :M 4.488 :m 1.488 .149( )J 230 116 :M 8.976 :m 1.417 .142( )J gR gS 0 0 552 730 rC 234 116 :M f1_12 sf 1.066A f3_12 sf .666 .067(* )J f4_12 sf .651(R)A f3_12 sf .567 .057( in )J f5_12 sf (p)S 288 116 :M f4_12 sf S 292 118 :M f4_10 sf .475(n)A f3_12 sf 0 -2 rm 1.022 .102(, then in )J 0 2 rm 344 116 :M f4_12 sf (G)S 353 116 :M f3_12 sf <28>S 357 116 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 376 116 :M f3_12 sf .159(,)A f0_12 sf .426(L)A f3_12 sf .31 .031(\) )J f4_12 sf (M)S 405 116 :M f3_12 sf 1.044 .104( is an ancestor of)J 59 134 :M .305 .031(either )J f4_12 sf .125(P)A f3_12 sf .114 .011( or )J f4_12 sf .125(R)A f3_12 sf .119 .012( or )J 136 134 :M f0_12 sf (S)S 143 134 :M f3_12 sf .182 .018(. Because )J 193 134 :M f4_12 sf .069(P)A f3_12 sf .07 .007( *)J f1_12 sf S 221 134 :M f3_12 sf .055 .006( )J f4_12 sf (M)S 234 134 :M f3_12 sf .229 .023( in )J 250 134 :M f5_12 sf (p)S 257 134 :M f4_12 sf S 261 136 :M f4_10 sf .063(n)A f3_12 sf 0 -2 rm .063 .006(, )J 0 2 rm f4_12 sf 0 -2 rm (M)S 0 2 rm 282 134 :M f3_12 sf .207 .021( is not an ancestor of )J f4_12 sf .123(P)A f3_12 sf .116 .012( or )J 410 134 :M f4_12 sf .057(S)A f3_12 sf .134 .013(. Hence )J f4_12 sf (M)S 466 134 :M f3_12 sf .211 .021( is an)J 59 152 :M (ancestor of )S 115 152 :M f4_12 sf (R)S f3_12 sf (..)S 77 176 :M .137 .014(Case 4: If )J f4_12 sf .083(B)A f3_12 sf .032 .003( )J 144 176 :M f1_12 sf S 156 176 :M f3_12 sf .054 .005( * )J f4_12 sf .083(C)A f3_12 sf .052 .005(, )J f4_12 sf .076(B)A f3_12 sf ( )S f1_12 sf S 204 176 :M f3_12 sf ( )S f4_12 sf (D)S 216 176 :M f3_12 sf .1 .01(, and )J 243 176 :M f4_12 sf (D)S 252 176 :M f3_12 sf .047 .005( o)J f1_12 sf .075A f3_12 sf .047 .005(* )J f4_12 sf (C)S f3_12 sf .08 .008(, then orient as )J f4_12 sf (D)S 373 176 :M f3_12 sf .125 .012( )J 377 176 :M f1_12 sf S 389 176 :M f3_12 sf (* )S f4_12 sf (C)S f3_12 sf .089 .009(. By the induction)J 59 194 :M .166 .017(hypothesis, )J 117 194 :M f4_12 sf .14(B)A f3_12 sf .238 .024( is not an ancestor of )J 229 194 :M f4_12 sf .148(C)A f3_12 sf .235 .023(, but is an ancestor of )J 345 194 :M f4_12 sf (D)S 354 194 :M f3_12 sf .258 .026(. Hence )J 395 194 :M f4_12 sf (D)S 404 194 :M f3_12 sf .241 .024( is not an ancestor)J 59 212 :M (of )S 72 212 :M f4_12 sf (C)S f3_12 sf (.)S 77 236 :M .513 .051(Case 5: If )J 129 236 :M f4_12 sf (U)S 138 236 :M f3_12 sf .428 .043( is a definite discriminating path between )J 344 236 :M f4_12 sf .342(A)A f3_12 sf .435 .044( and )J f4_12 sf .374(C)A f3_12 sf .389 .039( for )J 405 236 :M f4_12 sf .343(B)A f3_12 sf .299 .03( in )J f5_12 sf (p)S 435 236 :M f4_12 sf S 439 238 :M f4_10 sf .171(n)A f3_12 sf 0 -2 rm .347 .035(, and )J 0 2 rm f4_12 sf 0 -2 rm (D)S 0 2 rm 480 236 :M f3_12 sf .57 .057( is)J 59 254 :M .183 .018(adjacent to )J f4_12 sf .085(C)A f3_12 sf .083 .008( on )J 141 254 :M f4_12 sf (U)S 150 254 :M f3_12 sf .125 .012(, and )J f4_12 sf (D)S 185 254 :M f3_12 sf .081 .008(, )J f4_12 sf .119(B)A f3_12 sf .172 .017(, and )J 225 254 :M f4_12 sf .099(C)A f3_12 sf .165 .016( form a triangle, then if )J f4_12 sf .091(B)A f3_12 sf .091 .009( is in )J 382 254 :M f0_12 sf (Sepset)S 415 254 :M f3_12 sf <28>S 419 254 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (C)S f3_12 sf .108 .011(\) then mark)J 59 272 :M f4_12 sf .131(B)A f3_12 sf .255 .026( as a non-collider on subpath )J f4_12 sf (D)S 222 272 :M f3_12 sf .146 .015( *)J f1_12 sf S 243 0 6 730 rC 243 272 :M f3_12 sf 12 f6_1 :p 3.409 :m .409 .041( )J 245 272 :M 6.818 :m .39 .039( )J gR gS 0 0 552 730 rC 243 272 :M f3_12 sf 12 f6_1 :p 9.409 :m .372 .037(* )J 249 0 4 730 rC 249 272 :M 3.409 :m .409 .041( )J 249 272 :M 6.818 :m .39 .039( )J gR gS 253 0 7 730 rC 253 272 :M f4_12 sf 12 f7_1 :p 3.409 :m .409 .041( )J 256 272 :M 6.818 :m .39 .039( )J gR gS 0 0 552 730 rC 253 272 :M f4_12 sf 12 f7_1 :p 7.33 :m (B)S 253 0 7 730 rC 253 272 :M 3.409 :m .409 .041( )J 256 272 :M 6.818 :m .39 .039( )J gR gS 260 0 3 730 rC 259 272 :M f3_12 sf 12 f6_1 :p 6.818 :m .39 .039( )J gR gS 0 0 552 730 rC 260 272 :M f3_12 sf 12 f6_1 :p 9.409 :m .372 .037( *)J 263 0 6 730 rC 263 272 :M 3.409 :m .409 .041( )J 265 272 :M 6.818 :m .39 .039( )J gR gS 0 0 552 730 rC 269 272 :M f1_12 sf .234A f3_12 sf .159 .016(* )J 291 272 :M f4_12 sf .174(C)A f3_12 sf .236 .024( else orient )J f4_12 sf (D)S 368 272 :M f3_12 sf .372 .037( *)J 378 272 :M f1_12 sf .253A f3_12 sf .158 .016(* )J f4_12 sf .154(B)A f3_12 sf .158 .016( *)J f1_12 sf .253A f3_12 sf .172 .017(* )J 437 272 :M f4_12 sf .273(C)A f3_12 sf .237 .024( as )J 462 272 :M f4_12 sf (D)S 471 272 :M f3_12 sf .147 .015( *)J f1_12 sf S 59 290 :M f4_12 sf (B)S f3_12 sf ( )S f1_12 sf S 84 290 :M f3_12 sf ( * )S f4_12 sf (C)S f3_12 sf (.)S 77 314 :M .487 .049(There are two cases. First suppose that )J 272 314 :M f4_12 sf .252(B)A f3_12 sf .244 .024( is in )J f0_12 sf .224(Sepset)A 340 314 :M f3_12 sf <28>S 344 314 :M f4_12 sf .153(A)A f3_12 sf .063(,)A f4_12 sf .168(C)A f3_12 sf .402 .04(\). Suppose, contrary to the)J 59 332 :M .167 .017(hypothesis that )J f4_12 sf .058(B)A f3_12 sf .1 .01( is not an ancestor of )J 245 332 :M f4_12 sf .123(C)A f3_12 sf .107 .011( or )J 270 332 :M f4_12 sf (D)S 279 332 :M f3_12 sf .108 .011( or )J f0_12 sf (S)S 302 332 :M f3_12 sf .098 .01( in )J f4_12 sf (G)S 326 332 :M f3_12 sf <28>S 330 332 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 349 332 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf .102 .01(\). Because )J 414 332 :M f0_12 sf (Sepset)S 447 332 :M f3_12 sf <28>S 451 332 :M f4_12 sf .065(A)A f3_12 sf (,)S f4_12 sf .071(C)A f3_12 sf .083 .008(\) is a)J 59 350 :M .691 .069(minimal d-connecting set, and )J f4_12 sf .239(B)A f3_12 sf .232 .023( is in )J f0_12 sf .213(Sepset)A 280 350 :M f3_12 sf <28>S 284 350 :M f4_12 sf .299(A)A f3_12 sf .122(,)A f4_12 sf .327(C)A f3_12 sf .481 .048(\) then )J 335 350 :M f4_12 sf .446(B)A f3_12 sf .744 .074( is an ancestor of )J f4_12 sf .446(A)A f3_12 sf .423 .042( or )J 456 350 :M f4_12 sf .667(C)A f3_12 sf .58 .058( or )J 482 350 :M f0_12 sf (S)S 489 350 :M f3_12 sf (.)S 59 368 :M .455 .045(Because it is not an ancestor of )J 216 368 :M f4_12 sf .346(C)A f3_12 sf .288 .029( or )J f0_12 sf (S)S 248 368 :M f3_12 sf .491 .049( it is an ancestor of )J 346 368 :M f4_12 sf .136(A)A f3_12 sf .375 .038(. Because there are inducing)J 59 386 :M .149 .015(paths between )J f4_12 sf .052(B)A f3_12 sf .069 .007( and )J 161 386 :M f4_12 sf .079(C)A f3_12 sf .1 .01(, and )J f4_12 sf .072(B)A f3_12 sf .092 .009( and )J f4_12 sf (D)S 234 386 :M f3_12 sf .158 .016( in )J 250 386 :M f4_12 sf (G)S 259 386 :M f3_12 sf <28>S 263 386 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 282 386 :M f3_12 sf (,)S f0_12 sf .078(L)A f3_12 sf .102 .01(\), but )J f4_12 sf .071(B)A f3_12 sf .122 .012( is not an ancestor of )J 432 386 :M f4_12 sf .121(C)A f3_12 sf .105 .011( or )J 457 386 :M f4_12 sf (D)S 466 386 :M f3_12 sf .107 .011( or )J f0_12 sf (S)S 489 386 :M f3_12 sf (,)S 59 404 :M .529 .053(it follows from Lemma 11)J 189 404 :M .543 .054( that the inducing paths between )J 352 404 :M f4_12 sf (D)S 361 404 :M f3_12 sf .694 .069( and )J 386 404 :M f4_12 sf .234(B)A f3_12 sf .503 .05(, and between )J f4_12 sf .234(B)A f3_12 sf .5 .05( and)J 59 422 :M f4_12 sf .2(C)A f3_12 sf .311 .031(, are both into )J f4_12 sf .183(B)A f3_12 sf .422 .042(. The directed edge from each vertex )J 330 422 :M f4_12 sf (X)S f4_7 sf 0 3 rm (i)S 0 -3 rm 339 422 :M f3_12 sf .337 .034( on )J f4_12 sf (U)S 367 422 :M f3_12 sf .496 .05( \(except for )J f4_12 sf .268(A)A f3_12 sf .283 .028(\) to )J f4_12 sf .293(C)A f3_12 sf .233 .023( in )J f5_12 sf (p)S 485 422 :M f4_12 sf S 489 422 :M f3_12 sf (,)S 59 440 :M 1.018 .102(entails that )J f4_12 sf .445(X)A f4_7 sf 0 3 rm (i)S 0 -3 rm 127 440 :M f4_12 sf 1.962 .196( )J 132 440 :M f3_12 sf 1.377 .138(is an ancestor of )J 220 440 :M f4_12 sf 1.035(C)A f3_12 sf 1.336 .134( but not of )J 289 440 :M f0_12 sf 1.783 .178(S )J 301 440 :M f3_12 sf .886 .089(in )J f4_12 sf (G)S 324 440 :M f3_12 sf <28>S 328 440 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 347 440 :M f3_12 sf .217(,)A f0_12 sf .58(L)A f3_12 sf 1.186 .119(\). So, the inducing path in)J 59 458 :M f4_12 sf (G)S 68 458 :M f3_12 sf <28>S 72 458 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 91 458 :M f3_12 sf (,)S f0_12 sf .072(L)A f3_12 sf .173 .017(\) corresponding to a bi-directed edge between )J f4_12 sf .066(X)A f4_7 sf 0 3 rm (i)S 0 -3 rm 336 458 :M f3_12 sf .26 .026( and )J 360 458 :M f4_12 sf (X)S f4_7 sf 0 3 rm (i)S 0 -3 rm 369 461 :M f3_7 sf .068(+1)A f3_12 sf 0 -3 rm .137 .014( on )J 0 3 rm f4_12 sf 0 -3 rm (U)S 0 3 rm 403 458 :M f3_12 sf .247 .025( is into )J 440 458 :M f4_12 sf (X)S f4_7 sf 0 3 rm (i)S 0 -3 rm 449 458 :M f3_12 sf .26 .026( and )J 473 458 :M f4_12 sf (X)S f4_7 sf 0 3 rm (i)S 0 -3 rm 482 461 :M f3_7 sf (+1)S f3_12 sf 0 -3 rm (.)S 0 3 rm 59 476 :M .19 .019(Hence in )J 106 476 :M f4_12 sf (G)S 115 476 :M f3_12 sf <28>S 119 476 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 138 476 :M f3_12 sf (,)S f0_12 sf .074(L)A f3_12 sf .162 .016(\) there is in inducing sequence between )J 342 476 :M f4_12 sf .128(A)A f3_12 sf .169 .017( and )J 373 476 :M f4_12 sf .098(C)A f3_12 sf .171 .017(. Hence, by )J 440 476 :M .157 .016(Lemma 10)J 59 494 :M (in )S f4_12 sf (G)S 80 494 :M f3_12 sf <28>S 84 494 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 103 494 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (\) there is an inducing path between )S 285 494 :M f4_12 sf (A)S f3_12 sf ( and )S f4_12 sf (C)S f3_12 sf (, which is a contradiction.)S 77 518 :M .505 .051(Suppose next that )J f4_12 sf .201(B)A f3_12 sf .245 .025( is not in )J 221 518 :M f0_12 sf (Sepset)S 254 518 :M f3_12 sf <28>S 258 518 :M f4_12 sf .14(A)A f3_12 sf .057(,)A f4_12 sf .153(C)A f3_12 sf .327 .033(\). First we will show that every vertex along)J 59 536 :M f4_12 sf (U)S 68 536 :M f3_12 sf .345 .035( except for the endpoints is an ancestor of )J 276 536 :M f0_12 sf (Sepset)S 309 536 :M f3_12 sf <28>S 313 536 :M f4_12 sf .161(A)A f3_12 sf .066(,)A f4_12 sf .176(C)A f3_12 sf .17 .017(\) in )J f4_12 sf (G)S 360 536 :M f3_12 sf <28>S 364 536 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 383 536 :M f3_12 sf (,)S f0_12 sf .076(L)A f3_12 sf .235 .023(\). Suppose, contrary)J 59 554 :M .59 .059(to the hypothesis that some vertex on )J 247 554 :M f4_12 sf (U)S 256 554 :M f3_12 sf .494 .049( is not an ancestor of )J f0_12 sf .261(Sepset)A 397 554 :M f3_12 sf <28>S 401 554 :M f4_12 sf .269(A)A f3_12 sf .11(,)A f4_12 sf .294(C)A f3_12 sf .406 .041(\), and let )J f4_12 sf (W)S 477 554 :M f3_12 sf .751 .075( be)J 59 572 :M 1.225 .122(the closest such vertex on )J f4_12 sf (U)S 204 572 :M f3_12 sf .81 .081( to )J f4_12 sf .929(B)A f3_12 sf 1.605 .161(. It follows that in )J 329 572 :M f4_12 sf (G)S 338 572 :M f3_12 sf <28>S 342 572 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 361 572 :M f3_12 sf .249(,)A f0_12 sf .665(L)A f3_12 sf 1.255 .126(\) there is a sequence of)J 59 590 :M .337 .034(vertices <)J 107 590 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (X)S f3_7 sf 0 3 rm (1)S 0 -3 rm 128 590 :M f3_12 sf (,...,)S f4_12 sf (X)S f3_7 sf 0 3 rm (n)S 0 -3 rm 154 590 :M f3_12 sf (,)S f4_12 sf (W)S 167 590 :M f3_12 sf .435 .044(> such that each pair of vertices )J 328 590 :M f4_12 sf (X)S f4_7 sf 0 3 rm (i)S 0 -3 rm 337 590 :M f3_12 sf .367 .037( and )J f4_12 sf .288(X)A f4_7 sf 0 3 rm (i)S 0 -3 rm 370 593 :M f3_7 sf .1(+1)A f3_12 sf 0 -3 rm .406 .041( that are adjacent in the)J 0 3 rm 59 608 :M 1.181 .118(sequence are d-connected given )J 224 608 :M f0_12 sf (Sepset)S 257 608 :M f3_12 sf <28>S 261 608 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (C)S f3_12 sf (\)\\{)S f4_12 sf (X)S f4_7 sf 0 3 rm (i)S 0 -3 rm 301 608 :M f3_12 sf (,)S f4_12 sf (X)S f4_7 sf 0 3 rm (i)S 0 -3 rm 313 611 :M f3_7 sf .276(+1)A f3_12 sf 0 -3 rm 1.274 .127(} \(because of the existence of the)J 0 3 rm 59 626 :M .204 .02(inducing path into )J f4_12 sf .081(X)A f3_7 sf 0 3 rm (i)S 0 -3 rm 159 626 :M f3_12 sf .211 .021( and )J f4_12 sf .166(X)A f4_7 sf 0 3 rm (i)S 0 -3 rm 192 629 :M f3_7 sf .063(+1)A f3_12 sf 0 -3 rm .243 .024(\), and if a pair of paths d-connects )J 0 3 rm 369 626 :M f4_12 sf (X)S f4_7 sf 0 3 rm (i)S 0 -3 rm 378 629 :M f3_7 sf (-1)S 384 626 :M f3_12 sf .211 .021( and )J f4_12 sf .166(X)A f4_7 sf 0 3 rm (i)S 0 -3 rm 417 626 :M f3_12 sf .295 .029(, and )J 443 626 :M f4_12 sf (X)S f4_7 sf 0 3 rm (i)S 0 -3 rm 452 626 :M f3_12 sf .211 .021( and )J f4_12 sf .166(X)A f4_7 sf 0 3 rm (i)S 0 -3 rm 485 629 :M f3_7 sf (+1)S 59 644 :M f3_12 sf 2.315 .232(respectively, they collide at )J f4_12 sf .88(X)A f4_7 sf 0 3 rm (i)S 0 -3 rm 220 644 :M f3_12 sf 2.936 .294(. By hypothesis, in )J 330 644 :M f4_12 sf (G)S 339 644 :M f3_12 sf <28>S 343 644 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 362 644 :M f3_12 sf .722(,)A f0_12 sf 1.926(L)A f3_12 sf 1.53 .153(\) )J 385 644 :M f4_12 sf (X)S f4_7 sf 0 3 rm (i)S 0 -3 rm 394 644 :M f3_12 sf 3.039 .304( is an ancestor of)J 59 662 :M f0_12 sf (Sepset)S 92 662 :M f3_12 sf <28>S 96 662 :M f4_12 sf .181(A)A f3_12 