%!PS-Adobe-3.0 %%Title: (Microsoft Word - mags3) %%Creator: (Microsoft Word: LaserWriter 8 8.3.4) %%CreationDate: (11:24 AM Tuesday, November 4, 1997) %%For: (peter) %%Pages: 18 %%DocumentFonts: Times-Roman Symbol Times-Bold Times-Italic %%DocumentNeededFonts: Times-Roman Symbol Times-Bold Times-Italic %%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 - mags3)def /Creator(Microsoft Word: LaserWriter 8 8.3.4)def /CreationDate(11:24 AM Tuesday, November 4, 1997)def /For(peter)def /Pages 18 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 216 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 $m setmatrix stroke }bd %%EndFile %%BeginFile: adobe_psp_customps %%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. All Rights Reserved. /$t Z /$p Z /$s Z /$o 1. def /2state? false def /ps Z level2 startnoload /pushcolor/currentrgbcolor ld /popcolor/setrgbcolor ld /setcmykcolor where { pop/currentcmykcolor where { pop/pushcolor/currentcmykcolor ld /popcolor/setcmykcolor ld }if }if level2 endnoload level2 not startnoload /pushcolor { currentcolorspace $c eq { currentcolor currentcolorspace true }{ currentcmykcolor false }ifelse }bd /popcolor { { setcolorspace setcolor }{ setcmykcolor }ifelse }bd level2 not endnoload /pushstatic { ps 2state? $o $t $p $s $cs }bd /popstatic { /$cs xs /$s xs /$p xs /$t xs /$o xs /2state? xs /ps xs }bd /pushgstate { save errordict/nocurrentpoint{pop 0 0}put currentpoint 3 -1 roll restore pushcolor currentlinewidth currentlinecap currentlinejoin currentdash exch aload length np clippath pathbbox $m currentmatrix aload pop }bd /popgstate { $m astore setmatrix 2 index sub exch 3 index sub exch rC array astore exch setdash setlinejoin setlinecap lw popcolor np :M }bd /bu { pushgstate gR pushgstate 2state? { gR pushgstate }if pushstatic pm restore mT concat }bd /bn { /pm save store popstatic popgstate gS popgstate 2state? { gS popgstate }if }bd /cpat{pop 64 div setgray 8{pop}repeat}bd %%EndFile %%BeginFile: adobe_psp_basic_text %%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. All Rights Reserved. /S/show ld /A{ 0.0 exch ashow }bd /R{ 0.0 exch 32 exch widthshow }bd /W{ 0.0 3 1 roll widthshow }bd /J{ 0.0 32 4 2 roll 0.0 exch awidthshow }bd /V{ 0.0 4 1 roll 0.0 exch awidthshow }bd /fcflg true def /fc{ fcflg{ vmstatus exch sub 50000 lt{ (%%[ Warning: Running out of memory ]%%\r)print flush/fcflg false store }if pop }if }bd /$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 { $m scale :mf setfont }bd /ps Z /fz{/ps xs}bd /sf/setfont ld /cF/currentfont ld /mbf { /makeblendedfont where { pop makeblendedfont /ABlend exch definefont }{ pop }ifelse def }def %%EndFile %%BeginFile: adobe_psp_derived_styles %%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. All Rights Reserved. /wi version(23.0)eq { { gS 0 0 0 0 rC stringwidth gR }bind }{ /stringwidth load }ifelse def /$o 1. def /gl{$o G}bd /ms{:M S}bd /condensedmtx[.82 0 0 1 0 0]def /:mc { condensedmtx :mf def }bd /extendedmtx[1.18 0 0 1 0 0]def /:me { extendedmtx :mf def }bd /basefont Z /basefonto Z /dxa Z /dxb Z /dxc Z /dxd Z /dsdx2 Z /bfproc Z /:fbase { dup/FontType get 0 eq{ dup length dict begin dup{1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse}forall /FDepVector exch/FDepVector get[exch/:fbase load forall]def }/bfproc load ifelse /customfont currentdict end definefont }bd /:mo { /bfproc{ dup dup length 2 add dict begin { 1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse }forall /PaintType 2 def /StrokeWidth .012 0 FontMatrix idtransform pop def /customfont currentdict end definefont 8 dict begin /basefonto xdf /basefont xdf /FontType 3 def /FontMatrix[1 0 0 1 0 0]def /FontBBox[0 0 1 1]def /Encoding StandardEncoding def /BuildChar { exch begin basefont setfont ( )dup 0 4 -1 roll put dup wi setcharwidth 0 0 :M gS gl dup show gR basefonto setfont show end }def }store :fbase }bd /:mso { /bfproc{ 7 dict begin /basefont xdf /FontType 3 def /FontMatrix[1 0 0 1 0 0]def /FontBBox[0 0 1 1]def /Encoding StandardEncoding def /BuildChar { exch begin sD begin /dxa 1 ps div def basefont setfont ( )dup 0 4 -1 roll put dup wi 1 index 0 ne { exch dxa add exch }if setcharwidth dup 0 0 ms dup dxa 0 ms dup dxa dxa ms dup 0 dxa ms gl dxa 2. div dup ms end end }def }store :fbase }bd /:ms { /bfproc{ dup dup length 2 add dict begin { 1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse }forall /PaintType 2 def /StrokeWidth .012 0 FontMatrix idtransform pop def /customfont currentdict end definefont 8 dict begin /basefonto xdf /basefont xdf /FontType 3 def /FontMatrix[1 0 0 1 0 0]def /FontBBox[0 0 1 1]def /Encoding StandardEncoding def /BuildChar { exch begin sD begin /dxb .05 def basefont setfont ( )dup 0 4 -1 roll put dup wi exch dup 0 ne { dxb add }if exch setcharwidth dup dxb .01 add 0 ms 0 dxb :T gS gl dup 0 0 ms gR basefonto setfont 0 0 ms end end }def }store :fbase }bd /:mss { /bfproc{ 7 dict begin /basefont xdf /FontType 3 def /FontMatrix[1 0 0 1 0 0]def /FontBBox[0 0 1 1]def /Encoding StandardEncoding def /BuildChar { exch begin sD begin /dxc 1 ps div def /dsdx2 .05 dxc 2 div add def basefont setfont ( )dup 0 4 -1 roll put dup wi exch dup 0 ne { dsdx2 add }if exch setcharwidth dup dsdx2 .01 add 0 ms 0 .05 dxc 2 div sub :T dup 0 0 ms dup dxc 0 ms dup dxc dxc ms dup 0 dxc ms gl dxc 2 div dup ms end end }def }store :fbase }bd /:msb { /bfproc{ 7 dict begin /basefont xdf /FontType 3 def /FontMatrix[1 0 0 1 0 0]def /FontBBox[0 0 1 1]def /Encoding StandardEncoding def /BuildChar { exch begin sD begin /dxd .03 def basefont setfont ( )dup 0 4 -1 roll put dup wi 1 index 0 ne { exch dxd add exch }if setcharwidth dup 0 0 ms dup dxd 0 ms dup dxd dxd ms 0 dxd ms end end }def }store :fbase }bd /italicmtx[1 0 -.212557 1 0 0]def /:mi { italicmtx :mf def }bd /:v { [exch dup/FontMatrix get exch dup/FontInfo known { /FontInfo get dup/UnderlinePosition known { dup/UnderlinePosition get 2 index 0 3 1 roll transform exch pop }{ .1 }ifelse 3 1 roll dup/UnderlineThickness known { /UnderlineThickness get exch 0 3 1 roll transform exch pop abs }{ pop pop .067 }ifelse }{ pop pop .1 .067 }ifelse ] }bd /$t Z /$p Z /$s Z /:p { aload pop 2 index mul/$t xs 1 index mul/$p xs .012 mul/$s xs }bd /:m {gS 0 $p rm $t lw 0 rl stroke gR }bd /:n { gS 0 $p rm $t lw 0 rl gS gl stroke gR strokepath $s lw /setstrokeadjust where{pop currentstrokeadjust true setstrokeadjust stroke setstrokeadjust }{ stroke }ifelse gR }bd /:o {gS 0 $p rm $t 2 div dup rm $t lw dup 0 rl stroke gR :n }bd %%EndFile /currentpacking where {pop sc_oldpacking setpacking}if end %%EndProlog %%BeginSetup md begin countdictstack[{ %%BeginFeature: *ManualFeed False statusdict /manualfeed false put %%EndFeature }featurecleanup countdictstack[{ %%BeginFeature: *InputSlot Cassette %%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-Roman %%IncludeFont: Symbol %%IncludeFont: Times-Bold %%IncludeFont: Times-Italic /f0_1/Times-Roman :mre /f0_12 f0_1 12 scf /f0_10 f0_1 10 scf /f0_9 f0_1 9 scf /f0_7 f0_1 7 scf /f0_6 f0_1 6 scf /f1_1/Symbol :bsr 240/apple pd :esr /f1_12 f1_1 12 scf /f1_10 f1_1 10 scf /f1_7 f1_1 7 scf /f1_6 f1_1 6 scf /f2_1/Times-Bold :mre /f2_16 f2_1 16 scf /f2_12 f2_1 12 scf /f2_7 f2_1 7 scf /f3_1 f1_1 def /f3_12 f3_1 12 scf /f3_7 f3_1 7 scf /f4_1/Times-Italic :mre /f4_12 f4_1 12 scf /Courier findfont[10 0 0 -10 0 0]:mf setfont %%EndSetup %%Page: 1 1 %%BeginPageSetup initializepage (peter; page: 1 of 18)setjob %%EndPageSetup gS 0 0 552 730 rC 467 698 24 27 rC 485 721 :M f0_12 sf (1)S gR gS 0 0 552 730 rC 123 64 :M f2_16 sf -.158(The Dimensionality of Mixed Ancestral Graphs)A 156 91 :M f0_12 sf -.06(by Peter Spirtes, Thomas Richardson and Chris Meek)A 77 118 :M (First )S 105 118 :M (we )S 125 118 :M -.166(will )A 150 118 :M -.146(introduce )A 201 118 :M -.082(some )A 233 118 :M -.064(graph )A 267 118 :M -.055(terminology. )A 335 118 :M -.219(The )A 360 118 :M -.122(concepts )A 408 118 :M -.14(defined )A 450 118 :M -.161(here )A 477 118 :M -.323(are)A 59 139 :M -.053(illustrated in Figure 1. A graph consists of two parts, a set of vertices )A 390 139 :M f2_12 sf .25(V)A f0_12 sf .087 .009( )J 403 139 :M -.109(and )A 424 139 :M -.326(a )A 433 139 :M -.109(set )A 450 139 :M (of )S 464 139 :M -.08(edges)A 59 160 :M f2_12 sf -.11(E)A f0_12 sf -.065(. Each edge in )A f2_12 sf -.11(E)A f0_12 sf -.063( is between two distinct vertices in )A f2_12 sf -.119(V)A f0_12 sf -.067(. There are two kinds of edges in )A 479 160 :M f2_12 sf .724(E)A f0_12 sf (,)S 59 181 :M -.073(directed edges A )A f1_12 sf -.177A f0_12 sf -.072( B or A )A f1_12 sf -.177A f0_12 sf -.076( B, and double-headed edges A )A f1_12 sf -.186A f0_12 sf -.074( B; in )A 396 181 :M -.219(either )A 426 181 :M -.161(case )A 450 181 :M -.663(A )A 462 181 :M -.109(and )A 483 181 :M (B)S 59 202 :M .065 .007(are )J f2_12 sf .029(endpoints)A f0_12 sf .065 .007( of the edge; further, A and B are said to be )J f2_12 sf .028(adjacent)A f0_12 sf .061 .006(. In Figure 1 the set )J 481 202 :M (of)S 59 223 :M .101 .01(vertices is {A,B,C,D,E} and the set of edges is {A )J f1_12 sf .078A f0_12 sf .048 .005( B, B )J f1_12 sf .074A f0_12 sf .048 .005( C, C )J f1_12 sf .074A f0_12 sf .05 .005( D, E )J 427 223 :M f1_12 sf .126A f0_12 sf ( )S 443 223 :M .444 .044(D}. )J 465 223 :M .258 .026(For )J 486 223 :M (a)S 59 244 :M -.205(directed )A 100 244 :M -.163(edge )A 126 244 :M -.663(A )A 138 244 :M f1_12 sf .126A f0_12 sf ( )S 154 244 :M .83 .083(B, )J 170 244 :M -.663(A )A 182 244 :M (is )S 194 244 :M -.22(the )A 213 244 :M f2_12 sf .284(tail)A f0_12 sf .204 .02( )J 236 244 :M (of )S 251 244 :M -.22(the )A 270 244 :M -.163(edge )A 297 244 :M -.109(and )A 319 244 :M (B )S 332 244 :M (is )S 345 244 :M -.22(the )A 364 244 :M f2_12 sf .297(head)A f0_12 sf .144 .014( )J 395 244 :M (of )S 410 244 :M -.22(the )A 429 244 :M .232 .023(edge, )J 460 244 :M -.663(A )A 473 244 :M (is )S 486 244 :M (a)S 59 265 :M f2_12 sf .148(parent)A f0_12 sf .239 .024( of B, and B is a )J f2_12 sf .132(child)A f0_12 sf .288 .029( of A.)J 77 292 :M .599 .06(An )J f2_12 sf 1.439 .144(undirected path)J f0_12 sf .294 .029( U )J 197 292 :M -.139(between )A 240 292 :M .478(X)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 257 292 :M -.109(and )A 278 292 :M .478(X)A f0_7 sf 0 3 rm .193(n)A 0 -3 rm f0_12 sf .166 .017( )J 295 292 :M (is )S 307 292 :M -.326(a )A 316 292 :M -.121(sequence )A 363 292 :M (of )S 377 292 :M -.064(edges )A 408 292 :M .621( )J 469 292 :M (such)S 59 313 :M -.165(that one )A 100 313 :M -.124(endpoint )A 145 313 :M (of )S 159 313 :M .09(E)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf ( )S 174 313 :M (is )S 186 313 :M .876(X)A f0_7 sf 0 3 rm .354(1)A 0 -3 rm f0_12 sf .552 .055(, )J 207 313 :M -.109(one )A 228 313 :M -.124(endpoint )A 273 313 :M (of )S 287 313 :M -.36(E)A f0_7 sf 0 3 rm -.267(m)A 0 -3 rm f0_12 sf ( )S 303 313 :M (is )S 315 313 :M .876(X)A f0_7 sf 0 3 rm .354(n)A 0 -3 rm f0_12 sf .552 .055(, )J 336 313 :M -.109(and )A 357 313 :M (for )S 375 313 :M -.245(each )A 400 313 :M -.163(pair )A 422 313 :M (of )S 436 313 :M -.164(consecutive)A 59 334 :M -.036(edges E)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.03(, E)A f0_7 sf 0 3 rm -.021(i+1)A 0 -3 rm f0_12 sf -.032( in the sequence, E)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S f1_12 sf S f0_12 sf -.034( E)A f0_7 sf 0 3 rm -.021(i+1)A 0 -3 rm f0_12 sf -.032(, and one endpoint of E)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.033( equals one endpoint of )A 470 334 :M .475(E)A f0_7 sf 0 3 rm .203(i+1)A 0 -3 rm f0_12 sf (.)S 59 355 :M .077 .008(In Figure 1, A )J f1_12 sf .075A f0_12 sf ( )S 147 355 :M (B )S 159 355 :M f1_12 sf .126A f0_12 sf ( )S 175 355 :M (C )S 187 355 :M f1_12 sf .126A f0_12 sf ( )S 203 355 :M -.663(D )A 215 355 :M (is )S 227 355 :M -.163(an )A 242 355 :M -.235(example )A 285 355 :M (of )S 299 355 :M -.163(an )A 314 355 :M -.164(undirected )A 367 355 :M -.165(path )A 391 355 :M -.139(between )A 434 355 :M -.663(A )A 446 355 :M -.109(and )A 467 355 :M .281 .028(D. )J 483 355 :M (A)S 59 376 :M f2_12 sf .546 .055(directed path)J f0_12 sf .281 .028( P between X)J f0_7 sf 0 3 rm .064(1)A 0 -3 rm f0_12 sf .226 .023( and X)J f0_7 sf 0 3 rm .064(n)A 0 -3 rm f0_12 sf .294 .029( is a sequence of directed edges )J 448 376 :M (such )S 474 376 :M -.331(that)A 59 397 :M -.005(the tail of E)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf ( is X)S f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.005(, the head of E)A f0_7 sf 0 3 rm (m)S 0 -3 rm f0_12 sf ( is X)S f0_7 sf 0 3 rm (n)S 0 -3 rm f0_12 sf (, )S 253 397 :M -.109(and )A 274 397 :M (for )S 292 397 :M -.245(each )A 317 397 :M -.163(pair )A 339 397 :M (of )S 353 397 :M -.064(edges )A 384 397 :M .348(E)A f0_7 sf 0 3 rm .092(i)A 0 -3 rm f0_12 sf .259 .026(, )J 401 397 :M .103(E)A f0_7 sf 0 3 rm .044(i+1)A 0 -3 rm f0_12 sf ( )S 422 397 :M -.247(adjacent )A 464 397 :M -.167(in )A 477 397 :M -.33(the)A 59 418 :M (sequence, )S 110 418 :M -.164(E)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S 123 418 :M f1_12 sf .285A f0_12 sf .13 .013( )J 134 418 :M .412(E)A f0_7 sf 0 3 rm .176(i+1)A 0 -3 rm f0_12 sf .307 .031(, )J 159 418 :M -.109(and )A 180 418 :M -.22(the )A 198 418 :M -.163(head )A 224 418 :M (of )S 238 418 :M -.164(E)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S 251 418 :M (is )S 263 418 :M -.22(the )A 281 418 :M -.332(tail )A 299 418 :M (of )S 313 418 :M .412(E)A f0_7 sf 0 3 rm .176(i+1)A 0 -3 rm f0_12 sf .307 .031(. )J 338 418 :M .258 .026(For )J 359 418 :M -.081(example, )A 406 418 :M (B )S 418 418 :M f1_12 sf .126A f0_12 sf ( )S 434 418 :M (C )S 446 418 :M f1_12 sf .126A f0_12 sf ( )S 462 418 :M -.663(D )A 474 418 :M (is )S 486 418 :M (a)S 59 439 :M .904 .09(directed path. A )J f2_12 sf 1.006 .101(vertex occurs on a path)J f0_12 sf .159 .016( )J 273 439 :M -.164(if )A 284 439 :M -.334(it )A 294 439 :M (is )S 306 439 :M -.163(an )A 321 439 :M -.124(endpoint )A 366 439 :M (of )S 380 439 :M -.109(one )A 401 439 :M (of )S 415 439 :M -.22(the )A 433 439 :M -.064(edges )A 464 439 :M -.167(in )A 477 439 :M -.33(the)A 59 460 :M -.078(path. The set of vertices )A 176 460 :M (on )S 192 460 :M -.663(A )A 204 460 :M f1_12 sf .4A f0_12 sf .096 .01( )J 221 460 :M (B )S 233 460 :M f1_12 sf .126A f0_12 sf ( )S 249 460 :M (C )S 261 460 :M f1_12 sf .126A f0_12 sf ( )S 277 460 :M -.663(D )A 289 460 :M (is )S 301 460 :M .444 .044({A, )J 323 460 :M .83 .083(B, )J 339 460 :M .83 .083(C, )J 355 460 :M .444 .044(D}. )J 377 460 :M -.663(A )A 389 460 :M -.165(path )A 413 460 :M (is )S 425 460 :M f2_12 sf .572(acyclic)A f0_12 sf .347 .035( )J 468 460 :M -.164(if )A 479 460 :M (no)S 59 481 :M -.087(vertex occurs more than once on the path. )A 260 481 :M -.219(The )A 282 481 :M -.073(following )A 332 481 :M (is )S 344 481 :M -.326(a )A 353 481 :M -.167(list )A 371 481 :M (of )S 385 481 :M -.331(all )A 400 481 :M -.22(the )A 418 481 :M -.282(acyclic )A 454 481 :M -.234(directed)A 59 502 :M .058 .006(paths in Figure 1: B )J f1_12 sf S f0_12 sf .033 .003( C, C )J f1_12 sf S f0_12 sf .033 .003( D, E )J f1_12 sf S f0_12 sf .033 .003( D, B )J f1_12 sf S f0_12 sf .025 .003( C )J f1_12 sf S f0_12 sf .051 .005( D.)J 77 529 :M -.663(A )A 90 529 :M -.064(graph )A 122 529 :M (is )S 135 529 :M -.326(a )A 145 529 :M f2_12 sf 2.237 .224(directed )J 196 529 :M .243(graph)A f0_12 sf .119 .012( )J 233 529 :M -.164(if )A 245 529 :M -.334(it )A 256 529 :M -.123(contains )A 301 529 :M (no )S 319 529 :M -.125(double-headed )A 394 529 :M (edges.A )S 439 529 :M -.064(graph )A 472 529 :M (is )S 486 529 :M (a)S 59 550 :M f2_12 sf 2.237 .224(directed )J 109 550 :M 3.149 .315(acyclic )J 153 550 :M .243(graph)A f0_12 sf .119 .012( )J 189 550 :M -.195(\(DAG\) )A 226 550 :M -.164(if )A 237 550 :M -.334(it )A 247 550 :M -.123(contains )A 290 550 :M (no )S 306 550 :M -.125(double-headed )A 379 550 :M .425 .043(edges, )J 415 550 :M -.109(and )A 437 550 :M (no )S 454 550 :M -.234(directed)A 59 571 :M (cycles.)S 77 598 :M -.663(A )A 89 598 :M -.163(vertex )A 122 598 :M -.663(A )A 134 598 :M (is )S 146 598 :M -.163(an )A 161 598 :M f2_12 sf .431(ancestor)A f0_12 sf .239 .024( )J 212 598 :M (of )S 226 598 :M (B )S 238 598 :M -.08(\(and )A 263 598 :M (B )S 275 598 :M (is )S 287 598 :M -.326(a )A 296 598 :M f2_12 sf .445(descendant)A f0_12 sf .233 .023( )J 362 598 :M (of )S 376 598 :M -.328(A\) )A 393 598 :M -.164(if )A 405 598 :M -.109(and )A 427 598 :M -.083(only )A 453 598 :M -.164(if )A 465 598 :M -.263(either)A 59 619 :M -.155(there is a directed )A 145 619 :M -.165(path )A 169 619 :M -.08(from )A 196 619 :M -.663(A )A 208 619 :M -.167(to )A 221 619 :M (B )S 233 619 :M (or )S 247 619 :M -.663(A )A 259 619 :M .211 .021(= )J 270 619 :M .83 .083(B. )J 286 619 :M (Thus )S 314 619 :M -.22(the )A 332 619 :M -.122(ancestor )A 375 619 :M -.206(relation )A 414 619 :M (is )S 426 619 :M -.22(the )A 444 619 :M -.065(transitive,)A 59 640 :M -.181(reflexive )A 107 640 :M -.092(closure )A 148 640 :M (of )S 165 640 :M -.22(the )A 186 640 :M -.163(parent )A 222 640 :M -.072(relation. )A 268 640 :M -.219(The )A 293 640 :M -.073(following )A 346 640 :M -.264(table )A 375 640 :M -.067(lists )A 401 640 :M -.22(the )A 423 640 :M (child, )S 458 640 :M (parent,)S 59 661 :M -.08(descendant and ancestor relations in Figure 1.)A endp %%Page: 2 2 %%BeginPageSetup initializepage (peter; page: 2 of 18)setjob %%EndPageSetup gS 0 0 552 730 rC 467 698 24 27 rC 485 721 :M f0_12 sf (2)S gR gS 0 0 552 730 rC 82 86 :M f0_12 sf -.329(Vertex)A 166 86 :M -.142(Children)A 257 86 :M -.053(Parents)A 334 86 :M -.164(Descendants)A 428 86 :M -.122(Ancestors)A 53 62 1 1 rF 53 62 1 1 rF 54 62 87 1 rF 141 62 1 1 rF 142 62 88 1 rF 230 62 1 1 rF 231 62 87 1 rF 318 62 1 1 rF 319 62 88 1 rF 407 62 1 1 rF 408 62 87 1 rF 495 62 1 1 rF 495 62 1 1 rF 53 63 1 27 rF 141 63 1 27 rF 230 63 1 27 rF 318 63 1 27 rF 407 63 1 27 rF 495 63 1 27 rF 94 114 :M (A)S 182 114 :M f1_12 sf S 270 114 :M S 354 114 :M f0_12 sf -.091({A})A 442 114 :M -.091({A})A 53 90 1 1 rF 54 90 87 1 rF 141 90 1 1 rF 142 90 88 1 rF 230 90 1 1 rF 231 90 87 1 rF 318 90 1 1 rF 319 90 88 1 rF 407 90 1 1 rF 408 90 87 1 rF 495 90 1 1 rF 53 91 1 27 rF 141 91 1 27 rF 230 91 1 27 rF 318 91 1 27 rF 407 91 1 27 rF 495 91 1 27 rF 94 142 :M (B)S 177 142 :M .238({C})A 270 142 :M f1_12 sf S 342 142 :M f0_12 sf .302({B,C,D})A 442 142 :M .238({B})A 53 118 1 1 rF 54 118 87 1 rF 141 118 1 1 rF 142 118 88 1 rF 230 118 1 1 rF 231 118 87 1 rF 318 118 1 1 rF 319 118 88 1 rF 407 118 1 1 rF 408 118 87 1 rF 495 118 1 1 rF 53 119 1 27 rF 141 119 1 27 rF 230 119 1 27 rF 318 119 1 27 rF 407 119 1 27 rF 495 119 1 27 rF 94 170 :M (C)S 177 170 :M -.091({D})A 265 170 :M .238({B})A 348 170 :M .203({C,D})A 434 170 :M .982 .098({B,C })J 53 146 1 1 rF 54 146 87 1 rF 141 146 1 1 rF 142 146 88 1 rF 230 146 1 1 rF 231 146 87 1 rF 318 146 1 1 rF 319 146 88 1 rF 407 146 1 1 rF 408 146 87 1 rF 495 146 1 1 rF 53 147 1 27 rF 141 147 1 27 rF 230 147 1 27 rF 318 147 1 27 rF 407 147 1 27 rF 495 147 1 27 rF 94 198 :M (D)S 182 198 :M f1_12 sf S 259 198 :M f0_12 sf .287({C,E})A 354 198 :M -.091({D})A 424 198 :M .31({B,C,D,E})A 53 174 1 1 rF 54 174 87 1 rF 141 174 1 1 rF 142 174 88 1 rF 230 174 1 1 rF 231 174 87 1 rF 318 174 1 1 rF 319 174 88 1 rF 407 174 1 1 rF 408 174 87 1 rF 495 174 1 1 rF 53 175 1 27 rF 141 175 1 27 rF 230 175 1 27 rF 318 175 1 27 rF 407 175 1 27 rF 495 175 1 27 rF 94 226 :M (E)S 177 226 :M -.091({D})A 270 226 :M f1_12 sf S 348 226 :M f0_12 sf .122({D,E})A 442 226 :M .075({E})A 53 202 1 1 rF 54 202 87 1 rF 141 202 1 1 rF 142 202 88 1 rF 230 202 1 1 rF 231 202 87 1 rF 318 202 1 1 rF 319 202 88 1 rF 407 202 1 1 rF 408 202 87 1 rF 495 202 1 1 rF 53 203 1 27 rF 53 230 1 1 rF 53 230 1 1 rF 54 230 87 1 rF 141 203 1 27 rF 141 230 1 1 rF 142 230 88 1 rF 230 203 1 27 rF 230 230 1 1 rF 231 230 87 1 rF 318 203 1 27 rF 318 230 1 1 rF 319 230 88 1 rF 407 203 1 27 rF 407 230 1 1 rF 408 230 87 1 rF 495 203 1 27 rF 495 230 1 1 rF 495 230 1 1 rF 77 254 :M -.023(A vertex X is a )A f2_12 sf -.023(collider)A f0_12 sf -.023( on undirected path U if and only if U contains a subpath )A 466 254 :M -.663(Y )A 478 254 :M f1_12 sf S 59 275 :M f0_12 sf .154 .015(X )J f1_12 sf .199A f0_12 sf .133 .013( Z, or Y )J f1_12 sf .188A f0_12 sf .101 .01( X )J f1_12 sf .199A f0_12 sf .133 .013( Z, or Y )J f1_12 sf .188A f0_12 sf .101 .01( X )J f1_12 sf .188A f0_12 sf .113 .011( Z, )J 263 275 :M (or )S 277 275 :M -.663(Y )A 289 275 :M f1_12 sf .4A f0_12 sf .096 .01( )J 306 275 :M .306 .031(X )J 319 275 :M f1_12 sf .126A f0_12 sf ( )S 335 275 :M -.332(Z; )A 349 275 :M -.072(otherwise )A 399 275 :M -.164(if )A 410 275 :M .306 .031(X )J 423 275 :M (is )S 435 275 :M (on )S 451 275 :M .306 .031(U )J 464 275 :M -.334(it )A 474 275 :M (is )S 486 275 :M (a)S 59 296 :M f2_12 sf .121(non-collider)A f0_12 sf .316 .032( on U. For example, D is a collider )J 297 296 :M (on )S 313 296 :M (C )S 325 296 :M f1_12 sf .126A f0_12 sf ( )S 341 296 :M -.663(D )A 353 296 :M f1_12 sf .126A f0_12 sf ( )S 369 296 :M -.33(E )A 380 296 :M -.109(and )A 401 296 :M (C )S 413 296 :M (is )S 425 296 :M -.326(a )A 434 296 :M -.149(non-collider)A 59 317 :M (on )S 75 317 :M (B )S 87 317 :M f1_12 sf .126A f0_12 sf ( )S 103 317 :M (C )S 115 317 :M f1_12 sf .126A f0_12 sf ( )S 131 317 :M .281 .028(D. )J 147 317 :M .306 .031(X )J 160 317 :M (is )S 172 317 :M -.163(an )A 187 317 :M f2_12 sf 2.606 .261(ancestor )J 239 317 :M 2.506 .251(of )J 256 317 :M .909 .091(a )J 267 317 :M .553(set)A f0_12 sf .356 .036( )J 287 317 :M (of )S 301 317 :M -.163(vertices )A 341 317 :M f2_12 sf (Z)S f0_12 sf ( )S 353 317 :M -.164(if )A 364 317 :M .306 .031(X )J 378 317 :M (is )S 391 317 :M -.163(an )A 407 317 :M -.122(ancestor )A 451 317 :M (of )S 466 317 :M -.109(some)A 59 338 :M -.026(member of )A f2_12 sf (Z)S f0_12 sf (.)S 77 365 :M .414 .041(For disjoint sets of vertices, )J f2_12 sf .213(X)A f0_12 sf .123 .012(, )J f2_12 sf .213(Y)A f0_12 sf .249 .025(, and )J f2_12 sf .197(Z)A f0_12 sf .123 .012(, )J f2_12 sf .213(X)A f0_12 sf .144 .014( is )J f2_12 sf .139(d-connected)A f0_12 sf .157 .016( to )J f2_12 sf .213(Y)A f0_12 sf .298 .03( given )J f2_12 sf .197(Z)A f0_12 sf .224 .022( if and )J 470 365 :M -.111(only)A 59 386 :M -.155(if there is an acyclic undirected path )A 231 386 :M .306 .031(U )J 244 386 :M -.139(between )A 287 386 :M -.082(some )A 316 386 :M -.219(member )A 358 386 :M .306 .031(X )J 371 386 :M (of )S 385 386 :M f2_12 sf 1.381(X)A f0_12 sf .869 .087(, )J 403 386 :M -.109(and )A 424 386 :M -.082(some )A 453 386 :M -.263(member)A 59 407 :M -.041(Y of )A f2_12 sf -.072(Y)A f0_12 sf -.04(, such that every )A 174 407 :M -.206(collider )A 213 407 :M (on )S 229 407 :M .306 .031(U )J 242 407 :M (is )S 254 407 :M -.163(an )A 269 407 :M -.122(ancestor )A 312 407 :M (of )S 326 407 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(, )J 342 407 :M -.109(and )A 363 407 :M -.129(every )A 393 407 :M -.137(non-collider )A 454 407 :M (on )S 470 407 :M .306 .031(U )J 483 407 :M (is)S 59 428 :M .224 .022(not in )J f2_12 sf .158(Z)A f0_12 sf .301 .03(. For disjoint sets of vertices, )J f2_12 sf .171(X)A f0_12 sf .099 .01(, )J f2_12 sf .171(Y)A f0_12 sf .2 .02(, and )J f2_12 sf .158(Z)A f0_12 sf .099 .01(, )J f2_12 sf .171(X)A f0_12 sf .059 .006( )J 321 428 :M (is )S 333 428 :M f2_12 sf .377(d-separated)A f0_12 sf .205 .021( )J 402 428 :M -.08(from )A 429 428 :M f2_12 sf .25(Y)A f0_12 sf .087 .009( )J 442 428 :M -.132(given )A 472 428 :M f2_12 sf (Z)S f0_12 sf ( )S 484 428 :M -.327(if)A 59 449 :M -.026(and only if )A f2_12 sf (X)S f0_12 sf -.027( is not d-connected to )A f2_12 sf (Y)S f0_12 sf -.027( given )A f2_12 sf (Z)S f0_12 sf (.)S 251 608 :M f2_12 sf 2.949 .295(Figure 1)J 77 635 :M f0_12 sf -.048(For example, the path A )A f1_12 sf -.12A f0_12 sf -.045( B )A f1_12 sf -.114A f0_12 sf -.067( C )A 248 635 :M -.097(d-connects )A 303 635 :M -.663(A )A 315 635 :M -.109(and )A 336 635 :M (C )S 348 635 :M -.132(given )A 378 635 :M f1_12 sf -.128A f0_12 sf -.082(; )A 395 635 :M -.334(it )A 405 635 :M -.082(also )A 428 635 :M -.097(d-connects )A 483 635 :M (A)S 59 656 :M -.109(and )A 80 656 :M (C )S 92 656 :M -.132(given )A 122 656 :M .584 .058({D}, )J 150 656 :M .822 .082({E}, )J 177 656 :M (or )S 191 656 :M .93 .093({D,E}. )J 230 656 :M -.33(E )A 242 656 :M f1_12 sf .126A f0_12 sf ( )S 259 656 :M -.663(D )A 272 656 :M f1_12 sf .126A f0_12 sf ( )S 289 656 :M (C )S 302 656 :M -.097(d-connects )A 358 656 :M -.33(E )A 370 656 :M -.109(and )A 392 656 :M (C )S 405 656 :M -.132(given )A 436 656 :M .584 .058({D}, )J 465 656 :M -.165(given)A 59 677 :M 1.133 .113({D,B}, )J 101 677 :M .721 .072({D,A}, )J 143 677 :M (or )S 159 677 :M 1.195 .119({D,A,B}. )J 213 677 :M -.219(The )A 237 677 :M -.073(following )A 289 677 :M (is )S 303 677 :M -.326(a )A 314 677 :M -.167(list )A 334 677 :M (of )S 350 677 :M -.331(all )A 367 677 :M -.22(the )A 387 677 :M -.08(pairwise )A 433 677 :M -.118(d-separation)A 186 469 44 31 rC 204 492 :M (A)S gR gS 324 468 43 31 rC 342 491 :M f0_12 sf (C)S gR gS 253 468 50 37 rC 271 491 :M f0_12 sf (B)S gR gS 320 536 48 32 rC 338 559 :M f0_12 sf (D)S gR gS 255 535 48 35 rC 273 558 :M f0_12 sf (E)S gR gS 0 0 552 730 rC 224 491.75 -.75 .75 263.75 491 .75 224 491 @a np 263 489 :M 263 493 :L 266 491 :L 263 489 :L .75 lw eofill -.75 -.75 263.75 493.75 .75 .75 263 489 @b -.75 -.75 263.75 493.75 .75 .75 266 491 @b 263 489.75 -.75 .75 266.75 491 .75 263 489 @a np 225 493 :M 225 489 :L 221 491 :L 225 493 :L eofill -.75 -.75 225.75 493.75 .75 .75 225 489 @b -.75 -.75 221.75 491.75 .75 .75 225 489 @b 221 491.75 -.75 .75 225.75 493 .75 221 491 @a 288 490.75 -.75 .75 330.75 490 .75 288 490 @a np 330 488 :M 330 492 :L 333 490 :L 330 488 :L eofill -.75 -.75 330.75 492.75 .75 .75 330 488 @b -.75 -.75 330.75 492.75 .75 .75 333 490 @b 330 488.75 -.75 .75 333.75 490 .75 330 488 @a 285 554.75 -.75 .75 327.75 554 .75 285 554 @a np 327 552 :M 327 556 :L 330 554 :L 327 552 :L eofill -.75 -.75 327.75 556.75 .75 .75 327 552 @b -.75 -.75 327.75 556.75 .75 .75 330 554 @b 327 552.75 -.75 .75 330.75 554 .75 327 552 @a -.75 -.75 345.75 538.75 .75 .75 345 499 @b np 347 538 :M 343 538 :L 345 541 :L 347 538 :L eofill 343 538.75 -.75 .75 347.75 538 .75 343 538 @a 343 538.75 -.75 .75 345.75 541 .75 343 538 @a -.75 -.75 345.75 541.75 .75 .75 347 538 @b endp %%Page: 3 3 %%BeginPageSetup initializepage (peter; page: 3 of 18)setjob %%EndPageSetup gS 0 0 552 730 rC 467 698 24 27 rC 485 721 :M f0_12 sf (3)S gR gS 0 0 552 730 rC 59 58 :M f0_12 sf -.095(relations in Figure 1 \(where each pair is )A 250 58 :M -.081(followed )A 296 58 :M (by )S 312 58 :M -.326(a )A 321 58 :M -.167(list )A 339 58 :M (of )S 353 58 :M -.331(all )A 368 58 :M (of )S 382 58 :M -.22(the )A 400 58 :M (sets )S 422 58 :M -.249(that )A 443 58 :M -.144(d-separate)A 59 79 :M -.264(them\):)A 77 106 :M .264 .026({A} and {C} are d-separated given: {B}, {B,D}, {B,E}, {B,D,E})J 77 133 :M .453 .045({A} and {D} are d-separated given: {B}, {C}, {B,C}, {B,E}, {C,E}, {B,C,E})J 77 160 :M .405 .04({A} and {E} are d-separated given: )J f1_12 sf .218A f0_12 sf .468 .047(, {B}, {C}, {B,C}, {B,D}, {C,D}, {B,C,D})J 77 187 :M .332 .033({B} and {E} are d-separated given: )J f1_12 sf .179A f0_12 sf .389 .039(, {A}, {C}, {A,C}, {C,D}, {A,C,D}.)J 77 214 :M -.044(In a graph G, with a set of vertices )A 245 214 :M f2_12 sf .25(V)A f0_12 sf .087 .009( )J 258 214 :M -.165(containing )A 311 214 :M f2_12 sf .405(O)A f0_12 sf .237 .024(, )J 328 214 :M -.164(if )A 339 214 :M -.663(A )A 351 214 :M -.109(and )A 372 214 :M (B )S 384 214 :M -.215(are )A 402 214 :M -.167(in )A 415 214 :M f2_12 sf .405(O)A f0_12 sf .237 .024(, )J 432 214 :M -.165(then )A 456 214 :M -.196(there )A 483 214 :M (is)S 59 235 :M .888 .089(an )J f2_12 sf 2.134 .213(inducing )J 127 235 :M 1.189 .119(path )J 156 235 :M f0_12 sf -.139(between )A 199 235 :M -.663(A )A 211 235 :M -.109(and )A 232 235 :M (B )S 244 235 :M -.132(given )A 274 235 :M f2_12 sf -.253(O)A f0_12 sf ( )S 287 235 :M -.164(if )A 298 235 :M -.109(and )A 319 235 :M -.083(only )A 344 235 :M -.164(if )A 355 235 :M -.196(there )A 382 235 :M (is )S 394 235 :M -.326(a )A 403 235 :M -.165(path )A 427 235 :M .306 .031(U )J 440 235 :M -.139(between )A 483 235 :M (A)S 59 256 :M -.109(and )A 80 256 :M (B )S 92 256 :M (such )S 118 256 :M -.249(that )A 139 256 :M -.129(every )A 169 256 :M -.219(member )A 212 256 :M (of )S 227 256 :M f2_12 sf -.253(O)A f0_12 sf ( )S 241 256 :M -.249(that )A 263 256 :M (is )S 276 256 :M (on )S 293 256 :M .306 .031(U )J 307 256 :M (is )S 320 256 :M -.326(a )A 330 256 :M -.072(collider, )A 374 256 :M -.109(and )A 396 256 :M -.129(every )A 427 256 :M -.206(collider )A 467 256 :M (is )S 480 256 :M -.326(an)A 59 277 :M -.041(ancestor of A or B. \(If )A f2_12 sf -.077(V)A f0_12 sf -.038( = )A f2_12 sf -.083(O)A f0_12 sf -.041(, we will simply say that there is )A 356 277 :M -.163(an )A 371 277 :M -.124(inducing )A 416 277 :M -.165(path )A 440 277 :M -.139(between )A 483 277 :M (A)S 59 298 :M -.109(and )A 80 298 :M .771 .077(B.\) )J 100 298 :M -.164(It )A 111 298 :M (has )S 131 298 :M -.163(been )A 157 298 :M .447 .045(shown )J 193 298 :M -.167(in )A 206 298 :M -.328(Verma )A 241 298 :M -.109(and )A 262 298 :M -.058(Pearl\(1990\) )A 322 298 :M -.249(that )A 343 298 :M -.167(in )A 356 298 :M -.326(a )A 366 298 :M -.33(DAG )A 396 298 :M 1.114 .111(G, )J 414 298 :M -.663(A )A 427 298 :M -.109(and )A 449 298 :M (B )S 462 298 :M -.215(are )A 481 298 :M (d-)S 59 319 :M -.077(separated given some subset of )A f2_12 sf -.147(O)A f0_12 sf -.075(\\{A,B} if and only if there is no inducing path )A 440 319 :M -.139(between )A 483 319 :M (A)S 59 340 :M -.022(and B given )A f2_12 sf (O)S f0_12 sf (.)S 77 367 :M -.663(A )A 89 367 :M f2_12 sf .205(MAG)A f0_12 sf .063 .006( )J 123 367 :M (\(or )S 141 367 :M f2_12 sf 2.45 .245(mixed )J 180 367 :M 2.293 .229(ancestral )J 235 367 :M .218(graph)A f0_12 sf .226 .023(\) )J 275 367 :M (is )S 287 367 :M -.326(a )A 296 367 :M -.064(graph )A 327 367 :M -.083(with )A 352 367 :M (two )S 374 367 :M (kinds )S 404 367 :M (of )S 419 367 :M -.109(edges: )A 454 367 :M -.234(directed)A 59 388 :M -.064(edges )A 90 388 :M 1.12 .112(\(e.g. )J 117 388 :M -.663(A )A 129 388 :M f1_12 sf .126A f0_12 sf ( )S 145 388 :M .771 .077(B\), )J 165 388 :M -.109(and )A 186 388 :M -.179(bi-directed )A 240 388 :M .425 .043(edges, )J 276 388 :M 1.12 .112(\(e.g. )J 304 388 :M (C )S 317 388 :M f1_12 sf .4A f0_12 sf .096 .01( )J 335 388 :M .264 .026(D\). )J 356 388 :M -.219(The )A 379 388 :M -.331(MAG )A 411 388 :M -.249(that )A 433 388 :M -.063(represents )A 486 388 :M (a)S 59 409 :M -.097(DAG G \(also called MAG\(G,)A f2_12 sf -.159(O)A f0_12 sf -.084(\) with a )A 248 409 :M -.109(set )A 265 409 :M (of )S 279 409 :M -.039(observed )A 326 409 :M -.145(variables )A 372 409 :M f2_12 sf -.187(O)A f0_12 sf -.14(\) )A 389 409 :M -.217(can )A 409 409 :M -.163(be )A 424 409 :M -.119(constructed )A 482 409 :M -.334(in)A 59 430 :M -.097(the following way:)A 77 457 :M f1_12 sf S 83 457 :M 9 .9( )J 95 457 :M f0_12 sf -.094(Place the edge A )A f1_12 sf -.226A f0_12 sf -.111( B in MAG\(G,)A f2_12 sf -.178(O)A f0_12 sf -.087(\) if and only if A is an )A 375 457 :M -.122(ancestor )A 418 457 :M (of )S 432 457 :M (B )S 444 457 :M -.167(in )A 457 457 :M 1.114 .111(G, )J 474 457 :M -.163(and)A 95 478 :M -.089(there is an inducing path between A and B given )A f2_12 sf -.168(O)A f0_12 sf -.097( in G.)A 77 505 :M f1_12 sf S 83 505 :M 9 .9( )J 95 505 :M f0_12 sf -.09(Place the edge A )A f1_12 sf -.229A f0_12 sf -.103( B in MAG\(G,)A f2_12 sf -.171(O)A f0_12 sf -.084(\) if and only if A is not an ancestor )A 439 505 :M (of )S 453 505 :M (B )S 465 505 :M -.167(in )A 478 505 :M 1.337(G,)A 95 526 :M -.088(B is not an ancestor of A in G, and there is an inducing path between A and B given)A 95 547 :M f2_12 sf .172(O)A f0_12 sf .199 .02( in G.)J 77 574 :M -.083(Some )A 108 574 :M -.164(examples )A 156 574 :M (of )S 170 574 :M -.165(MAGs )A 206 574 :M -.215(are )A 224 574 :M .447 .045(shown )J 260 574 :M -.167(in )A 273 574 :M -.054(Figure )A 308 574 :M .833 .083(2, )J 322 574 :M -.062(where )A 356 574 :M f2_12 sf -.253(O)A f0_12 sf ( )S 370 574 :M .211 .021(= )J 382 574 :M 1.573 .157({A,B,C,D}. )J 447 574 :M (\(In )S 466 574 :M -.078(cases)A 59 595 :M -.062(where )A 93 595 :M -.22(the )A 112 595 :M -.151(distinction )A 166 595 :M -.139(between )A 210 595 :M -.276(latent )A 240 595 :M -.145(variables )A 287 595 :M -.109(and )A 309 595 :M -.122(measured )A 360 595 :M -.145(variables )A 408 595 :M (is )S 422 595 :M -.066(important, )A 477 595 :M (we)S 59 616 :M -.106(enclose latent variables in ovals.\))A endp %%Page: 4 4 %%BeginPageSetup initializepage (peter; page: 4 of 18)setjob %%EndPageSetup gS 0 0 552 730 rC 467 698 24 27 rC 485 721 :M f0_12 sf (4)S gR gS 135 44 280 173 rC 135 53 :M f0_12 sf ( )S 138 53 :M ( )S 141 53 :M ( )S 144 53 :M ( )S 147 53 :M ( )S 150 53 :M ( )S 153 53 :M ( )S 156 53 :M ( )S 159 53 :M ( )S 162 53 :M ( )S 165 53 :M ( )S 168 53 :M ( )S 171 53 :M ( )S 174 53 :M ( )S 177 53 :M ( )S 180 53 :M ( )S 183 53 :M ( )S 186 53 :M ( )S 189 53 :M ( )S 192 53 :M f4_12 sf ( )S 195 53 :M f0_12 sf (T)S 135 77 :M (A)S 143 77 :M ( )S 146 77 :M f1_12 sf ( )S 149 77 :M ( )S 152 77 :M ( )S 155 77 :M ( )S 158 77 :M ( )S 161 77 :M ( )S 164 77 :M ( )S 167 77 :M ( )S 170 77 :M ( )S 173 77 :M f0_12 sf (B)S 181 77 :M ( )S 184 77 :M ( )S 187 77 :M ( )S 190 77 :M ( )S 193 77 :M ( )S 196 77 :M ( )S 199 77 :M ( )S 202 77 :M ( )S 205 77 :M ( )S 208 77 :M ( )S 211 77 :M ( )S 214 77 :M (C)S 222 77 :M ( )S 225 77 :M ( )S 228 77 :M ( )S 231 77 :M ( )S 234 77 :M ( )S 237 77 :M ( )S 240 77 :M ( )S 243 77 :M ( )S 246 77 :M ( )S 249 77 :M ( )S 252 77 :M ( )S 255 77 :M (D)S 263 77 :M ( )S 266 77 :M ( )S 269 77 :M ( )S 272 77 :M ( )S 275 77 :M (A)S 283 77 :M ( )S 286 77 :M ( )S 289 77 :M ( )S 292 77 :M ( )S 295 77 :M ( )S 298 77 :M ( )S 301 77 :M ( )S 304 77 :M ( )S 307 77 :M ( )S 310 77 :M ( )S 313 77 :M ( )S 316 77 :M (B)S 324 77 :M ( )S 327 77 :M ( )S 330 77 :M ( )S 333 77 :M ( )S 336 77 :M ( )S 339 77 :M ( )S 342 77 :M ( )S 345 77 :M ( )S 348 77 :M ( )S 351 77 :M ( )S 354 77 :M (C)S 362 77 :M ( )S 365 77 :M ( )S 368 77 :M ( )S 371 77 :M ( )S 374 77 :M ( )S 377 77 :M ( )S 380 77 :M ( )S 383 77 :M ( )S 386 77 :M ( )S 389 77 :M ( )S 392 77 :M ( )S 395 77 :M (D)S 135 101 :M ( )S 138 101 :M ( )S 141 101 :M ( )S 144 101 :M ( )S 147 101 :M ( )S 150 101 :M ( )S 153 101 :M ( )S 156 101 :M ( )S 159 101 :M ( )S 162 101 :M ( )S 165 101 :M ( )S 168 101 :M ( )S 171 101 :M ( )S 174 101 :M ( )S 177 101 :M (D)S 185 101 :M (A)S 193 101 :M (G)S 202 101 :M ( )S 205 101 :M (G)S 214 104 :M f0_7 sf (1)S 218 101 :M f0_12 sf ( )S 221 101 :M ( )S 224 101 :M ( )S 227 101 :M ( )S 230 101 :M ( )S 233 101 :M ( )S 236 101 :M ( )S 239 101 :M ( )S 242 101 :M ( )S 245 101 :M ( )S 248 101 :M ( )S 251 101 :M ( )S 254 101 :M ( )S 257 101 :M ( )S 260 101 :M ( )S 263 101 :M ( )S 266 101 :M ( )S 269 101 :M ( )S 272 101 :M ( )S 275 101 :M ( )S 278 101 :M ( )S 281 101 :M ( )S 284 101 :M ( )S 287 101 :M ( )S 290 101 :M ( )S 293 101 :M ( )S 296 101 :M ( )S 299 101 :M ( )S 302 101 :M ( )S 305 101 :M ( )S 308 101 :M ( )S 311 101 :M ( )S 314 101 :M (M)S 324 101 :M (A)S 332 101 :M (G)S 341 101 :M <28>S 345 101 :M (G)S 354 104 :M f0_7 sf (1)S 358 101 :M f0_12 sf (,)S 362 101 :M (O)S 370 101 :M <29>S 135 125 :M (A)S 279 125 :M (A)S 135 149 :M ( )S 138 149 :M ( )S 141 149 :M ( )S 144 149 :M ( )S 147 149 :M ( )S 150 149 :M ( )S 153 149 :M ( )S 156 149 :M ( )S 159 149 :M ( )S 162 149 :M ( )S 165 149 :M ( )S 168 149 :M (B)S 176 149 :M ( )S 179 149 :M ( )S 182 149 :M ( )S 185 149 :M ( )S 188 149 :M ( )S 191 149 :M ( )S 194 149 :M ( )S 197 149 :M ( )S 200 149 :M (D)S 315 149 :M (B)S 351 149 :M (D)S 135 173 :M (C)S 279 173 :M (C)S 135 197 :M ( )S 138 197 :M ( )S 141 197 :M ( )S 144 197 :M ( )S 147 197 :M ( )S 150 197 :M ( )S 153 197 :M ( )S 156 197 :M ( )S 159 197 :M ( )S 162 197 :M ( )S 165 197 :M (D)S 173 197 :M (A)S 181 197 :M (G)S 190 197 :M ( )S 193 197 :M (G)S 202 200 :M f0_7 sf (2)S 206 197 :M f0_12 sf ( )S 209 197 :M ( )S 212 197 :M ( )S 215 197 :M ( )S 218 197 :M ( )S 221 197 :M ( )S 224 197 :M ( )S 227 197 :M ( )S 230 197 :M ( )S 233 197 :M ( )S 236 197 :M ( )S 239 197 :M ( )S 242 197 :M ( )S 245 197 :M ( )S 248 197 :M ( )S 251 197 :M ( )S 254 197 :M ( )S 257 197 :M ( )S 260 197 :M ( )S 263 197 :M ( )S 266 197 :M ( )S 269 197 :M ( )S 272 197 :M ( )S 275 197 :M ( )S 278 197 :M ( )S 281 197 :M ( )S 284 197 :M ( )S 287 197 :M ( )S 290 197 :M ( )S 293 197 :M (M)S 303 197 :M (A)S 311 197 :M (G)S 320 197 :M <28>S 324 197 :M (G)S 333 200 :M f0_7 sf (2)S 337 197 :M f0_12 sf (,)S 341 197 :M (O)S 349 197 :M <29>S gR gS 134 41 282 177 rC -.75 -.75 185.75 65.75 .75 .75 197 54 @b np 189 66 :M 184 61 :L 181 69 :L 189 66 :L .75 lw eofill 184 61.75 -.75 .75 189.75 66 .75 184 61 @a -.75 -.75 181.75 69.75 .75 .75 184 61 @b -.75 -.75 181.75 69.75 .75 .75 189 66 @b 201 55.75 -.75 .75 210.75 63 .75 201 55 @a np 211 59 :M 206 64 :L 214 67 :L 211 59 :L eofill -.75 -.75 206.75 64.75 .75 .75 211 59 @b 206 64.75 -.75 .75 214.75 67 .75 206 64 @a 211 59.75 -.75 .75 214.75 67 .75 211 59 @a 235 74.75 -.75 .75 252.75 74 .75 235 74 @a np 237 78 :M 237 70 :L 229 74 :L 237 78 :L eofill -.75 -.75 237.75 78.75 .75 .75 237 70 @b -.75 -.75 229.75 74.75 .75 .75 237 70 @b 229 74.75 -.75 .75 237.75 78 .75 229 74 @a 290 74.75 -.75 .75 304.75 74 .75 290 74 @a np 302 70 :M 302 78 :L 310 74 :L 302 70 :L eofill -.75 -.75 302.75 78.75 .75 .75 302 70 @b -.75 -.75 302.75 78.75 .75 .75 310 74 @b 302 70.75 -.75 .75 310.75 74 .75 302 70 @a 336 74.75 -.75 .75 346.75 74 .75 336 74 @a np 344 70 :M 344 78 :L 352 74 :L 344 70 :L eofill -.75 -.75 344.75 78.75 .75 .75 344 70 @b -.75 -.75 344.75 78.75 .75 .75 352 74 @b 344 70.75 -.75 .75 352.75 74 .75 344 70 @a np 338 78 :M 338 70 :L 330 74 :L 338 78 :L eofill -.75 -.75 338.75 78.75 .75 .75 338 70 @b -.75 -.75 330.75 74.75 .75 .75 338 70 @b 330 74.75 -.75 .75 338.75 78 .75 330 74 @a 375 74.75 -.75 .75 392.75 74 .75 375 74 @a np 377 78 :M 377 70 :L 369 74 :L 377 78 :L eofill -.75 -.75 377.75 78.75 .75 .75 377 70 @b -.75 -.75 369.75 74.75 .75 .75 377 70 @b 369 74.75 -.75 .75 377.75 78 .75 369 74 @a 147 125.75 -.75 .75 161.75 137 .75 147 125 @a np 162 133 :M 158 139 :L 166 141 :L 162 133 :L eofill -.75 -.75 158.75 139.75 .75 .75 162 133 @b 158 139.75 -.75 .75 166.75 141 .75 158 139 @a 162 133.75 -.75 .75 166.75 141 .75 162 133 @a -.75 -.75 148.75 168.75 .75 .75 164 156 @b np 160 154 :M 165 160 :L 169 152 :L 160 154 :L eofill 160 154.75 -.75 .75 165.75 160 .75 160 154 @a -.75 -.75 165.75 160.75 .75 .75 169 152 @b -.75 -.75 160.75 154.75 .75 .75 169 152 @b 179 146.75 -.75 .75 191.75 146 .75 179 146 @a np 189 142 :M 189 150 :L 197 146 :L 189 142 :L eofill -.75 -.75 189.75 150.75 .75 .75 189 142 @b -.75 -.75 189.75 150.75 .75 .75 197 146 @b 189 142.75 -.75 .75 197.75 146 .75 189 142 @a 329 145.75 -.75 .75 341.75 145 .75 329 145 @a np 339 141 :M 339 149 :L 347 145 :L 339 141 :L eofill -.75 -.75 339.75 149.75 .75 .75 339 141 @b -.75 -.75 339.75 149.75 .75 .75 347 145 @b 339 141.75 -.75 .75 347.75 145 .75 339 141 @a 295 126.75 -.75 .75 309.75 138 .75 295 126 @a np 310 134 :M 306 140 :L 314 142 :L 310 134 :L eofill -.75 -.75 306.75 140.75 .75 .75 310 134 @b 306 140.75 -.75 .75 314.75 142 .75 306 140 @a 310 134.75 -.75 .75 314.75 142 .75 310 134 @a -.75 -.75 293.75 165.75 .75 .75 309 153 @b np 305 151 :M 310 157 :L 314 149 :L 305 151 :L eofill 305 151.75 -.75 .75 310.75 157 .75 305 151 @a -.75 -.75 310.75 157.75 .75 .75 314 149 @b -.75 -.75 305.75 151.75 .75 .75 314 149 @b 153 74.75 -.75 .75 165.75 74 .75 153 74 @a np 163 70 :M 163 78 :L 171 74 :L 163 70 :L eofill -.75 -.75 163.75 78.75 .75 .75 163 70 @b -.75 -.75 163.75 78.75 .75 .75 171 74 @b 163 70.75 -.75 .75 171.75 74 .75 163 70 @a 38 13 202.5 48 @f gR gS 0 0 552 730 rC 251 239 :M f2_12 sf 2.949 .295(Figure 2)J 77 265 :M f0_12 sf -.053(It has )A 107 265 :M -.163(been )A 133 265 :M .447 .045(shown )J 169 265 :M -.167(in )A 182 265 :M -.046(Spirtes )A 219 265 :M -.33(et )A 231 265 :M .261 .026(al. )J 247 265 :M (\(1993\) )S 283 265 :M -.249(that )A 304 265 :M -.167(in )A 317 265 :M -.326(a )A 326 265 :M -.331(MAG )A 357 265 :M 1.114 .111(G, )J 374 265 :M -.663(A )A 386 265 :M -.109(and )A 407 265 :M (B )S 419 265 :M -.215(are )A 437 265 :M -.129(d-separated)A 59 286 :M -.073(given some subset of {A,B} if and only if there is no inducing path between A and B )A 465 286 :M -.167(in )A 478 286 :M 1.337(G.)A 59 307 :M (\(In )S 77 307 :M -.046(Spirtes )A 114 307 :M -.33(et )A 126 307 :M .261 .026(al. )J 143 307 :M (1993 )S 172 307 :M -.22(the )A 191 307 :M (proof )S 222 307 :M (is )S 235 307 :M (for )S 254 307 :M -.124(inducing )A 300 307 :M -.165(path )A 325 307 :M .596 .06(graphs, )J 366 307 :M -.111(but )A 386 307 :M -.22(the )A 405 307 :M .217 .022(proofs )J 441 307 :M -.128(carry )A 470 307 :M -.106(over)A 59 328 :M -.033(unchanged for MAGs.\))A 77 355 :M (Thus )S 108 355 :M -.326(a )A 120 355 :M -.331(MAG )A 154 355 :M -.667(M )A 171 355 :M -.22(may )A 198 355 :M -.163(be )A 216 355 :M -.097(considered )A 274 355 :M -.167(to )A 290 355 :M -.107(represent )A 340 355 :M -.109(any )A 364 355 :M -.33(DAG )A 396 355 :M .306 .031(G )J 412 355 :M (such )S 441 355 :M -.249(that )A 466 355 :M -.667(M )A 484 355 :M (=)S 59 376 :M .118(MAG\(G,)A f2_12 sf .152(O)A f0_12 sf .135 .014(\). )J 126 376 :M (MAG\(G,)S f2_12 sf (O)S f0_12 sf (\) )S 189 376 :M -.063(represents )A 243 376 :M -.22(the )A 264 376 :M -.073(following )A 317 376 :M -.121(features )A 361 376 :M (of )S 378 376 :M -.326(a )A 390 376 :M -.33(DAG )A 422 376 :M .306 .031(G )J 438 376 :M -.083(with )A 466 376 :M -.331(latent)A 59 397 :M -.182(variables:)A 77 424 :M f1_12 sf S 83 424 :M 9 .9( )J 95 424 :M f0_12 sf -.115(the ancestor relations among the members of )A f2_12 sf -.216(O)A f0_12 sf -.127( in G;)A 77 451 :M f1_12 sf S 83 451 :M 9 .9( )J 95 451 :M f0_12 sf -.089(the d-separation relations among the members of )A f2_12 sf -.167(O)A f0_12 sf -.097( in G.)A 77 478 :M -.094(Although we will not prove it in this paper, MAGs have the following useful features:)A 77 505 :M f1_12 sf S 83 505 :M 9 .9( )J 95 505 :M f0_12 sf -.126(DAG G)A f0_7 sf 0 3 rm -.059(1)A 0 -3 rm f0_12 sf -.085( in Figure 2 is an example of a DAG such )A 336 505 :M -.249(that )A 357 505 :M (as )S 371 505 :M -.22(the )A 389 505 :M -.165(sample )A 426 505 :M -.163(size )A 448 505 :M -.121(increases)A 95 526 :M -.095(without )A 135 526 :M -.112(limit, )A 164 526 :M -.22(the )A 182 526 :M -.162(difference )A 233 526 :M -.139(between )A 276 526 :M -.22(the )A 294 526 :M (BIC )S 318 526 :M (of )S 332 526 :M .067(MAG\(G)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf (,)S f2_12 sf .077(O)A f0_12 sf .052 .005(\) )J 397 526 :M -.109(and )A 418 526 :M -.22(the )A 436 526 :M (BIC )S 460 526 :M (of )S 474 526 :M -.163(any)A 95 547 :M -.33(DAG )A 125 547 :M .286 .029<47D520>J 143 547 :M -.249(that )A 165 547 :M -.123(contains )A 209 547 :M -.083(only )A 235 547 :M -.205(variable )A 277 547 :M -.167(in )A 291 547 :M f2_12 sf -.253(O)A f0_12 sf ( )S 305 547 :M -.107(increases )A 353 547 :M -.095(without )A 394 547 :M -.334(limit )A 421 547 :M -.166(almost )A 458 547 :M .113(surely.)A 95 568 :M -.128(Hence )A 129 568 :M -.167(in )A 142 568 :M -.082(some )A 171 568 :M -.062(cases )A 200 568 :M -.326(a )A 209 568 :M -.237(maximum )A 260 568 :M -.166(likelihood )A 311 568 :M -.248(estimate )A 353 568 :M (of )S 367 568 :M -.22(the )A 385 568 :M -.331(MAG )A 417 568 :M -.163(parameters )A 473 568 :M (is )S 486 568 :M (a)S 95 589 :M -.127(better estimator of some of the )A 242 589 :M -.133(population )A 296 589 :M -.163(parameters )A 351 589 :M -.165(than )A 375 589 :M -.22(the )A 393 589 :M -.237(maximum )A 444 589 :M -.185(likelihood)A 95 610 :M -.131(estimate of any DAG parameters.)A 77 637 :M f1_12 sf S 83 637 :M 9 .9( )J 95 637 :M f0_12 sf (In )S 109 637 :M -.22(the )A 128 637 :M -.196(large )A 156 637 :M -.165(sample )A 194 637 :M -.112(limit, )A 224 637 :M (for )S 243 637 :M -.228(multi-variate )A 307 637 :M (normal, )S 349 637 :M -.109(any )A 371 637 :M (\(possibly )S 420 637 :M -.276(latent )A 450 637 :M -.204(variable\))A 95 658 :M -.114(DAG with a maximum BIC score is represented )A 325 658 :M (by )S 341 658 :M -.22(the )A 359 658 :M -.331(MAG )A 390 658 :M -.083(with )A 415 658 :M -.22(the )A 433 658 :M -.094(highest )A 471 658 :M (BIC)S 95 679 :M -.081(score among all MAGs.)A endp %%Page: 5 5 %%BeginPageSetup initializepage (peter; page: 5 of 18)setjob %%EndPageSetup gS 0 0 552 730 rC 467 698 24 27 rC 485 721 :M f0_12 sf (5)S gR gS 0 0 552 730 rC 77 58 :M f0_12 sf (In )S 91 58 :M -.038(general, )A 133 58 :M -.245(each )A 158 58 :M -.331(MAG )A 189 58 :M -.063(represents )A 241 58 :M -.165(many )A 271 58 :M -.144(different )A 315 58 :M -.276(latent )A 345 58 :M -.205(variable )A 387 58 :M .199 .02(models. )J 430 58 :M -.663(A )A 443 58 :M -.331(MAG )A 475 58 :M -.326(can)A 59 79 :M (thus )S 84 79 :M -.082(also )A 108 79 :M -.163(be )A 124 79 :M -.097(considered )A 180 79 :M -.326(a )A 190 79 :M -.14(representation )A 261 79 :M (of )S 277 79 :M -.326(a )A 288 79 :M -.109(set )A 307 79 :M (of )S 323 79 :M -.181(conditional )A 381 79 :M -.164(independence )A 451 79 :M -.164(relations)A 59 100 :M -.092(among variables in )A f2_12 sf -.173(O)A f0_12 sf -.096( \(which in some cases )A 268 100 :M -.164(cannot )A 303 100 :M -.163(be )A 318 100 :M -.117(represented )A 376 100 :M (by )S 392 100 :M -.109(any )A 413 100 :M -.33(DAG )A 442 100 :M -.184(containing)A 59 121 :M -.076(just variables in )A f2_12 sf -.162(O)A f0_12 sf -.084(.\) A MAG imposes no restrictions on the set of distributions )A 433 121 :M -.334(it )A 443 121 :M -.07(represents)A 59 142 :M -.131(other )A 88 142 :M -.165(than )A 113 142 :M -.22(the )A 132 142 :M -.181(conditional )A 189 142 :M -.164(independence )A 258 142 :M -.146(relations )A 303 142 :M -.249(that )A 325 142 :M -.334(it )A 336 142 :M -.04(entails. )A 375 142 :M -.162(\(The )A 402 142 :M -.064(class )A 430 142 :M (of )S 445 142 :M -.165(MAGs )A 483 142 :M (is)S 59 163 :M -.188(neither )A 95 163 :M -.326(a )A 104 163 :M (subset )S 139 163 :M (nor )S 160 163 :M -.326(a )A 170 163 :M -.039(superset )A 214 163 :M (of )S 229 163 :M -.131(other )A 258 163 :M -.175(generalizations )A 333 163 :M (of )S 348 163 :M -.164(DAGs )A 383 163 :M (such )S 410 163 :M (as )S 425 163 :M -.197(chain )A 455 163 :M .169(graphs,)A 59 184 :M -.082(cyclic directed graphs, or cyclic chain graphs.