sf .074(,)A f4_12 sf .197(C)A f3_12 sf .276 .028(\), and )J f4_12 sf (W)S 155 662 :M f3_12 sf .531 .053( is an ancestor of )J 243 662 :M f4_12 sf .381(C)A f3_12 sf .492 .049( but not of )J 307 662 :M f0_12 sf (Sepset)S 340 662 :M f3_12 sf <28>S 344 662 :M f4_12 sf .22(A)A f3_12 sf .09(,)A f4_12 sf .24(C)A f3_12 sf .43 .043(\). It follows there is a path)J 59 680 :M f4_12 sf (D)S 68 680 :M f3_12 sf .141 .014( from )J 98 680 :M f4_12 sf (W)S 108 680 :M f3_12 sf .055 .006( to )J f4_12 sf .069(C)A f3_12 sf .148 .015( that d-connects )J 211 680 :M f4_12 sf (W)S 221 680 :M f3_12 sf .068 .007( and )J f4_12 sf .059(C)A f3_12 sf .088 .009( given )J f0_12 sf .048(Sepset)A 318 680 :M f3_12 sf <28>S 322 680 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (C)S f3_12 sf (\)\\{)S f4_12 sf (W)S 363 680 :M f3_12 sf .147 .015(} in )J 385 680 :M f4_12 sf (G)S 394 680 :M f3_12 sf <28>S 398 680 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 417 680 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf .1 .01(\). By Lemma)J endp %%Page: 43 43 %%BeginPageSetup initializepage (peter; page: 43 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (43)S gR gS 0 0 552 730 rC 59 56 :M f3_12 sf 1.293 .129(1 it follows that )J 146 56 :M f4_12 sf .881(A)A f3_12 sf 1.168 .117( and )J 179 56 :M f4_12 sf .492(C)A f3_12 sf 1.11 .111( are d-connected given )J 307 56 :M f0_12 sf (Sepset)S 340 56 :M f3_12 sf <28>S 344 56 :M f4_12 sf .717(A)A f3_12 sf .294(,)A f4_12 sf .783(C)A f3_12 sf .788 .079(\) in )J 384 56 :M f4_12 sf (G)S 393 56 :M f3_12 sf <28>S 397 56 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 416 56 :M f3_12 sf .238(,)A f0_12 sf .635(L)A f3_12 sf 1.108 .111(\), which is a)J 59 74 :M (contradiction.)S 77 98 :M .461 .046(Since every vertex along )J 202 98 :M f4_12 sf (U)S 211 98 :M f3_12 sf .427 .043( except for the endpoints is an ancestor of )J f0_12 sf .184(Sepset)A 453 98 :M f3_12 sf <28>S 457 98 :M f4_12 sf .156(A)A f3_12 sf .064(,)A f4_12 sf .17(C)A f3_12 sf .267 .027(\) in)J 59 116 :M f4_12 sf (G)S 68 116 :M f3_12 sf <28>S 72 116 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 91 116 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf .063 .006(\), it follows that there is a sequence of vertices <)J 336 116 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (X)S f3_7 sf 0 3 rm (1)S 0 -3 rm 357 116 :M f3_12 sf (,...,)S f4_12 sf (X)S f3_7 sf 0 3 rm (n)S 0 -3 rm 383 116 :M f3_12 sf (,)S f4_12 sf (B)S f3_12 sf .06 .006(> such that each pair)J 59 134 :M 2.601 .26(of vertices )J f4_12 sf 1.145(X)A f4_7 sf 0 3 rm (i)S 0 -3 rm 131 134 :M f3_12 sf 2.796 .28( and )J f4_12 sf 2.197(X)A f4_7 sf 0 3 rm (i)S 0 -3 rm 173 137 :M f3_7 sf .596(+1)A f3_12 sf 0 -3 rm 2.95 .295( that are adjacent in the sequence are d-connected given)J 0 3 rm 59 152 :M f0_12 sf (Sepset)S 92 152 :M f3_12 sf <28>S 96 152 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (C)S f3_12 sf (\)\\{)S f4_12 sf (X)S f4_7 sf 0 3 rm (i)S 0 -3 rm 136 152 :M f3_12 sf (,)S f4_12 sf (X)S f4_7 sf 0 3 rm (i)S 0 -3 rm 148 155 :M f3_7 sf .568(+1)A f3_12 sf 0 -3 rm 2.217 .222(}, and if a pair of paths d-connects )J 0 3 rm 351 152 :M f4_12 sf (X)S f4_7 sf 0 3 rm (i)S 0 -3 rm 360 155 :M f3_7 sf (-1)S 366 152 :M f3_12 sf 1.922 .192( and )J f4_12 sf 1.51(X)A f4_7 sf 0 3 rm (i)S 0 -3 rm 405 152 :M f3_12 sf 2.686 .269(, and )J 437 152 :M f4_12 sf (X)S f4_7 sf 0 3 rm (i)S 0 -3 rm 446 152 :M f3_12 sf 1.922 .192( and )J f4_12 sf 1.51(X)A f4_7 sf 0 3 rm (i)S 0 -3 rm 485 155 :M f3_7 sf (+1)S 59 170 :M f3_12 sf .689 .069(respectively, they collide at )J f4_12 sf .262(X)A f4_7 sf 0 3 rm (i)S 0 -3 rm 208 170 :M f3_12 sf .835 .084(. Since each of the )J f4_12 sf .47(X)A f4_7 sf 0 3 rm (i)S 0 -3 rm 314 170 :M f3_12 sf .92 .092( has a descendant in )J 420 170 :M f0_12 sf (Sepset)S 453 170 :M f3_12 sf <28>S 457 170 :M f4_12 sf .272(A)A f3_12 sf .112(,)A f4_12 sf .298(C)A f3_12 sf .467 .047(\) in)J 59 188 :M f4_12 sf (G)S 68 188 :M f3_12 sf <28>S 72 188 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 91 188 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (\), )S 112 188 :M f4_12 sf (A)S f3_12 sf ( and )S f4_12 sf (B)S f3_12 sf ( are d-connected given )S 261 188 :M f0_12 sf (Sepset)S 294 188 :M f3_12 sf <28>S 298 188 :M f4_12 sf (A)S f3_12 sf (,)S f4_12 sf (C)S f3_12 sf (\) in )S f4_12 sf (G)S 344 188 :M f3_12 sf <28>S 348 188 