\))A 77 211 :M -.102(It is not the )A 134 211 :M -.161(case )A 158 211 :M -.249(that )A 179 211 :M -.163(an )A 194 211 :M -.144(arbitrary )A 238 211 :M -.064(graph )A 269 211 :M (is )S 281 211 :M -.326(a )A 290 211 :M (MAG, )S 325 211 :M .957 .096(i.e. )J 345 211 :M -.196(there )A 372 211 :M -.22(may )A 396 211 :M -.163(be )A 411 211 :M (no )S 427 211 :M -.33(DAG )A 456 211 :M .306 .031(G )J 469 211 :M (such)S 59 232 :M -.085(that M = MAG\(G,)A f2_12 sf -.135(O)A f0_12 sf -.072(\). The following theorem states necessary and )A 376 232 :M -.131(sufficient )A 424 232 :M -.1(conditions )A 477 232 :M (for)S 59 253 :M -.667(M )A 73 253 :M -.167(to )A 86 253 :M -.163(be )A 101 253 :M -.326(a )A 110 253 :M (MAG. )S 145 253 :M -.33(Let )A 165 253 :M -.22(the )A 184 253 :M f2_12 sf 2.809 .281(canonical )J 243 253 :M .243(graph)A f0_12 sf .119 .012( )J 280 253 :M -.079(G\(M\) )A 312 253 :M (for )S 331 253 :M -.326(a )A 341 253 :M -.331(MAG )A 373 253 :M -.667(M )A 388 253 :M -.163(be )A 404 253 :M -.119(constructed )A 463 253 :M -.167(in )A 477 253 :M -.33(the)A 59 274 :M -.09(following way: If )A f2_12 sf -.174(O)A f0_12 sf -.082( is the set of vertices in )A 266 274 :M .277 .028(M, )J 284 274 :M -.165(then )A 308 274 :M -.22(the )A 326 274 :M -.109(set )A 343 274 :M (of )S 357 274 :M -.163(vertices )A 397 274 :M -.167(in )A 410 274 :M -.079(G\(M\) )A 441 274 :M (is )S 453 274 :M f2_12 sf -.253(O)A f0_12 sf ( )S 466 274 :M f1_12 sf -.161A f0_12 sf ( )S 479 274 :M f2_12 sf .724(T)A f0_12 sf (,)S 59 295 :M -.019(where T)A f0_7 sf 0 3 rm (i,j)S 0 -3 rm f0_12 sf ( )S f1_12 sf S f0_12 sf ( )S f2_12 sf (T)S f0_12 sf -.015( if there is an edge X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S f1_12 sf S f0_12 sf ( X)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.016( in M, for )A 307 295 :M .726(X)A f0_7 sf 0 3 rm .163(i)A 0 -3 rm f0_12 sf .457 .046(, )J 326 295 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 341 295 :M -.167(in )A 354 295 :M f2_12 sf .405(O)A f0_12 sf .237 .024(, )J 371 295 :M -.196(there )A 398 295 :M -.215(are )A 416 295 :M -.064(edges )A 447 295 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 462 295 :M f1_12 sf .126A f0_12 sf ( )S 478 295 :M (T)S f0_7 sf 0 3 rm (i,j)S 0 -3 rm 59 316 :M f1_12 sf -.094A f0_12 sf -.046( X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.037( in G\(M\) if and only if there is an edge X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S f1_12 sf -.099A f0_12 sf -.046( X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.041( in M, and )A 364 316 :M -.163(an )A 379 316 :M -.163(edge )A 405 316 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 420 316 :M f1_12 sf .126A f0_12 sf ( )S 436 316 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 451 316 :M -.167(in )A 464 316 :M -.105(G\(M\))A 59 337 :M -.043(only if there is an edge X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S f1_12 sf -.11A f0_12 sf -.054( X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.054( in M.)A 59 364 :M f2_12 sf -.015(Lemma 1:)A f0_12 sf -.012( If M is a MAG, then G\(M\) has the same )A 311 364 :M -.122(ancestor )A 354 364 :M -.146(relations )A 398 364 :M -.132(among )A 434 364 :M -.14(members )A 481 364 :M (of)S 59 385 :M f2_12 sf (O)S f0_12 sf ( as M does.)S 77 412 :M -.042(Proof. By the algorithm )A 194 412 :M (for )S 212 412 :M -.109(constructing )A 274 412 :M .456 .046(G\(M\), )J 309 412 :M -.22(the )A 327 412 :M -.109(set )A 344 412 :M (of )S 358 412 :M -.205(directed )A 399 412 :M -.064(edges )A 430 412 :M -.167(in )A 443 412 :M -.079(G\(M\) )A 474 412 :M (is )S 486 412 :M (a)S 59 433 :M -.039(superset )A 104 433 :M (of )S 120 433 :M -.205(directed )A 163 433 :M -.064(edges )A 196 433 :M -.167(in )A 211 433 :M .277 .028(M. )J 231 433 :M -.128(Hence )A 267 433 :M -.164(if )A 280 433 :M -.196(there )A 309 433 :M (is )S 323 433 :M -.326(a )A 334 433 :M -.205(directed )A 377 433 :M -.165(path )A 404 433 :M -.167(in )A 420 433 :M .277 .028(M, )J 441 433 :M -.196(there )A 471 433 :M (is )S 486 433 :M (a)S 59 454 :M -.074(corresponding directed path in G\(M\).)A 77 481 :M -.071(Suppose there is a directed path P in G\(M\) between X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.076( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.073( in )A 384 481 :M f2_12 sf .405(O)A f0_12 sf .237 .024(. )J 401 481 :M .299 .03(P )J 412 481 :M (does )S 438 481 :M -.111(not )A 457 481 :M -.22(contain)A 59 502 :M -.096(any member of )A f2_12 sf -.142(T)A f0_12 sf -.088(, because by the algorithm for constructing G\(M\), every edge containing a)A 59 523 :M -.219(member )A 101 523 :M (of )S 116 523 :M .123(T)A f0_7 sf 0 3 rm .047(a,b)A 0 -3 rm f0_12 sf .05 .005( )J 137 523 :M (of )S 152 523 :M f2_12 sf (T)S f0_12 sf ( )S 165 523 :M (is )S 178 523 :M -.111(out )A 198 523 :M (of )S 213 523 :M .444(T)A f0_7 sf 0 3 rm .169(a,b)A 0 -3 rm f0_12 sf .33 .033(. )J 238 523 :M -.128(Hence )A 273 523 :M -.129(every )A 304 523 :M -.163(edge )A 331 523 :M (on )S 348 523 :M .299 .03(P )J 360 523 :M (is )S 373 523 :M -.326(a )A 383 523 :M -.205(directed )A 425 523 :M -.163(edge )A 452 523 :M -.163(between)A 59 544 :M -.018(members of )A f2_12 sf (O)S f0_12 sf -.019(. By )A 152 544 :M -.22(the )A 170 544 :M -.184(algorithm )A 219 544 :M (for )S 237 544 :M -.109(constructing )A 299 544 :M .456 .046(G\(M\), )J 334 544 :M -.196(there )A 361 544 :M (is )S 373 544 :M -.326(a )A 382 544 :M -.049(corresponding )A 454 544 :M -.234(directed)A 59 565 :M -.077(edge in M. Hence there is a directed path between X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.083( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.076( in G\(M\). )A f1_12 sf <5C>S 77 592 :M f0_12 sf -.081(Note )A 104 592 :M -.249(that )A 125 592 :M -.079(G\(M\) )A 156 592 :M (is )S 168 592 :M -.282(acyclic )A 204 592 :M -.139(because )A 245 592 :M -.196(there )A 272 592 :M (is )S 284 592 :M (no )S 300 592 :M -.205(directed )A 341 592 :M -.163(cycles )A 374 592 :M -.08(from )A 401 592 :M -.326(a )A 410 592 :M -.219(member )A 453 592 :M (of )S 468 592 :M f2_12 sf -.253(O)A f0_12 sf ( )S 482 592 :M -.334(to)A 59 613 :M -.125(itself in M, and hence no directed cycles from a member )A 327 613 :M (of )S 341 613 :M f2_12 sf -.253(O)A f0_12 sf ( )S 354 613 :M -.167(to )A 367 613 :M -.165(itself )A 394 613 :M -.167(in )A 407 613 :M .456 .046(G\(M\). )J 442 613 :M -.166(Also )A 468 613 :M -.245(there)A 59 634 :M -.104(is no directed cycle from a member of )A f2_12 sf -.171(T)A f0_12 sf -.101( to itself in G\(M\), because there are no edges )A 464 634 :M -.167(into )A 486 634 :M (a)S 59 655 :M -.049(member of )A f2_12 sf -.071(T)A f0_12 sf -.05( in G\(M\).)A endp %%Page: 6 6 %%BeginPageSetup initializepage (peter; page: 6 of 18)setjob %%EndPageSetup gS 0 0 552 730 rC 467 698 24 27 rC 485 721 :M f0_12 sf (6)S gR gS 0 0 552 730 rC 59 58 :M f2_12 sf -.069(Lemma 2:)A f0_12 sf -.054( If M = MAG\(G\) has vertices )A f2_12 sf -.098(O)A f0_12 sf -.053(, then M and G have the same ancestor relations)A 59 79 :M -.05(among members of )A f2_12 sf -.084(O)A f0_12 sf (.)S 77 106 :M -.038(Proof. Suppose there is a directed path P from X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.038( to X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.041( in M. Then by the )A 428 106 :M -.184(algorithm )A 477 106 :M (for)S 59 127 :M -.109(constructing )A 121 127 :M .226 .023(MAGs, )J 161 127 :M -.245(each )A 186 127 :M -.163(vertex )A 219 127 :M -.132(along )A 249 127 :M .299 .03(P )J 260 127 :M (is )S 272 127 :M -.163(an )A 287 127 :M -.122(ancestor )A 330 127 :M (of )S 344 127 :M -.112(its )A 359 127 :M .196 .02(sucessor )J 404 127 :M (on )S 420 127 :M -.22(the )A 439 127 :M -.165(path )A 464 127 :M -.167(in )A 478 127 :M 1.337(G.)A 59 148 :M -.016(Hence X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.012( is an an ancestor of X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.014( in G.)A 77 175 :M -.046(Suppose there is a directed path P from X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.045( to X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.046( in G. Then if )A 371 175 :M .135(X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf ( )S 387 175 :M -.109(and )A 408 175 :M .478(X)A f0_7 sf 0 3 rm .193(b)A 0 -3 rm f0_12 sf .166 .017( )J 425 175 :M -.215(are )A 443 175 :M (on )S 459 175 :M 1.107 .111(P, )J 474 175 :M -.163(and)A 59 196 :M -.081(there is no other member of )A f2_12 sf -.155(O)A f0_12 sf -.091( between X)A f0_7 sf 0 3 rm -.052(a)A 0 -3 rm f0_12 sf -.088( and X)A f0_7 sf 0 3 rm -.058(b)A 0 -3 rm f0_12 sf -.077( on P, there is an inducing path relative )A 482 196 :M -.334(to)A 59 217 :M f2_12 sf -.253(O)A f0_12 sf ( )S 74 217 :M -.167(in )A 89 217 :M .306 .031(G )J 104 217 :M -.139(between )A 149 217 :M .135(X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf ( )S 167 217 :M -.109(and )A 190 217 :M .876(X)A f0_7 sf 0 3 rm .354(b)A 0 -3 rm f0_12 sf .552 .055(. )J 213 217 :M -.164(It )A 226 217 :M (follows )S 268 217 :M -.249(that )A 291 217 :M -.196(there )A 320 217 :M (is )S 334 217 :M -.163(an )A 351 217 :M -.163(edge )A 379 217 :M .135(X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf ( )S 397 217 :M f1_12 sf .126A f0_12 sf ( )S 416 217 :M .478(X)A f0_7 sf 0 3 rm .193(b)A 0 -3 rm f0_12 sf .166 .017( )J 436 217 :M -.167(in )A 452 217 :M .277 .028(M. )J 473 217 :M -.328(The)A 59 238 :M -.101(concatenation of these edges in M is a directed path from X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.1( to X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.096( in M. )A f1_12 sf <5C>S 59 271 :M f2_12 sf -.007(Theorem 1:)A f0_12 sf -.006( A graph M is a MAG if and only if:)A 77 298 :M (1)S 83 298 :M (.)S 87 298 :M 5 .5( )J 95 298 :M -.089(If there is an inducing path between X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.108( and )A 301 298 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 316 298 :M -.167(in )A 329 298 :M .277 .028(M, )J 347 298 :M -.165(then )A 371 298 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 386 298 :M -.109(and )A 407 298 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 422 298 :M -.215(are )A 440 298 :M -.247(adjacent )A 482 298 :M -.334(in)A 95 319 :M .333(M.)A 77 346 :M (2)S 83 346 :M (.)S 87 346 :M 5 .5( )J 95 346 :M -.041(If there is an edge X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S f1_12 sf -.104A f0_12 sf -.051( X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.044( in M, then X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.041( is not an ancestor of X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.045( in )A 416 346 :M .277 .028(M, )J 434 346 :M -.111(but )A 453 346 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 468 346 :M (is )S 480 346 :M -.326(an)A 95 367 :M -.03(ancestor of X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.035( in M.)A 77 394 :M (3)S 83 394 :M (.)S 87 394 :M 5 .5( )J 95 394 :M -.019(If there is an edge X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S f1_12 sf -.052A f0_12 sf ( X)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.022( in M, )A 256 394 :M -.165(then )A 280 394 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 295 394 :M (is )S 307 394 :M -.111(not )A 326 394 :M -.163(an )A 341 394 :M -.122(ancestor )A 384 394 :M (of )S 398 394 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 413 394 :M -.109(and )A 434 394 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 449 394 :M (is )S 461 394 :M -.111(not )A 480 394 :M -.326(an)A 95 415 :M -.03(ancestor of X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.035( in M.)A 77 442 :M .127 .013(Proof. Suppose that M satisfies 1\), 2\), )J 264 442 :M -.109(and )A 285 442 :M .775 .077(3\). )J 303 442 :M -.326(We )A 323 442 :M -.166(will )A 345 442 :M .479 .048(show )J 375 442 :M -.249(that )A 396 442 :M -.196(there )A 423 442 :M (is )S 435 442 :M -.082(some )A 464 442 :M -.08(graph)A 59 463 :M -.073(G\(M\) such that MAG\(G\(M\)\) = M.)A 77 490 :M -.079(First we will show that if there is an inducing path U between )A 372 490 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 387 490 :M -.109(and )A 408 490 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 423 490 :M -.167(in )A 436 490 :M .456 .046(G\(M\), )J 471 490 :M -.22(then)A 59 511 :M -.159(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.098( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.088( are adjacent in M. Let U\325 be the path in M corresponding to U \(that is )A 438 511 :M -.164(if )A 449 511 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 464 511 :M f1_12 sf .126A f0_12 sf ( )S 480 511 :M .32(X)A f0_7 sf 0 3 rm (j)S 0 -3 rm 59 532 :M f0_12 sf .081 .008(occurs on U in G\(M\), X)J f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S f1_12 sf .06A f0_12 sf .049 .005( X)J f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .066 .007( occurs on )J 259 532 :M .286 .029<55D520>J 276 532 :M -.167(in )A 289 532 :M .277 .028(M, )J 307 532 :M -.109(and )A 328 532 :M -.164(if )A 339 532 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 354 532 :M f1_12 sf .126A f0_12 sf ( )S 370 532 :M (T)S f0_7 sf 0 3 rm (i,j)S 0 -3 rm f0_12 sf ( )S 387 532 :M f1_12 sf .126A f0_12 sf ( )S 403 532 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 418 532 :M -.052(occurs )A 453 532 :M (on )S 469 532 :M .306 .031(U )J 482 532 :M -.334(in)A 59 553 :M -.039(G\(M\) then X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S f1_12 sf -.089A f0_12 sf -.042( X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.035( occurs on U\325 in M.\) By definition, every member of )A f2_12 sf -.067(O)A f0_12 sf ( )S 422 553 :M (on )S 438 553 :M .306 .031(U )J 451 553 :M -.167(in )A 464 553 :M -.105(G\(M\))A 59 574 :M -.059(is a collider on U. From the algorithm for constructing G\(M\), it follows )A 402 574 :M -.249(that )A 423 574 :M -.129(every )A 453 574 :M -.263(member)A 59 595 :M (of )S 73 595 :M f2_12 sf -.253(O)A f0_12 sf ( )S 86 595 :M (on )S 102 595 :M .286 .029<55D520>J 119 595 :M -.167(in )A 132 595 :M -.667(M )A 146 595 :M (is )S 158 595 :M -.326(a )A 167 595 :M -.072(collider. )A 210 595 :M -.13(Every )A 242 595 :M -.206(collider )A 281 595 :M (on )S 297 595 :M .306 .031(U )J 310 595 :M -.167(in )A 323 595 :M -.079(G\(M\) )A 354 595 :M (is )S 366 595 :M -.163(an )A 381 595 :M -.122(ancestor )A 424 595 :M (of )S 438 595 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 453 595 :M (or )S 467 595 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 482 595 :M -.334(in)A 59 616 :M -.079(G\(M\). Because by Lemma 1, G\(M\) and M have the )A 307 616 :M -.163(same )A 335 616 :M -.122(ancestor )A 378 616 :M -.031(relations, )A 426 616 :M -.129(every )A 456 616 :M -.235(collider)A 59 637 :M (on )S 75 637 :M .286 .029<55D520>J 92 637 :M -.167(in )A 105 637 :M -.667(M )A 119 637 :M (is )S 131 637 :M -.163(an )A 146 637 :M -.122(ancestor )A 189 637 :M (of )S 203 637 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 218 637 :M (or )S 232 637 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 247 637 :M -.167(in )A 260 637 :M .277 .028(M. )J 278 637 :M (By )S 296 637 :M .32 .032(hypothesis, )J 356 637 :M -.164(if )A 368 637 :M -.196(there )A 396 637 :M (is )S 409 637 :M -.163(an )A 425 637 :M -.124(inducing )A 471 637 :M -.22(path)A 59 658 :M -.061(between X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.056( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.054( in M, X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.056( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.053( are adjacent in M.)A endp %%Page: 7 7 %%BeginPageSetup initializepage (peter; page: 7 of 18)setjob %%EndPageSetup gS 0 0 552 730 rC 467 698 24 27 rC 485 721 :M f0_12 sf (7)S gR gS 0 0 552 730 rC 77 58 :M f0_12 sf -.031(Next we will show that if X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.029( is an ancestor of X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.033( in G\(M\), and X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.034( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.033( are )A 440 58 :M -.247(adjacent )A 482 58 :M -.334(in)A 59 79 :M -.042(M, then the edge between X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.043( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.037( is oriented as X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S f1_12 sf -.096A f0_12 sf -.047( X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.043( in M. )A 369 79 :M -.139(Because )A 412 79 :M -.079(G\(M\) )A 443 79 :M (is )S 455 79 :M -.139(acyclic,)A 59 100 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 74 100 :M (is )S 86 100 :M -.111(not )A 105 100 :M -.163(an )A 120 100 :M -.122(ancestor )A 163 100 :M (of )S 177 100 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 192 100 :M -.167(in )A 205 100 :M .456 .046(G\(M\). )J 240 100 :M (By )S 258 100 :M -.33(Lemma )A 297 100 :M .833 .083(1, )J 311 100 :M -.079(G\(M\) )A 342 100 :M -.109(and )A 363 100 :M -.667(M )A 377 100 :M -.163(have )A 404 100 :M -.22(the )A 423 100 :M -.163(same )A 452 100 :M -.139(ancestor)A 59 121 :M -.031(relations, )A 107 121 :M .277 .028(so )J 122 121 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 137 121 :M (is )S 149 121 :M -.163(an )A 164 121 :M -.122(ancestor )A 207 121 :M (of )S 221 121 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 236 121 :M -.167(in )A 249 121 :M .277 .028(M. )J 267 121 :M (By )S 285 121 :M (2\) )S 299 121 :M -.109(and )A 320 121 :M .775 .077(3\), )J 339 121 :M -.22(the )A 358 121 :M -.163(edge )A 385 121 :M -.139(between )A 429 121 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 445 121 :M -.109(and )A 467 121 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 483 121 :M (is)S 59 142 :M -.015(oriented as X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S f1_12 sf S f0_12 sf ( X)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.017( in M.)A 77 169 :M -.046(Finally we will show that if X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.045( is not an ancestor of X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.05( in G\(M\), X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.046( is not an ancestor of)A 59 190 :M -.099(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.061( in G\(M\),and X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.061( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.056( are adjacent in M, then the edge between X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.061( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf ( )S 427 190 :M (is )S 439 190 :M -.164(oriented )A 481 190 :M (as)S 59 211 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 75 211 :M f1_12 sf .4A f0_12 sf .096 .01( )J 93 211 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 109 211 :M -.167(in )A 123 211 :M .277 .028(M. )J 142 211 :M (By )S 161 211 :M -.33(Lemma )A 202 211 :M .833 .083(1, )J 218 211 :M -.079(G\(M\) )A 251 211 :M -.109(and )A 274 211 :M -.667(M )A 290 211 :M -.163(have )A 318 211 :M -.22(the )A 338 211 :M -.163(same )A 368 211 :M -.122(ancestor )A 413 211 :M -.146(relations )A 459 211 :M -.165(among)A 59 232 :M -.042(members of )A f2_12 sf -.072(O)A f0_12 sf -.036(, so X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.036( is not an ancestor of X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.041( in G\(M\) and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.036( is not an ancestor of )A 449 232 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 464 232 :M -.167(in )A 477 232 :M .333(M.)A 59 253 :M -.02(By 3\), the edge between X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.021( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.018( is oriented as X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S f1_12 sf S f0_12 sf ( X)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.023( in M.)A 77 280 :M -.07(It follows that MAG\(G\(M\)\) = M.)A 77 307 :M -.054(Next we will show that conversely, if M is )A 283 307 :M -.326(a )A 292 307 :M (MAG, )S 327 307 :M -.165(then )A 351 307 :M .775 .077(1\), )J 369 307 :M .775 .077(2\), )J 387 307 :M -.109(and )A 408 307 :M (3\) )S 422 307 :M .444 .044(hold. )J 451 307 :M -.131(Since )A 481 307 :M (M)S 59 328 :M -.051(is a MAG, there is some graph G such that M = MAG\(G\).)A 77 355 :M .151 .015(First we will show 1\). Suppose )J 231 355 :M (on )S 247 355 :M -.22(the )A 265 355 :M -.122(contrary )A 308 355 :M -.249(that )A 329 355 :M -.196(there )A 356 355 :M (is )S 368 355 :M -.163(an )A 383 355 :M -.124(inducing )A 428 355 :M -.165(path )A 452 355 :M -.163(between)A 59 376 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 74 376 :M -.109(and )A 95 376 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 110 376 :M -.167(in )A 123 376 :M .277 .028(M, )J 141 376 :M -.111(but )A 160 376 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 175 376 :M -.109(and )A 196 376 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 211 376 :M -.215(are )A 229 376 :M -.111(not )A 248 376 :M -.247(adjacent )A 291 376 :M -.167(in )A 305 376 :M .277 .028(M. )J 324 376 :M -.139(Because )A 368 376 :M -.196(there )A 396 376 :M (is )S 409 376 :M -.163(an )A 425 376 :M -.124(inducing )A 471 376 :M -.22(path)A 59 397 :M -.068(between X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.064( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.061( in M, X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.064( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.06( are d-connected given every )A 362 397 :M (subset )S 396 397 :M (of )S 410 397 :M f2_12 sf .29(O)A f0_12 sf .184(\\{X)A f0_7 sf 0 3 rm .06(i)A 0 -3 rm f0_12 sf .181(,X)A f0_7 sf 0 3 rm .06(j)A 0 -3 rm f0_12 sf .247 .025(} )J 464 397 :M -.167(in )A 477 397 :M .333(M.)A 59 418 :M -.037(In Spirtes and Richardson\(1996\) it was shown that if X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.041( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.039( are )A 379 418 :M -.148(d-connected )A 440 418 :M -.132(given )A 470 418 :M f2_12 sf (Z)S f0_12 sf ( )S 482 418 :M -.334(in)A 59 439 :M .21 .021(MAG\(G\), )J 111 439 :M -.165(then )A 135 439 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 150 439 :M -.109(and )A 171 439 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 186 439 :M -.215(are )A 204 439 :M -.148(d-connected )A 265 439 :M -.132(given )A 295 439 :M f2_12 sf (Z)S f0_12 sf ( )S 307 439 :M -.167(in )A 320 439 :M 1.114 .111(G. )J 337 439 :M -.164(It )A 348 439 :M (follows )S 388 439 :M -.249(that )A 409 439 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 424 439 :M -.109(and )A 446 439 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 462 439 :M -.215(are )A 481 439 :M (d-)S 59 460 :M -.182(connected )A 110 460 :M -.132(given )A 140 460 :M -.129(every )A 170 460 :M (subset )S 204 460 :M (of )S 218 460 :M f2_12 sf .29(O)A f0_12 sf .184(\\{X)A f0_7 sf 0 3 rm .06(i)A 0 -3 rm f0_12 sf .181(,X)A f0_7 sf 0 3 rm .06(j)A 0 -3 rm f0_12 sf .247 .025(} )J 272 460 :M -.167(in )A 285 460 :M 1.114 .111(G. )J 302 460 :M (By )S 320 460 :M -.328(Verma )A 355 460 :M -.109(and )A 376 460 :M .165 .016(Pearl\(1990\), )J 440 460 :M -.196(there )A 467 460 :M (is )S 480 460 :M -.326(an)A 59 481 :M -.124(inducing )A 104 481 :M -.165(path )A 128 481 :M -.139(between )A 172 481 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 188 481 :M -.109(and )A 210 481 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 226 481 :M -.247(relative )A 265 481 :M -.167(to )A 279 481 :M f2_12 sf -.253(O)A f0_12 sf ( )S 293 481 :M -.167(in )A 307 481 :M 1.114 .111(G. )J 325 481 :M (By )S 344 481 :M -.22(the )A 363 481 :M -.166(method )A 403 481 :M (of )S 418 481 :M -.109(construction )A 481 481 :M (of)S 59 502 :M -.071(MAGs, X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.056( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.052( are adjacent in M.)A 77 529 :M -.081(Next )A 104 529 :M (we )S 122 529 :M .479 .048(show )J 152 529 :M .775 .077(2\). )J 170 529 :M .197 .02(Suppose )J 215 529 :M -.249(that )A 236 529 :M -.196(there )A 263 529 :M (is )S 275 529 :M -.163(an )A 290 529 :M -.163(edge )A 317 529 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 333 529 :M f1_12 sf .126A f0_12 sf ( )S 350 529 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 366 529 :M -.167(in )A 380 529 :M .277 .028(M, )J 399 529 :M -.111(but )A 419 529 :M -.122(contrary )A 463 529 :M -.167(to )A 477 529 :M -.33(the)A 59 550 :M .32 .032(hypothesis, )J 118 550 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 133 550 :M (is )S 145 550 :M -.163(an )A 160 550 :M -.122(ancestor )A 203 550 :M (of )S 217 550 :M .726(X)A f0_7 sf 0 3 rm .163(i)A 0 -3 rm f0_12 sf .457 .046(, )J 236 550 :M (or )S 250 550 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 265 550 :M -.111(not )A 284 550 :M -.163(an )A 299 550 :M -.122(ancestor )A 342 550 :M (of )S 356 550 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 371 550 :M -.167(in )A 384 550 :M .277 .028(M. )J 403 550 :M (By )S 422 550 :M -.22(the )A 441 550 :M -.166(method )A 481 550 :M (of)S 59 571 :M .037 .004(construction of MAGs, in G, X)J f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .027 .003( is ancestor of X)J f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .025 .003(, and X)J f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .017 .002( is not )J 362 571 :M -.122(ancestor )A 405 571 :M (of )S 419 571 :M .726(X)A f0_7 sf 0 3 rm .163(i)A 0 -3 rm f0_12 sf .457 .046(. )J 438 571 :M (By )S 456 571 :M -.413(Lemma)A 59 592 :M -.059(2, M has the same ancestor relations as G. This is a contradiction.)A 77 619 :M (Finally, )S 118 619 :M (we )S 136 619 :M .479 .048(show )J 167 619 :M .775 .077(3\). )J 186 619 :M .197 .02(Suppose )J 232 619 :M -.196(there )A 260 619 :M (is )S 273 619 :M -.163(an )A 289 619 :M -.163(edge )A 316 619 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 332 619 :M f1_12 sf .4A f0_12 sf .096 .01( )J 350 619 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 366 619 :M -.167(in )A 380 619 :M .277 .028(M, )J 399 619 :M -.111(but )A 419 619 :M -.122(contrary )A 463 619 :M -.167(to )A 477 619 :M -.33(the)A 59 640 :M -.033(hypothesis )A 115 640 :M -.249(that )A 137 640 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 153 640 :M (is )S 166 640 :M -.163(an )A 182 640 :M -.122(ancestor )A 226 640 :M (of )S 241 640 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 257 640 :M (or )S 272 640 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 288 640 :M (is )S 301 640 :M -.163(an )A 317 640 :M -.122(ancestor )A 361 640 :M (of )S 376 640 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 392 640 :M -.167(in )A 406 640 :M .277 .028(M. )J 425 640 :M (In )S 440 640 :M 1.114 .111(G, )J 459 640 :M (by )S 477 640 :M -.33(the)A 59 661 :M -.184(algorithm )A 108 661 :M (for )S 126 661 :M -.109(construction )A 188 661 :M .226 .023(MAGs, )J 228 661 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 243 661 :M (is )S 255 661 :M -.111(not )A 274 661 :M -.122(ancestor )A 317 661 :M (of )S 331 661 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 346 661 :M -.109(and )A 367 661 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 382 661 :M -.111(not )A 401 661 :M -.122(ancestor )A 444 661 :M (of )S 458 661 :M .726(X)A f0_7 sf 0 3 rm .163(i)A 0 -3 rm f0_12 sf .457 .046(. )J 477 661 :M (By)S 59 682 :M -.077(Lemma 2, M has the same ancestor relations as G. This is a contradiction. )A f1_12 sf <5C>S endp %%Page: 8 8 %%BeginPageSetup initializepage (peter; page: 8 of 18)setjob %%EndPageSetup gS 0 0 552 730 rC 467 698 24 27 rC 485 721 :M f0_12 sf (8)S gR gS 0 0 552 730 rC 77 58 :M f0_12 sf .645 .064(A )J f2_12 sf 1.336 .134(linear parameterization of a MAG G )J f0_12 sf .534 .053(is a )J 315 58 :M -.219(linear )A 345 58 :M -.131(structural )A 393 58 :M -.165(equation )A 437 58 :M -.199(model )A 470 58 :M -.11(with)A 59 79 :M -.132(correlated errors parameterized in the following way:)A 77 106 :M f1_12 sf S 83 106 :M 9 .9( )J 95 106 :M f0_12 sf -.205( Each variable A )A 176 106 :M -.167(in )A 189 106 :M -.22(the )A 207 106 :M -.064(graph )A 238 106 :M (is )S 250 106 :M -.326(a )A 259 106 :M -.219(linear )A 289 106 :M -.123(function )A 332 106 :M (of )S 346 106 :M -.112(its )A 361 106 :M -.092(parents )A 399 106 :M -.167(in )A 412 106 :M -.22(the )A 430 106 :M .425 .043(graph, )J 465 106 :M -.109(and )A 486 106 :M (a)S 95 127 :M -.013(unique error term, )A f1_12 sf (e)S f0_7 sf 0 3 rm (A)S 0 -3 rm f0_12 sf (.)S 77 154 :M f1_12 sf S 83 154 :M 9 .9( )J 95 154 :M f0_12 sf -.06( Two error terms )A 179 154 :M f1_12 sf .055(e)A f0_7 sf 0 3 rm .053(A)A 0 -3 rm f0_12 sf .175 .017(.and )J 214 154 :M f1_12 sf (e)S f0_7 sf 0 3 rm (B)S 0 -3 rm f0_12 sf ( )S 228 154 :M -.163(have )A 254 154 :M -.326(a )A 263 154 :M -.08(non-zero )A 309 154 :M -.179(correlation )A 363 154 :M ( )S 367 154 :M -.083(only )A 392 154 :M -.164(if )A 403 154 :M -.196(there )A 430 154 :M (is )S 442 154 :M -.163(an )A 457 154 :M -.163(edge )A 483 154 :M (A)S 95 175 :M f1_12 sf S f0_12 sf .028 .003( B in the graph.)J 59 209 :M -.087(For notational convenience, we )A 211 209 :M -.166(will )A 233 209 :M -.053(assume )A 272 209 :M -.249(that )A 293 209 :M -.22(the )A 311 209 :M -.145(variables )A 357 209 :M -.167(in )A 370 209 :M -.326(a )A 379 209 :M -.331(MAG )A 410 209 :M .306 .031(G )J 423 209 :M -.215(are )A 441 209 :M 1.571(X)A f0_7 sf 0 3 rm .635(1)A 0 -3 rm f0_12 sf .715(,...,X)A f0_7 sf 0 3 rm .635(n)A 0 -3 rm f0_12 sf (,)S 59 229 :M -.068(where X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.055( is not an ancestor )A 191 229 :M (of )S 205 229 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 220 229 :M -.167(in )A 233 229 :M .306 .031(G )J 246 229 :M -.164(if )A 257 229 :M -.334(i )A 264 229 :M .211 .021(> )J 275 229 :M .555 .055(j. )J 286 229 :M -.326(We )A 306 229 :M -.166(will )A 328 229 :M -.126(refer )A 354 229 :M -.167(to )A 367 229 :M -.22(the )A 385 229 :M -.061(error )A 412 229 :M -.247(term )A 437 229 :M (of )S 451 229 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 466 229 :M (as )S 480 229 :M f1_12 sf .406(e)A f0_7 sf 0 3 rm .15(i)A 0 -3 rm f0_12 sf (,)S 59 249 :M -.162(rather )A 90 249 :M -.165(than )A 114 249 :M f1_12 sf .212(e)A f0_7 sf 0 3 rm .203(X)A 0 -3 rm 0 5 rm .078(i)A 0 -5 rm f0_12 sf .219 .022(. )J 134 249 :M -.663(A )A 146 249 :M -.331(MAG )A 177 249 :M -.667(M )A 191 249 :M f2_12 sf 2.597 .26(linearly )J 239 249 :M .482(entails)A f0_12 sf .303 .03( )J 280 249 :M -.249(that )A 301 249 :M f2_12 sf .993(X)A f0_12 sf .344 .034( )J 315 249 :M (is )S 327 249 :M -.15(independent )A 389 249 :M (of )S 404 249 :M f2_12 sf .25(Y)A f0_12 sf .087 .009( )J 