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 367 188 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf <29>S 77 212 :M .137 .014(Suppose, contrary to the hypothesis, that )J f4_12 sf .052(B)A f3_12 sf .086 .009( is an ancestor of )J f4_12 sf .057(C)A f3_12 sf .11 .011(. There are two cases. If)J 59 230 :M f4_12 sf .947(B)A f3_12 sf 1.577 .158( is an ancestor of )J f0_12 sf .843(Sepset)A 199 230 :M f3_12 sf <28>S 203 230 :M f4_12 sf 1.097(A)A f3_12 sf .449(,)A f4_12 sf 1.198(C)A f3_12 sf 1.721 .172(\), then in )J 277 230 :M f4_12 sf (G)S 286 230 :M f3_12 sf <28>S 290 230 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 309 230 :M f3_12 sf .35(,)A f0_12 sf .935(L)A f3_12 sf .68 .068(\) )J f4_12 sf .856(A)A f3_12 sf 1.089 .109( and )J f4_12 sf .856(B)A f3_12 sf 2.494 .249( are d-connected given)J 59 248 :M f0_12 sf (Sepset)S 92 248 :M f3_12 sf <28>S 96 248 :M f4_12 sf .668(A)A f3_12 sf .273(,)A f4_12 sf .729(C)A f3_12 sf 1.023 .102(\), and )J f4_12 sf .668(B)A f3_12 sf .886 .089( and )J 181 248 :M f4_12 sf .379(C)A f3_12 sf .842 .084( are d-connected given )J f0_12 sf .309(Sepset)A 341 248 :M f3_12 sf <28>S 345 248 :M f4_12 sf .544(A)A f3_12 sf .223(,)A f4_12 sf .594(C)A f3_12 sf .866 .087(\), and )J 397 248 :M f4_12 sf .598(B)A f3_12 sf 1.163 .116( is an ancestor of)J 59 266 :M f0_12 sf (Sepset)S 92 266 :M f3_12 sf <28>S 96 266 :M f4_12 sf .761(A)A f3_12 sf .311(,)A f4_12 sf .831(C)A f3_12 sf 1.096 .11(\) but not in )J f0_12 sf .678(Sepset)A 214 266 :M f3_12 sf <28>S 218 266 :M f4_12 sf .783(A)A f3_12 sf .32(,)A f4_12 sf .855(C)A f3_12 sf 1.577 .158(\). It follows from )J 334 266 :M 1.983 .198(Lemma 1 that )J 412 266 :M f4_12 sf 1.175(A)A f3_12 sf 1.495 .149( and )J f4_12 sf 1.282(C)A f3_12 sf 1.888 .189( are d-)J 59 284 :M 1.831 .183(connected given )J 147 284 :M f0_12 sf (Sepset)S 180 284 :M f3_12 sf <28>S 184 284 :M f4_12 sf .907(A)A f3_12 sf .371(,)A f4_12 sf .99(C)A f3_12 sf 1.818 .182(\), which is a contradiction. If in )J f4_12 sf (G)S 387 284 :M f3_12 sf <28>S 391 284 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 410 284 :M f3_12 sf .557(,)A f0_12 sf 1.487(L)A f3_12 sf 1.083 .108(\) )J f4_12 sf 1.362(B)A f3_12 sf 2.08 .208( is not an)J 59 302 :M .226 .023(ancestor of )J 116 302 :M f0_12 sf (Sepset)S 149 302 :M f3_12 sf <28>S 153 302 :M f4_12 sf .123(A)A f3_12 sf .05(,)A f4_12 sf .134(C)A f3_12 sf .216 .022(\) but is an ancestor of )J 280 302 :M f4_12 sf .143(C)A f3_12 sf .188 .019(, then in )J f4_12 sf (G)S 340 302 :M f3_12 sf <28>S 344 302 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 363 302 :M f3_12 sf (,)S f0_12 sf .111(L)A f3_12 sf .213 .021(\) there is a directed path)J 59 320 :M f4_12 sf (D)S 68 320 :M f3_12 sf 1.021 .102( from )J f4_12 sf .664(B)A f3_12 sf .603 .06( to )J 124 320 :M f4_12 sf .318(C)A f3_12 sf .619 .062( that contains no member of )J f0_12 sf .259(Sepset)A 309 320 :M f3_12 sf <28>S 313 320 :M f4_12 sf .284(A)A f3_12 sf .116(,)A f4_12 sf .31(C)A f3_12 sf .671 .067(\). By Lemma 1)J 407 320 :M .84 .084( it follows that )J 485 320 :M f4_12 sf (B)S 59 338 :M f3_12 sf .09 .009(and )J f4_12 sf (C)S f3_12 sf .112 .011( are d-connected given )J 200 338 :M f0_12 sf (Sepset)S 233 338 :M f3_12 sf <28>S 237 338 :M f4_12 sf .055(A)A f3_12 sf (,)S f4_12 sf .06(C)A f3_12 sf .058 .006(\) in )J f4_12 sf (G)S 283 338 :M f3_12 sf <28>S 287 338 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 306 338 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf .102 .01(\) which is a contradiction. It follows)J 59 356 :M .261 .026(that )J f4_12 sf .137(B)A f3_12 sf .233 .023( is not an ancestor of )J 192 356 :M f4_12 sf .107(C)A f3_12 sf .191 .019(. Because in )J f4_12 sf (G)S 271 356 :M f3_12 sf <28>S 275 356 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 294 356 :M f3_12 sf .058(,)A f0_12 sf .155(L)A f3_12 sf .123 .012(\) )J 313 356 :M f4_12 sf .14(B)A f3_12 sf .238 .024( is not an ancestor of )J 425 356 :M f4_12 sf .173(C)A f3_12 sf .21 .021(, but )J 457 356 :M f4_12 sf (D)S 466 356 :M f3_12 sf .279 .028( is an)J 59 374 :M (ancestor of )S 115 374 :M f4_12 sf (C )S f3_12 sf (by hypothesis, )S f4_12 sf (B)S f3_12 sf ( is not an ancestor of )S 308 374 :M f4_12 sf (D)S 317 374 :M f3_12 sf (. )S 323 365 9 9 rC gS 1.286 1 scale 251.224 374 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 77 398 :M f0_12 sf .383 .038(Theorem 1: )J 141 398 :M f3_12 