418 249 :M -.132(given )A 449 249 :M f2_12 sf (Z)S f0_12 sf ( )S 462 249 :M -.164(if )A 474 249 :M -.163(and)A 59 272 :M -.083(only )A 86 272 :M -.164(if )A 99 272 :M -.167(in )A 114 272 :M -.129(every )A 146 272 :M -.219(linear )A 178 272 :M -.226(parameterization )A 261 272 :M (of )S 277 272 :M .277 .028(M, )J 297 272 :M f2_12 sf .993(X)A f0_12 sf .344 .034( )J 313 272 :M (is )S 327 272 :M -.15(independent )A 390 272 :M (of )S 406 272 :M f2_12 sf .25(Y)A f0_12 sf .087 .009( )J 421 272 :M -.132(given )A 454 272 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(. )J 473 272 :M -.328(The)A 59 293 :M -.053(following theorems are proved in Spirtes and Richardson\(1997\).)A 59 325 :M f2_12 sf .309 .031(Theorem 2: )J f0_12 sf .257 .026(A MAG G linearly )J 219 325 :M -.189(entails )A 253 325 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 268 325 :M (is )S 280 325 :M -.15(independent )A 341 325 :M (of )S 355 325 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 370 325 :M -.132(given )A 400 325 :M f2_12 sf (Z)S f0_12 sf ( )S 412 325 :M -.164(if )A 423 325 :M -.109(and )A 444 325 :M -.083(only )A 469 325 :M -.164(if )A 480 325 :M .32(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm 59 346 :M f0_12 sf -.025(is d-separated from X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.024( given )A f2_12 sf (Z)S f0_12 sf -.027( in G.)A 59 379 :M f2_12 sf 1.277 .128(Theorem 3: )J f0_12 sf .67 .067(A )J 139 379 :M -.331(MAG )A 170 379 :M .306 .031(G )J 183 379 :M -.206(linearly )A 222 379 :M -.189(entails )A 256 379 :M .102(cov\(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .099(,X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf (|)S f2_12 sf .136(Z)A f0_12 sf .108 .011(\) )J 321 379 :M .211 .021(= )J 332 379 :M (0 )S 342 379 :M -.164(if )A 353 379 :M -.109(and )A 374 379 :M -.083(only )A 399 379 :M -.164(if )A 410 379 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 425 379 :M (is )S 437 379 :M -.129(d-separated)A 59 400 :M .086 .009(from X)J f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .047 .005( given )J f2_12 sf (Z)S f0_12 sf .042 .004( in G.)J 77 427 :M (A )S f2_12 sf -.009(complete MAG)A f0_12 sf -.007( is a MAG in which every pair of variables is adjacent.)A 59 460 :M f2_12 sf .081 .008(Lemma 3: )J f0_12 sf .063 .006(If G is a MAG, there is a complete MAG G)J f0_7 sf 0 3 rm (C)S 0 -3 rm f0_12 sf .057 .006(, such that G is a subgraph of G)J f0_7 sf 0 3 rm (C)S 0 -3 rm f0_12 sf (.)S 77 487 :M -.023(Proof. Suppose that G is a MAG. Form G)A f0_7 sf 0 3 rm (C)S 0 -3 rm f0_12 sf -.021( in the following way: if X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.021( is an ancestor of)A 59 508 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 74 508 :M -.167(in )A 87 508 :M 1.114 .111(G, )J 104 508 :M -.109(add )A 125 508 :M -.163(an )A 140 508 :M -.163(edge )A 166 508 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 181 508 :M f1_12 sf .126A f0_12 sf ( )S 197 508 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 212 508 :M -.167(in )A 225 508 :M .747(G)A f0_7 sf 0 3 rm .403(C)A 0 -3 rm f0_12 sf .471 .047(, )J 247 508 :M -.109(and )A 268 508 :M -.164(if )A 279 508 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 294 508 :M (is )S 306 508 :M -.111(not )A 325 508 :M -.163(an )A 340 508 :M -.122(ancestor )A 383 508 :M (of )S 397 508 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 412 508 :M -.109(and )A 433 508 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 448 508 :M (is )S 460 508 :M -.111(not )A 480 508 :M -.326(an)A 59 529 :M -.034(ancestor of X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.034(, then add an )A 190 529 :M -.163(edge )A 216 529 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 231 529 :M f1_12 sf .4A f0_12 sf .096 .01( )J 248 529 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 263 529 :M -.167(to )A 276 529 :M .747(G)A f0_7 sf 0 3 rm .403(C)A 0 -3 rm f0_12 sf .471 .047(. )J 298 529 :M -.13(Every )A 330 529 :M -.163(pair )A 352 529 :M (of )S 366 529 :M -.163(vertices )A 406 529 :M -.167(in )A 419 529 :M .354(G)A f0_7 sf 0 3 rm .191(C)A 0 -3 rm f0_12 sf .123 .012( )J 437 529 :M (is )S 449 529 :M -.122(adjacent.)A 59 550 :M -.087(Note that G and G)A f0_7 sf 0 3 rm -.079(C)A 0 -3 rm f0_12 sf -.083( have the same ancestor relations. By the method of construction )A 459 550 :M (of )S 473 550 :M .885(G)A f0_7 sf 0 3 rm .477(C)A 0 -3 rm f0_12 sf (,)S 59 571 :M -.006(if there is an edge X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( *)S f1_12 sf S f0_12 sf ( X)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf ( in G)S f0_7 sf 0 3 rm (C)S 0 -3 rm f0_12 sf (, then X)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf ( is )S 276 571 :M -.111(not )A 295 571 :M -.163(an )A 310 571 :M -.122(ancestor )A 353 571 :M (of )S 367 571 :M .726(X)A f0_7 sf 0 3 rm .163(i)A 0 -3 rm f0_12 sf .457 .046(. )J 386 571 :M -.139(Because )A 429 571 :M -.129(every )A 459 571 :M -.163(pair )A 481 571 :M (of)S 59 592 :M -.084(vertices in G)A f0_7 sf 0 3 rm -.083(C)A 0 -3 rm f0_12 sf -.076( is adjacent, it is )A 204 592 :M -.221(trivially )A 244 592 :M -.163(true )A 266 592 :M -.249(that )A 287 592 :M -.164(if )A 298 592 :M -.196(there )A 325 592 :M (is )S 337 592 :M -.163(an )A 352 592 :M -.124(inducing )A 397 592 :M -.165(path )A 421 592 :M -.139(between )A 464 592 :M -.326(a )A 473 592 :M -.218(pair)A 59 613 :M -.091(of vertices, then there is an edge between that pair of vertices. )A f1_12 sf <5C>S endp %%Page: 9 9 %%BeginPageSetup initializepage (peter; page: 9 of 18)setjob %%EndPageSetup gS 0 0 552 730 rC 467 698 24 27 rC 485 721 :M f0_12 sf (9)S gR gS 0 0 552 730 rC 59 58 :M f2_12 sf -.014(Lemma 4)A f0_12 sf -.011(: In a MAG G with an edge X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S f1_12 sf S f0_12 sf ( X)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf (, if )S f1_12 sf (q)S f0_12 sf -.01(\(0\) is a parameterization of G )A 449 58 :M -.167(in )A 462 58 :M -.081(which)A 59 79 :M .031(cov\()A f1_12 sf (e)S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf (,)S f1_12 sf (e)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf (\) )S 106 79 :M .211 .021(= )J 117 79 :M .833 .083(0, )J 131 79 :M -.109(and )A 152 79 :M f1_12 sf .121(q)A f0_12 sf .242 .024(\(c\) )J 176 79 :M (is )S 188 79 :M -.326(a )A 197 79 :M -.226(parameterization )A 278 79 :M (of )S 292 79 :M .306 .031(G )J 305 79 :M -.257(identical )A 348 79 :M -.167(to )A 361 79 :M f1_12 sf .205(q)A f0_12 sf .428 .043(\(0\) )J 387 79 :M -.219(except )A 422 79 :M .031(cov\()A f1_12 sf (e)S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf (,)S f1_12 sf (e)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf (\) )S 470 79 :M .211 .021(= )J 482 79 :M .674(c,)A 59 100 :M -.22(then)A 65 133 :M .153(cov)A f1_7 sf 0 3 rm .096(q)A 0 -3 rm f0_7 sf 0 3 rm .068(\(c\))A 0 -3 rm f0_12 sf .167(\(X)A f0_7 sf 0 3 rm .051(i)A 0 -3 rm f0_12 sf .154(,X)A f0_7 sf 0 3 rm .051(j)A 0 -3 rm f0_12 sf .063(|)A f2_12 sf .203(An)A f0_12 sf .167(\(X)A f0_7 sf 0 3 rm .051(i)A 0 -3 rm f0_12 sf .154 .015(\) )J f1_12 sf .243A f0_12 sf .072 .007( )J f2_12 sf .203(An)A f0_12 sf .167(\(X)A f0_7 sf 0 3 rm .051(j)A 0 -3 rm f0_12 sf .144(\)\\{X)A f0_7 sf 0 3 rm .051(i)A 0 -3 rm f0_12 sf .154(,X)A f0_7 sf 0 3 rm .051(j)A 0 -3 rm f0_12 sf .29 .029(}\) = cov)J f1_7 sf 0 3 rm .096(q)A 0 -3 rm f0_7 sf 0 3 rm .072(\(0\))A 0 -3 rm f0_12 sf .167(\(X)A f0_7 sf 0 3 rm .051(i)A 0 -3 rm f0_12 sf .154(,X)A f0_7 sf 0 3 rm .051(j)A 0 -3 rm f0_12 sf .063(|)A f2_12 sf .203(An)A f0_12 sf .167(\(X)A f0_7 sf 0 3 rm .051(i)A 0 -3 rm f0_12 sf .154 .015(\) )J f1_12 sf .243A f0_12 sf .072 .007( )J f2_12 sf .203(An)A f0_12 sf .167(\(X)A f0_7 sf 0 3 rm .051(j)A 0 -3 rm f0_12 sf .144(\)\\{X)A f0_7 sf 0 3 rm .051(i)A 0 -3 rm f0_12 sf .154(,X)A f0_7 sf 0 3 rm .051(j)A 0 -3 rm f0_12 sf .313 .031(}\) + c.)J 77 160 :M (Proof. First we will show that if )S 234 160 :M .478(X)A f0_7 sf 0 3 rm .193(k)A 0 -3 rm f0_12 sf .166 .017( )J 251 160 :M (is )S 263 160 :M -.167(in )A 276 160 :M f2_12 sf .223(An)A f0_12 sf .184(\(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .184 .018(\) )J 315 160 :M f1_12 sf -.161A f0_12 sf ( )S 328 160 :M f2_12 sf .303(An)A f0_12 sf .25(\(X)A f0_7 sf 0 3 rm .077(j)A 0 -3 rm f0_12 sf .215(\)\\{X)A f0_7 sf 0 3 rm .077(i)A 0 -3 rm f0_12 sf .23(,X)A f0_7 sf 0 3 rm .077(j)A 0 -3 rm f0_12 sf .314 .031(} )J 408 160 :M -.165(then )A 432 160 :M .074(var)A f1_7 sf 0 3 rm .053(q)A 0 -3 rm f0_7 sf 0 3 rm .039(\(0\))A 0 -3 rm f0_12 sf .091(\(X)A f0_7 sf 0 3 rm .051(k)A 0 -3 rm f0_12 sf .092 .009(\) )J 484 160 :M (=)S 59 181 :M -.024(var)A f1_7 sf 0 3 rm (q)S 0 -3 rm f0_7 sf 0 3 rm (\(c\))S 0 -3 rm f0_12 sf -.03(\(X)A f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf -.022(\). Note that in )A 172 181 :M .177(G\()A f1_12 sf .174(q)A f0_12 sf .418 .042(\(0\)\) )J 214 181 :M -.109(and )A 235 181 :M .268(G\()A f1_12 sf .265(q)A f0_12 sf .657 .066(\(c\)\), )J 280 181 :M -.22(the )A 298 181 :M -.163(coefficients )A 356 181 :M (of )S 370 181 :M -.22(the )A 388 181 :M -.139(reduced )A 429 181 :M -.08(form )A 456 181 :M (of )S 470 181 :M -.326(each)A 59 202 :M -.14(variable is exactly the same in each of the parameterizations, because the structual )A 446 202 :M -.123(equations)A 59 223 :M -.13(in each parameterization are identical. X)A f0_7 sf 0 3 rm -.097(k)A 0 -3 rm f0_12 sf -.129( is a function )A 319 223 :M (of )S 333 223 :M -.16(\(a )A 346 223 :M (possibly )S 390 223 :M -.052(proper )A 425 223 :M (subset )S 459 223 :M (of\) )S 477 223 :M -.33(the)A 59 244 :M .195 .02(error terms in )J f2_12 sf .099(An)A f0_12 sf .081(\(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .075 .007(\) )J f1_12 sf .119A f0_12 sf ( )S f2_12 sf .099(An)A f0_12 sf .081(\(X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .07(\)\\{X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .075(,X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .321 .032(}. Hence)J 146 254 275 56 rC 421 310 :M psb currentpoint pse 146 254 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 8800 div 1792 3 -1 roll exch div scale currentpoint translate 64 34 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (X) 3324 346 sh (a) 4639 346 sh (var) -3 1297 sh (\(X) 871 1297 sh (\)) 1464 1297 sh (a) 2441 1297 sh (var) 2858 1297 sh (\() 3732 1297 sh (\)) 4185 1297 sh (2) 4678 1297 sh (a) 5894 1297 sh (a) 6300 1297 sh (cov) 6713 1297 sh (\() 7686 1297 sh (,) 8126 1297 sh (\)) 8565 1297 sh 224 ns (k) 3627 442 sh (kr) 4840 442 sh (r) 5240 442 sh (r) 4212 719 sh (k) 4462 719 sh (q\(0\)) 466 1393 sh (k) 1301 1393 sh (kr) 2642 1393 sh (2) 2636 1125 sh (q\(0\)) 3327 1393 sh (r) 4059 1393 sh (kr) 6095 1393 sh (ks) 6501 1393 sh (q\(0\)) 7281 1393 sh (r) 8013 1393 sh (r) 5346 1670 sh (s) 5586 1670 sh (k) 5838 1670 sh (r) 2014 1670 sh (k) 2264 1670 sh (s) 8437 1393 sh 384 /Symbol f1 (=) 3879 346 sh (=) 1681 1297 sh (+) 4387 1297 sh 224 ns (\243) 4313 719 sh (<) 5448 1670 sh (\243) 5689 1670 sh (\243) 2115 1670 sh 576 ns (\345) 4189 433 sh (\345) 5444 1384 sh (\345) 4882 1384 sh (\345) 1991 1384 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) 5043 346 sh (e) 3862 1297 sh (e) 7816 1297 sh (e) 8251 1297 sh end MTsave restore pse gR gS 0 0 552 730 rC 77 333 :M f0_12 sf -.023(In the formula for var)A f1_7 sf 0 3 rm (q)S 0 -3 rm f0_7 sf 0 3 rm (\(0\))S 0 -3 rm f0_12 sf -.031(\(X)A f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf -.022(\), r and s are less than or equal to k, and k is less )A 443 333 :M -.165(than )A 467 333 :M -.334(i )A 474 333 :M -.163(and)A 59 354 :M -.334(j )A 67 354 :M (by )S 84 354 :M .32 .032(hypothesis. )J 144 354 :M .224 .022(Hence, )J 183 354 :M -.22(the )A 202 354 :M -.14(formula )A 244 354 :M (is )S 257 354 :M -.257(identical )A 301 354 :M (for )S 320 354 :M .108(var)A f1_7 sf 0 3 rm .077(q)A 0 -3 rm f0_7 sf 0 3 rm .055(\(c\))A 0 -3 rm f0_12 sf .134(\(X)A f0_7 sf 0 3 rm .074(k)A 0 -3 rm f0_12 sf .176 .018(\), )J 376 354 :M -.109(and )A 398 354 :M -.196(hence )A 430 354 :M .074(var)A f1_7 sf 0 3 rm .053(q)A 0 -3 rm f0_7 sf 0 3 rm .039(\(0\))A 0 -3 rm f0_12 sf .091(\(X)A f0_7 sf 0 3 rm .051(k)A 0 -3 rm f0_12 sf .092 .009(\) )J 484 354 :M (=)S 59 375 :M .115(var)A f1_7 sf 0 3 rm .082(q)A 0 -3 rm f0_7 sf 0 3 rm .058(\(c\))A 0 -3 rm f0_12 sf .142(\(X)A f0_7 sf 0 3 rm .079(k)A 0 -3 rm f0_12 sf .157(\).)A 77 402 :M -.081(Next )A 104 402 :M (we )S 122 402 :M -.166(will )A 144 402 :M .479 .048(show )J 175 402 :M -.249(that )A 197 402 :M -.164(if )A 209 402 :M -.166(distinct )A 248 402 :M -.163(vertices )A 289 402 :M .478(X)A f0_7 sf 0 3 rm .193(k)A 0 -3 rm f0_12 sf .166 .017( )J 307 402 :M -.109(and )A 329 402 :M .25(X)A f0_7 sf 0 3 rm .056(l)A 0 -3 rm f0_12 sf .086 .009( )J 345 402 :M -.215(are )A 364 402 :M -.167(in )A 378 402 :M f2_12 sf .223(An)A f0_12 sf .184(\(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .184 .018(\) )J 418 402 :M f1_12 sf -.161A f0_12 sf ( )S 432 402 :M f2_12 sf .398(An)A f0_12 sf .328(\(X)A f0_7 sf 0 3 rm .101(j)A 0 -3 rm f0_12 sf .432 .043(\), )J 476 402 :M -.167(but)A 59 423 :M .485({X)A f0_7 sf 0 3 rm .235(k)A 0 -3 rm f0_12 sf .392(,X)A f0_7 sf 0 3 rm .131(l)A 0 -3 rm f0_12 sf .535 .054(} )J 104 423 :M .387 .039(<> )J 124 423 :M .564({X)A f0_7 sf 0 3 rm .152(i)A 0 -3 rm f0_12 sf .456(,X)A f0_7 sf 0 3 rm .152(j)A 0 -3 rm f0_12 sf .766 .077(}, )J 172 423 :M -.165(then )A 198 423 :M .185(cov)A f1_7 sf 0 3 rm .117(q)A 0 -3 rm f0_7 sf 0 3 rm .087(\(0\))A 0 -3 rm f0_12 sf .203(\(X)A f0_7 sf 0 3 rm .112(k)A 0 -3 rm f0_12 sf .187(,X)A f0_7 sf 0 3 rm .062(l)A 0 -3 rm f0_12 sf .204 .02(\) )J 269 423 :M .211 .021(= )J 282 423 :M .211(cov)A f1_7 sf 0 3 rm .133(q)A 0 -3 rm f0_7 sf 0 3 rm .095(\(c\))A 0 -3 rm f0_12 sf .231(\(X)A f0_7 sf 0 3 rm .128(k)A 0 -3 rm f0_12 sf .213(,X)A f0_7 sf 0 3 rm .071(l)A 0 -3 rm f0_12 sf .304 .03(\). )J 356 423 :M .197 .02(Suppose )J 403 423 :M .478(X)A f0_7 sf 0 3 rm .193(k)A 0 -3 rm f0_12 sf .166 .017( )J 422 423 :M -.109(and )A 445 423 :M .25(X)A f0_7 sf 0 3 rm .056(l)A 0 -3 rm f0_12 sf .086 .009( )J 462 423 :M -.215(are )A 482 423 :M -.334(in)A 59 444 :M f2_12 sf .306(An)A f0_12 sf .253(\(X)A f0_7 sf 0 3 rm .078(i)A 0 -3 rm f0_12 sf .233 .023(\) )J f1_12 sf .368A f0_12 sf .109 .011( )J f2_12 sf .306(An)A f0_12 sf .253(\(X)A f0_7 sf 0 3 rm .078(j)A 0 -3 rm f0_12 sf .589 .059(\), but {X)J f0_7 sf 0 3 rm .14(k)A 0 -3 rm f0_12 sf .233(,X)A f0_7 sf 0 3 rm .078(l)A 0 -3 rm f0_12 sf .587 .059(} <> {X)J f0_7 sf 0 3 rm .078(i)A 0 -3 rm f0_12 sf .233(,X)A f0_7 sf 0 3 rm .078(j)A 0 -3 rm f0_12 sf .576 .058(}. Also, )J 304 444 :M .2 .02(suppose )J 347 444 :M -.095(without )A 387 444 :M .237 .024(loss )J 410 444 :M (of )S 424 444 :M -.197(generality )A 474 444 :M -.331(that)A 59 465 :M (X)S f0_7 sf 0 3 rm (l)S 0 -3 rm f0_12 sf ( is not )S 103 465 :M -.197(equal )A 132 465 :M -.167(to )A 145 465 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 160 465 :M (or )S 174 465 :M -.167(to )A 187 465 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 202 465 :M -.109(and )A 223 465 :M -.196(hence )A 254 465 :M .25(X)A f0_7 sf 0 3 rm .056(l)A 0 -3 rm f0_12 sf .086 .009( )J 269 465 :M (is )S 281 465 :M -.167(in )A 294 465 :M f2_12 sf .223(An)A f0_12 sf .184(\(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .184 .018(\) )J 333 465 :M f1_12 sf -.161A f0_12 sf ( )S 346 465 :M f2_12 sf .388(An)A f0_12 sf .321(\(X)A f0_7 sf 0 3 rm .099(j)A 0 -3 rm f0_12 sf .275(\)\\{X)A f0_7 sf 0 3 rm .099(i)A 0 -3 rm f0_12 sf .295(,X)A f0_7 sf 0 3 rm .099(j)A 0 -3 rm f0_12 sf .496 .05(}. )J 430 465 :M ( )S 434 465 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 449 465 :M (is )S 461 465 :M -.111(not )A 480 465 :M -.326(an)A 59 486 :M -.025(ancestor of any member of )A f2_12 sf -.039(An)A f0_12 sf -.032(\(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf (\) )S f1_12 sf S f0_12 sf ( )S f2_12 sf -.039(An)A f0_12 sf -.032(\(X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.027(\)\\{X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.021(}; if it is an ancestor of)A f2_12 sf ( )S 406 486 :M .198(An)A f0_12 sf .163(\(X)A f0_7 sf 0 3 rm .05(i)A 0 -3 rm f0_12 sf .14(\)\\{X)A f0_7 sf 0 3 rm .05(i)A 0 -3 rm f0_12 sf .206 .021(} )J 471 486 :M -.22(then)A 59 507 :M -.104(there is a directed cycle in G, and if it is an )A 264 507 :M -.122(ancestor )A 307 507 :M (of )S 321 507 :M f2_12 sf .198(An)A f0_12 sf .163(\(X)A f0_7 sf 0 3 rm .05(j)A 0 -3 rm f0_12 sf .14(\)\\{X)A f0_7 sf 0 3 rm .05(i)A 0 -3 rm f0_12 sf .206 .021(} )J 386 507 :M -.165(then )A 410 507 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 425 507 :M (is )S 437 507 :M -.163(an )A 452 507 :M -.139(ancestor)A 59 528 :M -.049(of X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.044(, even though there is an edge )A 228 528 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 243 528 :M f1_12 sf .4A f0_12 sf .096 .01( )J 260 528 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 275 528 :M -.167(in )A 288 528 :M 1.114 .111(G. )J 305 528 :M -.066(Similarly, )A 356 528 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 371 528 :M (is )S 383 528 :M -.111(not )A 402 528 :M -.163(an )A 417 528 :M -.122(ancestor )A 460 528 :M (of )S 474 528 :M -.163(any)A 59 549 :M .019 .002(member of )J f2_12 sf (An)S f0_12 sf (\(X)S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf (\) )S f1_12 sf S f0_12 sf ( )S f2_12 sf (An)S f0_12 sf (\(X)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf (\)\\{X)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .017 .002(}. It follows that neither X)J f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( nor X)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .013 .001( is an ancestor of X)J f0_7 sf 0 3 rm (l)S 0 -3 rm f0_12 sf (.)S 77 576 :M .615 .061(For X)J f0_7 sf 0 3 rm .114(k)A 0 -3 rm f0_12 sf .401 .04( and X)J f0_7 sf 0 3 rm .063(l)A 0 -3 rm f0_12 sf (,)S 103 586 361 56 rC 464 642 :M psb currentpoint pse 103 586 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 11552 div 1792 3 -1 roll exch div scale currentpoint translate 64 34 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (X) 3505 346 sh (a) 4820 346 sh (,) 5540 346 sh (X) 5997 346 sh (b) 7265 346 sh (cov) -9 1297 sh (\(X) 964 1297 sh (,) 1544 1297 sh (X) 1681 1297 sh (\)) 2089 1297 sh (a) 3066 1297 sh (b) 3483 1297 sh (var) 3859 1297 sh (\() 4733 1297 sh (\)) 5192 1297 sh (2) 5685 1297 sh (\(a) 7224 1297 sh (b) 7768 1297 sh (a) 8530 1297 sh (b) 8949 1297 sh (\)cov) 9347 1297 sh (\() 10447 1297 sh (,) 10893 1297 sh (\)) 11338 1297 sh 224 ns (k) 3808 442 sh (kr) 5021 442 sh (r) 5427 442 sh (r) 4393 719 sh (k) 4643 719 sh (l) 6296 442 sh (lr) 7475 442 sh (r) 7831 442 sh (r) 6856 719 sh (l) 7102 719 sh (q\(0\)) 559 1393 sh (k) 1394 1393 sh (l) 1980 1393 sh (kr) 3267 1393 sh (lr) 3693 1393 sh (q\(0\)) 4328 1393 sh (r) 5066 1393 sh (kr) 7552 1393 sh (ls) 7978 1393 sh (ks) 8731 1393 sh (lr) 9159 1393 sh (q\(0\)) 10042 1393 sh (r) 10780 1393 sh (r) 6353 1670 sh (s) 6593 1670 sh (max\(k,) 6843 1670 sh (l\)) 7487 1670 sh (r) 2639 1670 sh (k) 2889 1670 sh (s) 11210 1393 sh 384 /Symbol f1 (=) 4060 346 sh (=) 6494 346 sh (=) 2306 1297 sh (+) 5394 1297 sh (+) 8242 1297 sh 224 ns (\243) 4494 719 sh (\243) 6957 719 sh (<) 6455 1670 sh (\243) 6696 1670 sh (\243) 2740 1670 sh 576 ns (\345) 4370 433 sh (\345) 6804 433 sh (\345) 6778 1384 sh (\345) 5889 1384 sh (\345) 2616 1384 sh 384 /Symbol f1 (e) 5232 346 sh (e) 7636 346 sh (e) 4871 1297 sh (e) 10585 1297 sh (e) 11026 1297 sh end MTsave restore pse endp %%Page: 10 10 %%BeginPageSetup initializepage (peter; page: 10 of 18)setjob %%EndPageSetup gS 0 0 552 730 rC 461 698 30 27 rC 479 721 :M f0_12 sf (10)S gR gS 0 0 552 730 rC 77 58 :M f0_12 sf -.219(The )A 100 58 :M -.14(formula )A 142 58 :M (for )S 161 58 :M .133(cov)A f1_7 sf 0 3 rm .084(q)A 0 -3 rm f0_7 sf 0 3 rm .059(\(c\))A 0 -3 rm f0_12 sf .145(\(X)A f0_7 sf 0 3 rm .08(k)A 0 -3 rm f0_12 sf .134(,X)A f0_7 sf 0 3 rm (l)S 0 -3 rm f0_12 sf .146 .015(\) )J 230 58 :M (is )S 243 58 :M -.235(exactly )A 281 58 :M -.22(the )A 300 58 :M -.163(same )A 329 58 :M (as )S 344 58 :M -.22(the )A 363 58 :M -.14(formula )A 406 58 :M (for )S 426 58 :M .194(cov)A f1_7 sf 0 3 rm .122(q)A 0 -3 rm f0_7 sf 0 3 rm .091(\(0\))A 0 -3 rm f0_12 sf .212(\(X)A f0_7 sf 0 3 rm .117(k)A 0 -3 rm f0_12 sf .196(,X)A f0_7 sf 0 3 rm .065(l)A 0 -3 rm f0_12 sf <29>S 59 79 :M -.074(except for terms of the form \(a)A f0_7 sf 0 3 rm -.042(ki)A 0 -3 rm f0_12 sf -.093(b)A f0_7 sf 0 3 rm -.03(lj)A 0 -3 rm f0_12 sf ( )S 223 79 :M .211 .021(+ )J 234 79 :M (a)S f0_7 sf 0 3 rm (kj)S 0 -3 rm f0_12 sf (b)S f0_7 sf 0 3 rm (li)S 0 -3 rm f0_12 sf .016(\)cov)A f1_12 sf (q)S f0_7 sf 0 3 rm (\(c\))S 0 -3 rm f0_12 sf <28>S f1_12 sf (e)S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf (,)S f1_12 sf (e)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf (\); )S 323 79 :M -.331(all )A 338 79 :M -.049(corresponding )A 410 79 :M -.131(terms )A 440 79 :M -.215(are )A 458 79 :M -.161(zero )A 482 79 :M -.334(in)A 59 100 :M .185(cov)A f1_7 sf 0 3 rm .117(q)A 0 -3 rm f0_7 sf 0 3 rm .087(\(0\))A 0 -3 rm f0_12 sf .203(\(X)A f0_7 sf 0 3 rm .112(k)A 0 -3 rm f0_12 sf .187(,X)A f0_7 sf 0 3 rm .062(l)A 0 -3 rm f0_12 sf .204 .02(\) )J 128 100 :M -.139(because )A 169 100 :M .082(cov)A f1_12 sf .089(q)A f0_7 sf 0 3 rm .038(\(0\))A 0 -3 rm f0_12 sf .057<28>A f1_12 sf .075(e)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf (,)S f1_12 sf .075(e)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .09 .009(\) )J 232 100 :M .211 .021(= )J 244 100 :M (0 )S 255 100 :M (by )S 272 100 :M -.06(definition. )A 326 100 :M .571 .057(However, )J 379 100 :M -.331(all )A 395 100 :M (such )S 422 100 :M -.131(terms )A 453 100 :M -.215(are )A 472 100 :M -.109(also)A 59 121 :M -.14(zero in )A 95 121 :M -.22(the )A 113 121 :M -.14(formula )A 154 121 :M (for )S 172 121 :M .211(cov)A f1_7 sf 0 3 rm .133(q)A 0 -3 rm f0_7 sf 0 3 rm .095(\(c\))A 0 -3 rm f0_12 sf .231(\(X)A f0_7 sf 0 3 rm .128(k)A 0 -3 rm f0_12 sf .213(,X)A f0_7 sf 0 3 rm .071(l)A 0 -3 rm f0_12 sf .304 .03(\), )J 244 121 :M -.139(because )A 285 121 :M -.188(neither )A 321 121 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 336 121 :M (nor )S 356 121 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 371 121 :M (is )S 383 121 :M -.163(an )A 398 121 :M -.122(ancestor )A 441 121 :M (of )S 455 121 :M .726(X)A f0_7 sf 0 3 rm .163(l)A 0 -3 rm f0_12 sf .457 .046(, )J 474 121 :M -.163(and)A 59 142 :M .009(henceb)A f0_7 sf 0 3 rm (lj)S 0 -3 rm f0_12 sf ( and b)S f0_7 sf 0 3 rm (li)S 0 -3 rm f0_12 sf ( = 0.)S 77 169 :M -.081(Next )A 108 169 :M (we )S 130 169 :M -.166(will )A 156 169 :M .479 .048(show )J 190 169 :M -.249(that )A 215 169 :M .15(cov)A f1_7 sf 0 3 rm .094(q)A 0 -3 rm f0_7 sf 0 3 rm .07(\(0\))A 0 -3 rm f0_12 sf .164(\(X)A f0_7 sf 0 3 rm .05(i)A 0 -3 rm f0_12 sf .151(,X)A f0_7 sf 0 3 rm .05(j)A 0 -3 rm f0_12 sf .164 .016(\) )J 286 169 :M .211 .021(= )J 301 169 :M .096(cov)A f1_7 sf 0 3 rm .06(q)A 0 -3 rm f0_7 sf 0 3 rm .043(\(c\))A 0 -3 rm f0_12 sf .105(\(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .097(,X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .105 .011(\) )J 371 169 :M .211 .021(+ )J 386 169 :M .562 .056(c. )J 404 169 :M -.219(The )A 431 169 :M -.14(formula )A 477 169 :M (for)S 59 190 :M .15(cov)A f1_7 sf 0 3 rm .094(q)A 0 -3 rm f0_7 sf 0 3 rm .07(\(0\))A 0 -3 rm f0_12 sf .164(\(X)A f0_7 sf 0 3 rm .05(i)A 0 -3 rm f0_12 sf .151(,X)A f0_7 sf 0 3 rm .05(j)A 0 -3 rm f0_12 sf .164 .016(\) )J 131 190 :M (is )S 148 190 :M -.257(identical )A 196 190 :M -.167(to )A 214 190 :M -.22(the )A 237 190 :M -.14(formula )A 283 190 :M (for )S 306 190 :M .178(cov)A f1_7 sf 0 3 rm .112(q)A 0 -3 rm f0_7 sf 0 3 rm .08(\(c\))A 0 -3 rm f0_12 sf .195(\(X)A f0_7 sf 0 3 rm .06(i)A 0 -3 rm f0_12 sf .179(,X)A f0_7 sf 0 3 rm .06(j)A 0 -3 rm f0_12 sf .256 .026(\), )J 381 190 :M -.219(except )A 420 190 :M -.249(that )A 446 190 :M -.22(the )A 470 190 :M -.329(term)A 59 211 :M (a)S f0_7 sf 0 3 rm (ii)S 0 -3 rm f0_12 sf (b)S f0_7 sf 0 3 rm (jj)S 0 -3 rm f0_12 sf .036(cov)A f1_12 sf (q)S f0_7 sf 0 3 rm (\(c\))S 0 -3 rm f0_12 sf <28>S f1_12 sf (e)S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf (,)S f1_12 sf (e)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .079 .008(\) = cov)J f1_12 sf (q)S f0_7 sf 0 3 rm (\(c\))S 0 -3 rm f0_12 sf <28>S f1_12 sf (e)S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf (,)S f1_12 sf (e)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .061 .006(\) = c, and )J 251 211 :M -.22(the )A 269 211 :M -.247(term )A 294 211 :M .05(a)A f0_7 sf 0 3 rm (ii)S 0 -3 rm f0_12 sf .056(b)A f0_7 sf 0 3 rm (jj)S 0 -3 rm f0_12 sf .054(cov)A f1_12 sf .059(q)A f0_7 sf 0 3 rm .026(\(0\))A 0 -3 rm f0_12 sf <28>S f1_12 sf (e)S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf (,)S f1_12 sf (e)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .06 .006(\) )J 375 211 :M .211 .021(= )J 386 211 :M .082(cov)A f1_12 sf .089(q)A f0_7 sf 0 3 rm .038(\(0\))A 0 -3 rm f0_12 sf .057<28>A f1_12 sf .075(e)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf (,)S f1_12 sf .075(e)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .09 .009(\) )J 448 211 :M .211 .021(= )J 459 211 :M .833 .083(0. )J 473 211 :M (\(By)S 59 232 :M -.049(convention, the error terms are scaled so that a)A f0_7 sf 0 3 rm (ii)S 0 -3 rm f0_12 sf -.049( = b)A f0_7 sf 0 3 rm (ii)S 0 -3 rm f0_12 sf -.057( = 1.)A 183 255 :M .158(cov)A f1_7 sf 0 3 rm .1(q)A 0 -3 rm f0_7 sf 0 3 rm .074(\(0\))A 0 -3 rm f0_12 sf .173(\(X)A f0_7 sf 0 3 rm .053(i)A 0 -3 rm f0_12 sf .16(,X)A f0_7 sf 0 3 rm .053(j)A 0 -3 rm f0_12 sf .066(|)A f2_12 sf .21(An)A f0_12 sf .173(\(X)A f0_7 sf 0 3 rm .053(i)A 0 -3 rm f0_12 sf .16 .016(\) )J f1_12 sf .252A f0_12 sf .075 .007( )J f2_12 sf .21(An)A f0_12 sf .173(\(X)A f0_7 sf 0 3 rm .053(j)A 0 -3 rm f0_12 sf .149(\)\\{X)A f0_7 sf 0 3 rm .053(i)A 0 -3 rm f0_12 sf .16(,X)A f0_7 sf 0 3 rm .053(j)A 0 -3 rm f0_12 sf .411 .041(}\) =)J 151 278 :M .207(cov)A f1_7 sf 0 3 rm .131(q)A 0 -3 rm f0_7 sf 0 3 rm .098(\(0\))A 0 -3 rm f0_12 sf .227(\(X)A f0_7 sf 0 3 rm .07(i)A 0 -3 rm f0_12 sf .209(,X)A f0_7 sf 0 3 rm .07(j)A 0 -3 rm f0_12 sf .416 .042(\) - cov)J f1_7 sf 0 3 rm .131(q)A 0 -3 rm f0_7 sf 0 3 rm .098(\(0\))A 0 -3 rm f0_12 sf .227(\(X)A f0_7 sf 0 3 rm .07(i)A 0 -3 rm f2_12 sf .547 .055(, An)J f0_12 sf .227(\(X)A f0_7 sf 0 3 rm .07(i)A 0 -3 rm f0_12 sf .209 .021(\) )J f1_12 sf .331A f0_12 sf .098 .01( )J f2_12 sf .275(An)A f0_12 sf .227(\(X)A f0_7 sf 0 3 rm .07(j)A 0 -3 rm f0_12 sf .195(\)\\{X)A f0_7 sf 0 3 rm .07(i)A 0 -3 rm f0_12 sf .209(,X)A f0_7 sf 0 3 rm .07(j)A 0 -3 rm f0_12 sf .352 .035(}\) )J f1_12 sf S 195 301 :M f0_12 sf .128(var)A f1_7 sf 0 3 rm .092(q)A 0 -3 rm f0_7 sf 0 3 rm .068(\(0\))A 0 -3 rm 0 -5 rm .073(-1)A 0 5 rm f0_12 sf .1<28>A f2_12 sf .192(An)A f0_12 sf .159(\(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .146 .015(\) )J f1_12 sf .231A f0_12 sf .068 .007( )J f2_12 sf .192(An)A f0_12 sf .159(\(X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .136(\)\\{X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .146(,X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .246 .025(}\) )J f1_12 sf S 192 324 :M f0_12 sf .234(cov)A f1_7 sf 0 3 rm .148(q)A 0 -3 rm f0_7 sf 0 3 rm .11(\(0\))A 0 -3 rm f0_12 sf .257(\(X)A f0_7 sf 0 3 rm .079(j)A 0 -3 rm f2_12 sf .618 .062(, An)J f0_12 sf .257(\(X)A f0_7 sf 0 3 rm .079(i)A 0 -3 rm f0_12 sf .236 .024(\) )J f1_12 sf .374A f0_12 sf .111 .011( )J f2_12 sf .311(An)A f0_12 sf .257(\(X)A f0_7 sf 0 3 rm .079(j)A 0 -3 rm f0_12 sf .221(\)\\{X)A f0_7 sf 0 3 rm .079(i)A 0 -3 rm f0_12 sf .237(,X)A f0_7 sf 0 3 rm .079(j)A 0 -3 rm f0_12 sf .396(}\))A 77 351 :M -.219(The )A 104 351 :M -.14(formula )A 150 351 :M (for )S 173 351 :M .139(cov)A f1_7 sf 0 3 rm .088(q)A 0 -3 rm f0_7 sf 0 3 rm .066(\(0\))A 0 -3 rm f0_12 sf .153(\(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .141(,X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .058(|)A f2_12 sf .185(An)A f0_12 sf .153(\(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .153 .015(\) )J 278 351 :M f1_12 sf -.161A f0_12 sf ( )S 296 351 :M f2_12 sf .288(An)A f0_12 sf .238(\(X)A f0_7 sf 0 3 rm .073(j)A 0 -3 rm f0_12 sf .205(\)\\{X)A f0_7 sf 0 3 rm .073(i)A 0 -3 rm f0_12 sf .219(,X)A f0_7 sf 0 3 rm .073(j)A 0 -3 rm f0_12 sf .4 .04(}\) )J 385 351 :M (is )S 403 351 :M -.22(the )A 427 351 :M -.163(same )A 461 351 :M -.262(except)A 59 372 :M -.057(everywhere )A f1_12 sf -.067(q)A f0_12 sf -.048(\(0\) occurs it is replaced by )A f1_12 sf -.067(q)A f0_12 sf -.054(\(c\). We have just shown that:)A 94 395 :M .47 .047( cov)J f1_7 sf 0 3 rm .118(q)A 0 -3 rm f0_7 sf 0 3 rm .088(\(0\))A 0 -3 rm f0_12 sf .205(\(X)A f0_7 sf 0 3 rm .063(i)A 0 -3 rm f0_12 sf .097(,)A f2_12 sf .457 .046( An)J f0_12 sf .205(\(X)A f0_7 sf 0 3 rm .063(i)A 0 -3 rm f0_12 sf .189 .019(\) )J f1_12 sf .298A f0_12 sf .088 .009( )J f2_12 sf .248(An)A f0_12 sf .205(\(X)A f0_7 sf 0 3 rm .063(j)A 0 -3 rm f0_12 sf .176(\)\\{X)A f0_7 sf 0 3 rm .063(i)A 0 -3 rm f0_12 sf .189(,X)A f0_7 sf 0 3 rm .063(j)A 0 -3 rm f0_12 sf .461 .046(}\) = cov)J f1_7 sf 0 3 rm .118(q)A 0 -3 rm f0_7 sf 0 3 rm .084(\(c\))A 0 -3 rm f0_12 sf .205(\(X)A f0_7 sf 0 3 rm .063(i)A 0 -3 rm f0_12 sf .097(,)A f2_12 sf .457 .046( An)J f0_12 sf .205(\(X)A f0_7 sf 0 3 rm .063(i)A 0 -3 rm f0_12 sf .189 .019(\) )J f1_12 sf .298A f0_12 sf .088 .009( )J f2_12 sf .248(An)A f0_12 sf .205(\(X)A f0_7 sf 0 3 rm .063(j)A 0 -3 rm f0_12 sf .176(\)\\{X)A f0_7 sf 0 3 rm .063(i)A 0 -3 rm f0_12 sf .189(,X)A f0_7 sf 0 3 rm .063(j)A 0 -3 rm f0_12 sf .316(}\))A 109 418 :M .286 .029( var)J f1_7 sf 0 3 rm .08(q)A 0 -3 rm f0_7 sf 0 3 rm .059(\(0\))A 0 -3 rm 0 -5 rm .064(-1)A 0 5 rm f0_12 sf .087<28>A f2_12 sf .168(An)A f0_12 sf .138(\(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .127 .013(\) )J f1_12 sf .201A f0_12 sf .06 .006( )J f2_12 sf .168(An)A f0_12 sf .138(\(X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .119(\)\\{X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .127(,X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .295 .03(}\) = var)J f1_7 sf 0 3 rm .08(q)A 0 -3 rm f0_7 sf 0 3 rm .057(\(c\))A 0 -3 rm 0 -5 rm .064(-1)A 0 5 rm f0_12 sf .087<28>A f2_12 sf .168(An)A f0_12 sf .138(\(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .127 .013(\) )J f1_12 sf .201A f0_12 sf .06 .006( )J f2_12 sf .168(An)A f0_12 sf .138(\(X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .119(\)\\{X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .127(,X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .213(}\))A 93 441 :M .549 .055( cov)J f1_7 sf 0 3 rm .138(q)A 0 -3 rm f0_7 sf 0 3 rm .103(\(0\))A 0 -3 rm f0_12 sf .239(\(X)A f0_7 sf 0 3 rm .074(j)A 0 -3 rm f2_12 sf .577 .058(, An)J f0_12 sf .239(\(X)A f0_7 sf 0 3 rm .074(i)A 0 -3 rm f0_12 sf .22 .022(\) )J f1_12 sf .349A f0_12 sf .103 .01( )J f2_12 sf .29(An)A f0_12 sf .239(\(X)A f0_7 sf 0 3 rm .074(j)A 0 -3 