sf .276 .028(If )J f5_12 sf (p)S 159 398 :M f3_12 sf .385 .038( is a partial ancestral graph, and there is a directed path )J 433 398 :M f4_12 sf (U)S 442 398 :M f3_12 sf .396 .04( from )J f4_12 sf .258(A)A f3_12 sf .361 .036( to)J 59 416 :M f4_12 sf .058(B)A f3_12 sf .051 .005( in )J f5_12 sf (p)S 88 416 :M f3_12 sf .076 .008(, then in every DAG )J f4_12 sf (G)S 198 416 :M f3_12 sf <28>S 202 416 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 221 416 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf .073 .007(\) with PAG )J 291 416 :M f5_12 sf (p)S 298 416 :M f3_12 sf .081 .008(, there is a directed path from )J 443 416 :M f4_12 sf .051(A)A f3_12 sf .044 .004( to )J f4_12 sf .051(B)A f3_12 sf .108 .011( and)J 59 434 :M f4_12 sf (A)S f3_12 sf ( is not an ancestor of )S 169 434 :M f0_12 sf (S)S 176 434 :M f3_12 sf (.)S 77 458 :M (Proof)S 104 458 :M f0_12 sf (. )S f3_12 sf .081 .008(By Theorem 5, for each directed edge between )J 338 458 :M f4_12 sf (M)S 348 458 :M f3_12 sf .091 .009( and )J f4_12 sf .079(N)A f3_12 sf .065 .007( in )J 395 458 :M f4_12 sf (U)S 404 458 :M f3_12 sf .078 .008( there is a directed)J 59 476 :M .615 .062(path from )J f4_12 sf (M)S 121 476 :M f3_12 sf .732 .073( to )J f4_12 sf .917(N)A f3_12 sf .764 .076( in )J 164 476 :M f4_12 sf (G)S 173 476 :M f3_12 sf <28>S 177 476 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 196 476 :M f3_12 sf .138(,)A f0_12 sf .367(L)A f3_12 sf .482 .048(\) and )J f4_12 sf (M)S 246 476 :M f3_12 sf .811 .081( is not an ancestor of )J 356 476 :M f0_12 sf (S)S 363 476 :M f3_12 sf .682 .068(. The concatenation of the)J 59 494 :M 1.449 .145(directed paths in )J 147 494 :M f4_12 sf (G)S 156 494 :M f3_12 sf <28>S 160 494 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 179 494 :M f3_12 sf .282(,)A f0_12 sf .752(L)A f3_12 sf 1.454 .145(\) contains a subpath that is a directed path from )J 444 494 :M f4_12 sf 1.469(A)A f3_12 sf 1.336 .134( to )J 471 494 :M f4_12 sf .846(B)A f3_12 sf 1.187 .119( in)J 59 512 :M f4_12 sf (G)S 68 512 :M f3_12 sf <28>S 72 512 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 91 512 :M f3_12 sf (,)S f0_12 sf .08(L)A f3_12 sf .169 .017(\). Because there is a directed edge between )J 315 512 :M f4_12 sf .085(A)A f3_12 sf .164 .016( and its successor on )J f4_12 sf (U)S 434 512 :M f3_12 sf .096 .01(, )J f4_12 sf .141(A)A f3_12 sf .215 .022( is not an)J 59 530 :M (ancestor of )S 115 530 :M f0_12 sf (S)S 122 530 :M f3_12 sf (. )S 128 521 9 9 rC gS 1.286 1 scale 99.556 530 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 77 554 :M f3_12 sf (A )S 89 554 :M f0_12 sf (semi-directed path from )S 216 554 :M f4_12 sf (A)S f3_12 sf ( to )S f4_12 sf (B)S f3_12 sf ( in partial ancestral graph )S 370 554 :M f5_12 sf (p)S 377 554 :M f3_12 sf ( is an undirected path )S 483 554 :M f4_12 sf (U)S 59 572 :M f3_12 sf .497 .05(from )J f4_12 sf .208(A)A f3_12 sf .181 .018( to )J f4_12 sf .208(B)A f3_12 sf .511 .051( in which no edge contains an arrowhead pointing towards )J f4_12 sf .208(A)A f3_12 sf .371 .037(, that is, there is)J 59 590 :M .412 .041(no arrowhead at )J f4_12 sf .173(A)A f3_12 sf .177 .018( on )J f4_12 sf (U)S 175 590 :M f3_12 sf .426 .043(, and if )J 214 590 :M f4_12 sf .212(X)A f3_12 sf .27 .027( and )J f4_12 sf (Y)S 252 590 :M f3_12 sf .358 .036( are adjacent on the path, and )J f4_12 sf .185(X)A f3_12 sf .342 .034( is between )J f4_12 sf .185(A)A f3_12 sf .394 .039( and)J 59 608 :M f4_12 sf (Y)S 66 608 :M f3_12 sf ( on the path, then there is no arrowhead at the )S 288 608 :M f4_12 sf (X)S f3_12 sf ( end of the edge between )S 418 608 :M f4_12 sf (X)S f3_12 sf ( and )S f4_12 sf (Y)S 455 608 :M f3_12 sf (.)S 77 632 :M f0_12 sf .197 .02(Lemma 15:)J 135 632 :M f3_12 sf .309 .031( If )J 150 632 :M f5_12 sf (p)S 157 632 :M f3_12 sf .229 .023( is a partial ancestral graph of directed acyclic graph )J 415 632 :M f4_12 sf (G)S 424 632 :M f3_12 sf <28>S 428 632 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 447 632 :M f3_12 sf (,)S f0_12 sf .063(L)A f3_12 sf .158 .016(\), there)J 59 650 :M .269 .027(is a directed path )J f4_12 sf (D)S 154 650 :M f3_12 sf .356 .036( from )J f4_12 sf .231(A)A f3_12 sf .202 .02( to )J f4_12 sf .231(B)A f3_12 sf .202 .02( in )J f4_12 sf (G)S 239 650 :M f3_12 sf <28>S 243 650 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 262 650 :M f3_12 sf .054(,)A f0_12 sf .143(L)A f3_12 sf .283 .028(\) that does not contain any member of )J 462 650 :M f0_12 sf (S)S 469 650 :M f3_12 sf .313 .031(, and)J 59 668 :M f0_12 sf (C)S 68 668 :M f3_12 sf .158 .016( is the set of vertices in )J 184 668 :M f0_12 sf .113(O)A f3_12 sf .09 .009( on )J f4_12 sf (D)S 220 668 :M f3_12 sf .143 .014(, then there is a semi-directed path from )J 416 668 :M f4_12 sf .14(A)A f3_12 sf .122 .012( to )J f4_12 sf .14(B)A f3_12 sf .128 .013( in )J 461 668 :M f5_12 sf (p)S 468 668 :M f3_12 sf .174 .017( that)J 59 686 :M (contains just the members of )S 200 686 :M f0_12 sf (C)S 209 686 :M f3_12 sf (.)S endp %%Page: 44 44 %%BeginPageSetup initializepage (peter; page: 44 of 46)setjob %%EndPageSetup gS 0 0 552 730 rC 260 701 30 24 rC 278 722 :M f3_12 sf (44)S gR gS 0 0 552 730 rC 77 56 :M f3_12 sf 1.377 .138(Proof. Suppose there is a directed path )J 280 56 :M f4_12 sf (D)S 289 56 :M f3_12 sf 1.944 .194( from )J f4_12 sf 1.264(A)A f3_12 sf 1.101 .11( to )J f4_12 sf 1.264(B)A f3_12 sf 1.149 .115( in )J 375 56 :M f4_12 sf (G)S 384 56 :M f3_12 sf <28>S 388 56 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 407 56 :M f3_12 sf .243(,)A f0_12 sf .649(L)A f3_12 sf 1.259 .126(\) that does not)J 59 74 :M .928 .093(contain any member of )J f0_12 sf (S)S 186 74 :M f3_12 sf .87 .087(. Let )J f0_12 sf (C)S 223 74 :M f3_12 sf 1.111 .111( be the set of vertices in )J 351 74 :M f0_12 sf .79(O)A f3_12 sf .635 .063( on )J f4_12 sf (D)S 390 74 :M f3_12 sf .791 .079(, and )J f4_12 sf .989<44D5>A 432 74 :M f3_12 sf .888 .089( in )J f5_12 sf (p)S 457 74 :M f3_12 sf 1.171 .117( be the)J 59 92 :M .364 .036(sequence of edges between vertices in )J 249 92 :M f0_12 sf .261(O)A f3_12 sf .352 .035( along )J 292 92 :M f4_12 sf (D)S 301 92 :M f3_12 sf .397 .04( in the order in which they occur on )J f4_12 sf (D)S 489 92 :M f3_12 sf (.)S 59 110 :M .237 .024(Let )J 79 110 :M f4_12 sf .121(X)A f3_12 sf .154 .015( and )J f4_12 sf (Y)S 116 110 :M f3_12 sf .212 .021( be any pair of vertices adjacent on )J 290 110 :M f4_12 sf <44D5>S 303 110 :M f3_12 sf .213 .021( for which )J f4_12 sf .122(X)A f3_12 sf .227 .023( is between )J f4_12 sf .122(A)A f3_12 sf .162 .016( and )J 451 110 :M f4_12 sf (Y)S 458 110 :M f3_12 sf .267 .027( or )J 475 110 :M f4_12 sf .132(X)A f3_12 sf .16 .016( =)J 59 128 :M f4_12 sf .086(A)A f3_12 sf .199 .02(. Because )J 116 128 :M f4_12 sf .133(X)A f3_12 sf .224 .022( is an ancestor of )J 209 128 :M f4_12 sf (Y)S 216 128 :M f3_12 sf .177 .018( in )J f4_12 sf (G)S 240 128 :M f3_12 sf <28>S 244 128 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 263 128 :M f3_12 sf (,)S f0_12 sf .108(L)A f3_12 sf .197 .02(\), the edge in)J 338 128 :M f5_12 sf .301 .03( p)J 349 128 :M f3_12 sf .274 .027( between )J f4_12 sf .127(X)A f3_12 sf .168 .017( and )J 426 128 :M f4_12 sf (Y)S 433 128 :M f3_12 sf .237 .024( has either a)J 59 146 :M .027 .003J f1_12 sf (-)S 104 146 :M f3_12 sf .036 .004(\322 at the )J 142 146 :M f4_12 sf (X)S f3_12 sf .031 .003( end of the edge. Hence )J 266 146 :M f4_12 sf <44D5>S 279 146 :M f3_12 sf .03 .003( is a semi-directed path from )J 420 146 :M f4_12 sf (A)S f3_12 sf .022 .002( to )J f4_12 sf (B)S f3_12 sf .022 .002( in )J f5_12 sf (p)S 471 146 :M f3_12 sf ( that)S 59 164 :M (contains the members of )S 180 164 :M f0_12 sf (C)S 189 164 :M f3_12 sf (.)S 77 188 :M f0_12 sf (Theorem 2)S 133 188 :M (:)S 137 188 :M f3_12 sf ( If )S 151 188 :M f5_12 sf (p)S 158 188 :M f3_12 sf ( is a partial ancestral graph, and there is no semi-directed path from )S 485 188 :M f4_12 sf (A)S 59 206 :M f3_12 sf (to )S f4_12 sf (B)S f3_12 sf .022 .002( in )J f5_12 sf (p)S 100 206 :M f3_12 sf .034 .003( that contains a member of )J 231 206 :M f0_12 sf (C)S 240 206 :M f3_12 sf .032 .003(, then every directed path from )J 392 206 :M f4_12 sf (A)S f3_12 sf ( to )S f4_12 sf (B)S f3_12 sf .04 .004( in every DAG)J 59 224 :M f4_12 sf (G)S 68 224 :M f3_12 sf <28>S 72 224 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 91 224 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (\) with PAG )S f5_12 sf (p)S 167 224 :M f3_12 sf ( that contains a member of )S 298 224 :M f0_12 sf (C)S 307 224 :M f3_12 sf ( also contains a member of )S 442 224 :M f0_12 sf (S)S 449 224 :M f3_12 sf (.)S 77 248 :M (Proof. This follow from )S 195 248 :M (Lemma 15)S 247 248 :M (. )S 253 239 9 9 rC gS 1.286 1 scale 196.779 248 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 77 272 :M f0_12 sf .29 .029(Theorem 3: )J 141 272 :M f3_12 sf .209 .021(If )J f5_12 sf (p)S 159 272 :M f3_12 sf .3 .03( is a partial ancestral graph of DAG )J 337 272 :M f4_12 sf (G)S 346 272 :M f3_12 sf <28>S 350 272 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 369 272 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (\),)S 387 272 :M f0_12 sf .058 .006( )J f3_12 sf .352 .035(and there is no semi-)J 59 290 :M 1.353 .135(directed path from )J 158 290 :M f4_12 sf 1.223(A)A f3_12 sf 1.066 .107( to )J f4_12 sf 1.223(B)A f3_12 sf 1.066 .107( in )J f5_12 sf (p)S 217 290 :M f3_12 sf 1.427 .143(, then every directed path from )J 381 290 :M f4_12 sf 1.4(A)A f3_12 sf 1.274 .127( to )J 408 290 :M f4_12 sf .608(B)A f3_12 sf 1.402 .14( in every DAG)J 59 308 :M f4_12 sf (G)S 68 308 :M f3_12 sf <28>S 72 308 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 91 308 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (\) with PAG )S f5_12 sf (p)S 167 308 :M f3_12 sf ( contains a member of )S 277 308 :M f0_12 sf (S)S 284 308 :M f3_12 sf (.)S 77 332 :M .149 .015(Proof. By Lemma 15)J 180 332 :M .188 .019(, if there is a directed path from )J f4_12 sf .103(A)A f3_12 sf .09 .009( to )J f4_12 sf .103(B)A f3_12 sf .094 .009( in )J 381 332 :M f4_12 sf (G)S 390 332 :M f3_12 sf <28>S 394 332 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 413 332 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf .118 .012(\) that contains)J 59 350 :M (no member of )S f0_12 sf (S)S 136 350 :M f3_12 sf (, there is a semi-directed path from )S 307 350 :M f4_12 sf (A)S f3_12 sf ( to )S f4_12 sf (B)S f3_12 sf ( in )S f5_12 sf (p)S 358 350 :M f3_12 sf (. )S 367 341 9 9 rC gS 1.286 1 scale 285.446 350 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 77 374 :M f0_12 sf .069 .007(Theorem 4)J 133 374 :M .059 .006(: )J f3_12 sf .086 .009(If )J f5_12 sf (p)S 158 374 :M f3_12 sf .079 .008( is a partial ancestral graph,)J 291 374 :M f0_12 sf ( )S f3_12 sf .089 .009(and every semi-directed path from )J 463 374 :M f4_12 sf .061(A)A f3_12 sf .053 .005( to )J f4_12 sf (B)S 59 392 :M f3_12 sf .895 .09(contains some member of )J 192 392 :M f0_12 sf (C)S 201 392 :M f3_12 sf 1.226 .123( in )J 218 392 :M f5_12 sf (p)S 225 392 :M f3_12 sf .929 .093(, then every directed path from )J 385 392 :M f4_12 sf .512(A)A f3_12 sf .446 .045( to )J f4_12 sf .512(B)A f3_12 sf 1.18 .118( in every DAG)J 59 410 :M f4_12 sf (G)S 68 410 :M f3_12 sf <28>S 72 410 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 91 410 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (\) with PAG )S f5_12 sf (p)S 167 410 :M f3_12 sf ( contains a member of )S 280 410 :M f0_12 sf (S)S 287 410 :M f3_12 sf ( )S f1_12 sf S f3_12 sf ( )S f0_12 sf (C)S 311 410 :M f3_12 sf (.)S 77 434 :M (Proof.)S 107 434 :M f0_12 sf ( )S 110 434 :M f3_12 sf (Suppose that )S 175 434 :M f4_12 sf (U)S 184 434 :M f3_12 sf -.006( is a directed path in )A 284 434 :M f4_12 sf (G)S 293 434 :M f3_12 sf <28>S 297 434 :M f0_12 sf (O)S f3_12 sf (,)S f0_12 sf (S)S 316 434 :M f3_12 sf (,)S f0_12 sf (L)S f3_12 sf (\) from )S f4_12 sf (A)S f3_12 sf ( to )S f4_12 sf (B)S f3_12 sf -.005( that does not contain)A 59 452 :M .622 .062(a member of )J 126 452 :M f0_12 sf (C)S 135 452 :M f3_12 sf .5 .05( or )J f0_12 sf (S)S 159 452 :M f3_12 sf .654 .065(. Then by )J 211 452 :M .5 .05(Lemma 15)J 264 452 :M .567 .057( there is a semi-directed path from )J 438 452 :M f4_12 sf .55(A)A f3_12 sf .5 .05( to )J 461 452 :M f4_12 sf .426(B)A f3_12 sf .371 .037( in )J f5_12 sf (p)S 59 470 :M f3_12 sf (that does not contain any member of )S f0_12 sf (C)S 246 470 :M f3_12 sf (. )S 252 461 9 9 rC gS 1.286 1 scale 196.001 470 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 59 519 :M f0_14 sf (I)S 64 519 :M (X)S 74 519 :M (.)S 77 519 :M ( )S 95 519 :M (Bibliography)S 59 536 :M f3_12 sf .077 .008(Beinlich, I., Suermondt, H., Chavez, R., and Cooper, G. \(1989\). The ALARM monitoring)J 77 548 :M .284 .028(system: A case study with two probabilistic inference techniques for belief networks,)J 77 560 :M .137 .014(in Proc. Second European Conference on Artificial Intelligence in Medicine, London,)J 77 572 :M (England. 247-256.)S 59 590 :M (Bollen, K. \(1989\). Structural Equations with Latent Variables. 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