rm f0_12 sf .206(\)\\{X)A f0_7 sf 0 3 rm .074(i)A 0 -3 rm f0_12 sf .221(,X)A f0_7 sf 0 3 rm .074(j)A 0 -3 rm f0_12 sf .538 .054(}\) = cov)J f1_7 sf 0 3 rm .138(q)A 0 -3 rm f0_7 sf 0 3 rm .098(\(c\))A 0 -3 rm f0_12 sf .239(\(X)A f0_7 sf 0 3 rm .074(j)A 0 -3 rm f2_12 sf .577 .058(, An)J f0_12 sf .239(\(X)A f0_7 sf 0 3 rm .074(i)A 0 -3 rm f0_12 sf .22 .022(\) )J f1_12 sf .349A f0_12 sf .103 .01( )J f2_12 sf .29(An)A f0_12 sf .239(\(X)A f0_7 sf 0 3 rm .074(j)A 0 -3 rm f0_12 sf .206(\)\\{X)A f0_7 sf 0 3 rm .074(i)A 0 -3 rm f0_12 sf .221(,X)A f0_7 sf 0 3 rm .074(j)A 0 -3 rm f0_12 sf .369(}\))A 206 464 :M .108(cov)A f1_7 sf 0 3 rm .068(q)A 0 -3 rm f0_7 sf 0 3 rm .051(\(0\))A 0 -3 rm f0_12 sf .118(\(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .109(,X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .236 .024(\) = cov)J f1_7 sf 0 3 rm .068(q)A 0 -3 rm f0_7 sf 0 3 rm .048(\(c\))A 0 -3 rm f0_12 sf .118(\(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .109(,X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .172 .017(\) + c)J 77 491 :M .072(Hence,)A 59 524 :M .153(cov)A f1_7 sf 0 3 rm .096(q)A 0 -3 rm f0_7 sf 0 3 rm .068(\(c\))A 0 -3 rm f0_12 sf .167(\(X)A f0_7 sf 0 3 rm .051(i)A 0 -3 rm f0_12 sf .154(,X)A f0_7 sf 0 3 rm .051(j)A 0 -3 rm f0_12 sf .063(|)A f2_12 sf .203(An)A f0_12 sf .167(\(X)A f0_7 sf 0 3 rm .051(i)A 0 -3 rm f0_12 sf .154 .015(\) )J f1_12 sf .243A f0_12 sf .072 .007( )J f2_12 sf .203(An)A f0_12 sf .167(\(X)A f0_7 sf 0 3 rm .051(j)A 0 -3 rm f0_12 sf .144(\)\\{X)A f0_7 sf 0 3 rm .051(i)A 0 -3 rm f0_12 sf .154(,X)A f0_7 sf 0 3 rm .051(j)A 0 -3 rm f0_12 sf .29 .029(}\) = cov)J f1_7 sf 0 3 rm .096(q)A 0 -3 rm f0_7 sf 0 3 rm .072(\(0\))A 0 -3 rm f0_12 sf .167(\(X)A f0_7 sf 0 3 rm .051(i)A 0 -3 rm f0_12 sf .154(,X)A f0_7 sf 0 3 rm .051(j)A 0 -3 rm f0_12 sf .063(|)A f2_12 sf .203(An)A f0_12 sf .167(\(X)A f0_7 sf 0 3 rm .051(i)A 0 -3 rm f0_12 sf .154 .015(\) )J f1_12 sf .243A f0_12 sf .072 .007( )J f2_12 sf .203(An)A f0_12 sf .167(\(X)A f0_7 sf 0 3 rm .051(j)A 0 -3 rm f0_12 sf .144(\)\\{X)A f0_7 sf 0 3 rm .051(i)A 0 -3 rm f0_12 sf .154(,X)A f0_7 sf 0 3 rm .051(j)A 0 -3 rm f0_12 sf .313 .031(}\) + c.)J 59 545 :M f3_12 sf <5C>S 59 578 :M f2_12 sf 1.973 .197(Theorem )J 113 578 :M 1.672 .167(4: )J 129 578 :M f0_12 sf (If )S 141 578 :M .354(G)A f0_7 sf 0 3 rm .191(C)A 0 -3 rm f0_12 sf .123 .012( )J 159 578 :M (is )S 172 578 :M -.326(a )A 182 578 :M -.248(complete )A 229 578 :M -.331(MAG )A 261 578 :M -.08(over )A 287 578 :M -.326(a )A 297 578 :M -.109(set )A 315 578 :M (of )S 330 578 :M -.145(variables )A 377 578 :M f2_12 sf 1.381(X)A f0_12 sf .869 .087(, )J 396 578 :M -.109(and )A 418 578 :M f1_12 sf .632(S)A f0_12 sf .267 .027( )J 431 578 :M (is )S 444 578 :M -.326(a )A 454 578 :M -.142(positive)A 59 599 :M -.206(definite )A 98 599 :M -.196(covariance )A 152 599 :M -.22(matrix )A 186 599 :M (for )S 204 599 :M f2_12 sf 1.381(X)A f0_12 sf .869 .087(, )J 222 599 :M -.165(then )A 246 599 :M -.196(there )A 273 599 :M (is )S 285 599 :M -.326(a )A 294 599 :M -.219(linear )A 324 599 :M -.226(parameterization )A 405 599 :M f1_12 sf .506(q)A f0_12 sf .242 .024( )J 416 599 :M (of )S 430 599 :M .354(G)A f0_7 sf 0 3 rm .191(C)A 0 -3 rm f0_12 sf .123 .012( )J 448 599 :M (such )S 474 599 :M -.331(that)A 59 618 :M f1_12 sf .444(S)A f0_7 sf 0 3 rm .316(G)A 0 -3 rm 0 6 rm .292(C)A 0 -6 rm 0 3 rm .146<28>A 0 -3 rm f1_7 sf 0 3 rm .228(q)A 0 -3 rm f0_7 sf 0 3 rm .146<29>A 0 -3 rm f0_12 sf .347 .035( = )J f1_12 sf .444(S)A f0_12 sf (.)S 77 647 :M .338 .034(Proof. Let )J f1_12 sf .144(S)A f0_12 sf .061 .006( )J 141 647 :M -.163(be )A 156 647 :M -.22(the )A 174 647 :M -.196(covariance )A 228 647 :M -.22(matrix )A 262 647 :M (for )S 280 647 :M f2_12 sf 1.381(X)A f0_12 sf .869 .087(. )J 298 647 :M -.331(An )A 316 647 :M -.179(instantiation )A 377 647 :M (of )S 391 647 :M -.326(a )A 400 647 :M -.226(parameterization )A 481 647 :M (of)S 59 668 :M .527(G)A f0_7 sf 0 3 rm .355 .036(C )J 0 -3 rm 76 668 :M f0_12 sf (has )S 96 668 :M -.22(the )A 114 668 :M -.097(properties )A 165 668 :M -.249(that )A 186 668 :M -.245(each )A 211 668 :M -.205(variable )A 252 668 :M -.217(can )A 272 668 :M -.163(be )A 287 668 :M -.034(expressed )A 338 668 :M (as )S 352 668 :M -.219(linear )A 382 668 :M -.123(function )A 426 668 :M (of )S 441 668 :M -.112(its )A 457 668 :M -.108(parents)A endp %%Page: 11 11 %%BeginPageSetup initializepage (peter; page: 11 of 18)setjob %%EndPageSetup gS 0 0 552 730 rC 461 698 30 27 rC 479 721 :M f0_12 sf (11)S gR gS 0 0 552 730 rC 59 58 :M f0_12 sf -.069(and an error term, and that if cov\()A f1_12 sf -.079(e)A f0_10 sf 0 2 rm -.075(p)A 0 -2 rm f0_12 sf (,)S f1_12 sf -.079(e)A f0_10 sf 0 2 rm -.075(q)A 0 -2 rm f0_12 sf -.083(\) )A cF f1_12 sf -.083A sf -.083( )A 259 58 :M (0 )S 269 58 :M -.165(then )A 293 58 :M .175(X)A f0_10 sf 0 2 rm .101(p)A 0 -2 rm f0_12 sf .061 .006( )J 311 58 :M f1_12 sf .451 .045J 328 58 :M f0_12 sf -.345(X)A f0_10 sf 0 2 rm -.199(q)A 0 -2 rm f0_12 sf ( )S 345 58 :M -.167(in )A 358 58 :M .147(G)A f0_7 sf 0 3 rm .079(C)A 0 -3 rm f0_12 sf .098 .01(; )J 379 58 :M (we )S 397 58 :M -.166(will )A 419 58 :M .259 .026(now )J 444 58 :M .479 .048(show )J 474 58 :M -.331(that)A 59 79 :M -.105(there is a parameterization of G)A f0_7 sf 0 3 rm -.103(C)A 0 -3 rm f0_12 sf -.105( that has covariance matrix )A f1_12 sf -.157(S)A f0_12 sf (.)S 77 103 :M (Note that since G)S f0_7 sf 0 3 rm (C)S 0 -3 rm f3_12 sf ( )S f0_12 sf -.003(is a complete ancestral graph )A f2_12 sf (Parents)S f0_12 sf (\(X)S f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf (\) )S f1_12 sf S f0_12 sf ({X)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf (|j < k}, )S 438 103 :M -.109(and )A 459 103 :M -.107(further)A 59 121 :M .483 .048(if X)J f0_7 sf 0 3 rm .069(i)A 0 -3 rm f0_12 sf .107A f1_12 sf .305A f0_12 sf .477 .048( {X)J f0_7 sf 0 3 rm .069(j)A 0 -3 rm f0_12 sf .427 .043(|j < k}\\)J f2_12 sf .2(Parents)A f0_12 sf .225(\(X)A f0_10 sf 0 2 rm .178(k)A 0 -2 rm f0_12 sf .5 .05(\) then X)J f0_10 sf 0 2 rm .099(i)A 0 -2 rm f0_12 sf .097 .01( )J f1_12 sf .445A f0_12 sf .346 .035( X)J f0_10 sf 0 2 rm .099(j)A 0 -2 rm f0_12 sf .342 .034( in G)J f0_7 sf 0 3 rm .166(C)A 0 -3 rm f0_12 sf .194 .019(. )J 318 121 :M -.326(We )A 338 121 :M -.166(will )A 360 121 :M -.197(abbreviate )A 412 121 :M f2_12 sf .318(Parents)A f0_12 sf .358(\(X)A f0_7 sf 0 3 rm .198(k)A 0 -3 rm f0_12 sf .36 .036(\) )J 479 121 :M (by)S 59 140 :M f2_12 sf (P)S f2_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf .065 .006(. Take each variable X)J f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf .057 .006( in turn. Regress X)J f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf .028 .003( on )J f2_12 sf (P)S f2_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf .058 .006(. Let)J 234 162 :M ( )S 237 149 76 29 rC 313 178 :M psb currentpoint pse 237 149 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2432 div 928 3 -1 roll exch div scale currentpoint translate 64 50 translate 69 258 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (\303) show -4 366 moveto 384 /Times-Roman f1 (X) show 299 462 moveto 224 ns (k) show 551 366 moveto 384 /Symbol f1 (=) show 1485 366 moveto 384 /Symbol f1 (a) show 1755 462 moveto 224 /Times-Roman f1 (kj) show 1942 366 moveto 384 ns (X) show 2257 462 moveto 224 ns (j) show 869 739 moveto 224 /Times-Roman f1 ( ) show 925 739 moveto 224 /Times-Roman f1 (X) show 1123 796 moveto 160 /Times-Roman f1 (j) show 1212 739 moveto 192 /Symbol f1 (\316) show 1335 739 moveto 224 /Times-Bold f1 (P) show 1499 796 moveto 160 /Times-Roman f1 (k) show 1036 453 moveto 576 /Symbol f1 (\345) show end pse gR gS 0 0 552 730 rC 313 162 :M f0_12 sf -.109( and)A 59 199 :M -.164(be the )A 91 199 :M -.219(linear )A 121 199 :M -.145(predictor )A 167 199 :M (of )S 181 199 :M .478(X)A f0_7 sf 0 3 rm .193(k)A 0 -3 rm f0_12 sf .166 .017( )J 198 199 :M (on )S 214 199 :M f2_12 sf .401(P)A f2_7 sf 0 3 rm .213(k)A 0 -3 rm f0_12 sf .164 .016( )J 230 199 :M -.05(\(where )A 267 199 :M -.148(summation )A 323 199 :M -.08(over )A 348 199 :M -.163(an )A 363 199 :M -.199(empty )A 396 199 :M -.109(set )A 413 199 :M (is )S 425 199 :M -.197(equal )A 454 199 :M -.167(to )A 467 199 :M -.16(zero\))A 59 217 :M -.102(and the residuals)A 241 227 67 17 rC 308 244 :M psb currentpoint pse 241 227 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2144 div 544 3 -1 roll exch div scale currentpoint translate 64 50 translate -8 366 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Symbol f1 (e) show 187 462 moveto 224 /Times-Roman f1 (k) show 332 366 moveto 384 /Times-Roman f1 (:) show 467 366 moveto 384 /Symbol f1 (=) show 781 366 moveto 384 /Times-Roman f1 (X) show 1084 462 moveto 224 ns (k) show 1320 366 moveto 384 /Symbol f1 (-) show 1690 258 moveto 384 /Times-Roman f1 (\303) show 1617 366 moveto 384 /Times-Roman f1 (X) show 1920 462 moveto 224 ns (k) show end pse gR gS 0 0 552 730 rC 77 264 :M f0_12 sf -.326(We )A 98 264 :M -.166(will )A 121 264 :M .259 .026(now )J 147 264 :M .479 .048(show )J 178 264 :M -.249(that )A 200 264 :M -.22(the )A 220 264 :M f1_12 sf .466(a)A f0_7 sf 0 3 rm .167(kj)A 0 -3 rm f0_12 sf .184 .018( )J 240 264 :M -.109(and )A 263 264 :M -.22(the )A 283 264 :M -.136(correlations )A 344 264 :M -.139(between )A 389 264 :M -.22(the )A 409 264 :M -.072(residuals )A 457 264 :M -.08(form )A 486 264 :M (a)S 59 283 :M -.226(parameterization )A 140 283 :M ( )S 144 283 :M (of )S 158 283 :M -.22(the )A 176 283 :M -.248(complete )A 222 283 :M -.331(MAG )A 253 283 :M .747(G)A f0_7 sf 0 3 rm .403(C)A 0 -3 rm f0_12 sf .471 .047(. )J 275 283 :M (First )S 302 283 :M -.165(note )A 327 283 :M -.249(that )A 349 283 :M .478(X)A f0_7 sf 0 3 rm .193(k)A 0 -3 rm f0_12 sf .166 .017( )J 367 283 :M (is )S 380 283 :M -.326(a )A 390 283 :M -.219(linear )A 421 283 :M -.123(function )A 465 283 :M (of )S 480 283 :M -.168(its)A 59 301 :M -.06(parents in G)A f0_7 sf 0 3 rm -.057(C)A 0 -3 rm f0_12 sf -.072( because)A 207 310 153 29 rC 360 339 :M psb currentpoint pse 207 310 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4896 div 928 3 -1 roll exch div scale currentpoint translate 64 50 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (X) -4 366 sh (=) 544 366 sh (X) 857 366 sh (+) 1389 366 sh (=) 2122 366 sh (X) 3477 366 sh ( ) 3871 366 sh (+) 4052 366 sh ( ) 4353 366 sh 224 ns (k) 299 462 sh (k) 1160 462 sh (k) 1877 462 sh (kj) 3290 462 sh (j) 3792 462 sh (k) 4644 462 sh (X) 2437 739 sh 160 ns (j) 2635 796 sh /mt_vec StandardEncoding 256 array copy def /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 mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\303) 930 258 sh 384 /Symbol f1 (e) 1682 366 sh (a) 3020 366 sh (e) 4449 366 sh 224 /Symbol f1 (\316) 2722 739 sh 576 ns (\345) 2571 453 sh 224 /Times-Bold f1 (P) 2862 739 sh 160 ns (k) 2995 795 sh end MTsave restore pse gR gS 0 0 552 730 rC 77 362 :M f0_12 sf .398 .04(Second, )J 120 362 :M (we )S 138 362 :M -.166(will )A 160 362 :M .479 .048(show )J 190 362 :M -.249(that )A 212 362 :M -.164(if )A 224 362 :M .056(Cov\()A f1_12 sf (e)S f0_6 sf 0 2 rm (p)S 0 -2 rm f0_12 sf (,)S f1_12 sf (e)S f0_6 sf 0 2 rm (q)S 0 -2 rm f0_12 sf .06 .006(\) )J 277 362 :M cF f1_12 sf .038A sf .376 .038( )J 289 362 :M (0 )S 300 362 :M -.165(then )A 325 362 :M .199(X)A f0_6 sf 0 2 rm .069(p)A 0 -2 rm f0_12 sf .069 .007( )J 342 362 :M f1_12 sf .451 .045J 360 362 :M f0_12 sf .199(X)A f0_6 sf 0 2 rm .069(q)A 0 -2 rm f0_12 sf .069 .007( )J 377 362 :M -.167(in )A 391 362 :M .747(G)A f0_7 sf 0 3 rm .403(C)A 0 -3 rm f0_12 sf .471 .047(. )J 414 362 :M .197 .02(Suppose )J 460 362 :M (on )S 477 362 :M -.33(the)A 59 383 :M -.044(contrary that cov\()A f1_12 sf (e)S f0_6 sf 0 2 rm (p)S 0 -2 rm f0_12 sf -.028(, )A f1_12 sf (e)S f0_6 sf 0 2 rm (q)S 0 -2 rm f0_12 sf -.044(\) )A cF f1_12 sf -.044A sf -.044( 0, but there is no double headed arrow X)A f0_6 sf 0 2 rm (p)S 0 -2 rm f0_12 sf ( )S 383 383 :M f1_12 sf .451 .045J 400 383 :M f0_12 sf .199(X)A f0_6 sf 0 2 rm .069(q)A 0 -2 rm f0_12 sf .069 .007( )J 416 383 :M -.167(in )A 429 383 :M .747(G)A f0_7 sf 0 3 rm .403(C)A 0 -3 rm f0_12 sf .471 .047(. )J 451 383 :M -.326(We )A 471 383 :M -.33(may)A 59 404 :M -.057(suppose without loss of generality that p < q. Since there is no double )A 393 404 :M -.163(headed )A 430 404 :M (arrow )S 462 404 :M .199(X)A f0_6 sf 0 2 rm .069(p)A 0 -2 rm f0_12 sf .069 .007( )J 478 404 :M f1_12 sf S 59 425 :M f0_12 sf .056(X)A f0_6 sf 0 2 rm (q)S 0 -2 rm f0_12 sf .079 .008(, and p < q it follows that X)J f0_6 sf 0 2 rm (p)S 0 -2 rm f1_12 sf .05 .005<20AE20>J f0_12 sf .056(X)A f0_6 sf 0 2 rm (q)S 0 -2 rm f0_12 sf .062 .006( in G)J f0_7 sf 0 3 rm (C)S 0 -3 rm f0_12 sf .095 .01(. It then follows that X)J f0_6 sf 0 2 rm (p)S 0 -2 rm f0_12 sf ( )S f1_12 sf .055A f0_12 sf ( )S f2_12 sf (P)S f0_6 sf 0 2 rm (q)S 0 -2 rm f0_12 sf (.)S 127 435 314 28 rC 441 463 :M psb currentpoint pse 127 435 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 10048 div 896 3 -1 roll exch div scale currentpoint translate 64 38 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (cov\() -9 346 sh (,) 1022 346 sh (\)) 1506 346 sh (cov\() 2032 346 sh (,) 3063 346 sh (\)) 5036 346 sh (cov\() 5562 346 sh (,) 6593 346 sh (\)) 7189 346 sh (,) 9277 346 sh (\)) 9827 346 sh 384 /Symbol f1 (e) 683 346 sh (e) 1155 346 sh (e) 2724 346 sh (e) 6254 346 sh (e) 8938 346 sh 224 /Times-Roman f1 (q) 873 442 sh (p) 1350 442 sh (q) 2914 442 sh (p) 3503 442 sh (j) 4938 442 sh (X) 4031 719 sh (q) 6444 442 sh (p) 7033 442 sh (X) 7689 719 sh (q) 9128 442 sh (j) 9729 442 sh 384 ns (X) 3200 346 sh (X) 4623 346 sh (X) 6730 346 sh (cov\() 8276 346 sh (X) 9414 346 sh 160 ns (j) 4229 776 sh (j) 7887 776 sh 384 /Symbol f1 (=) 1723 346 sh (-) 3732 346 sh (=) 5253 346 sh (-) 7390 346 sh 224 ns (\316) 4316 719 sh (\316) 7974 719 sh 576 ns (\345) 4163 433 sh (\345) 7821 433 sh 224 /Times-Bold f1 (P) 4456 719 sh (P) 8114 719 sh 160 ns (p) 4588 775 sh (p) 8246 775 sh end MTsave restore pse gR gS 0 0 552 730 rC 77 483 :M f0_12 sf .051 .005(We will now show that cov\()J f1_12 sf (e)S f0_7 sf 0 3 rm (q)S 0 -3 rm f0_12 sf (,)S f1_12 sf (e)S f0_7 sf 0 3 rm (p)S 0 -3 rm f0_12 sf .042 .004(\) = 0 by showing that cov\()J f1_12 sf (e)S f0_7 sf 0 3 rm (q)S 0 -3 rm f0_12 sf (,X)S f0_7 sf 0 3 rm (p)S 0 -3 rm f0_12 sf .03 .003(\) = 0, and for all )J 467 483 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 482 483 :M -.334(in)A 59 501 :M f2_12 sf .35(P)A f2_7 sf 0 3 rm .186(p)A 0 -3 rm f0_12 sf .815 .082(, cov\()J f1_12 sf .251(e)A f0_6 sf 0 2 rm .143(q)A 0 -2 rm f0_12 sf .278(,X)A f0_6 sf 0 2 rm .08(j)A 0 -2 rm f0_12 sf .492 .049(\) = 0.)J 77 525 :M -.067(By construction, )A f1_12 sf -.073(e)A f0_6 sf 0 2 rm (q)S 0 -2 rm f0_12 sf -.067( is uncorrelated with X)A f0_6 sf 0 2 rm (p)S 0 -2 rm f0_12 sf ( )S f1_12 sf -.119A f0_12 sf ( )S f2_12 sf -.102(P)A f0_6 sf 0 2 rm (q)S 0 -2 rm f0_12 sf -.058(, \(since )A f1_12 sf -.073(e)A f0_6 sf 0 2 rm (q)S 0 -2 rm f0_9 sf 0 2 rm ( )S 0 -2 rm f0_12 sf -.067(is the residual remaining after)A 59 543 :M .819 .082(regressing X)J f0_6 sf 0 2 rm .089(q)A 0 -2 rm f0_10 sf 0 2 rm .067 .007( )J 0 -2 rm f0_12 sf .341 .034(on )J f2_12 sf .217(P)A f0_6 sf 0 2 rm .089(q)A 0 -2 rm f0_12 sf .443 .044(\), so cov\()J f1_12 sf .156(e)A f0_7 sf 0 3 rm .103(q)A 0 -3 rm f0_12 sf .172(,X)A f0_7 sf 0 3 rm .103(p)A 0 -3 rm f0_12 sf .28 .028(\) = 0. If X)J f0_6 sf 0 2 rm (j)S 0 -2 rm f0_12 sf .081 .008( )J f1_12 sf .253A f0_12 sf .089 .009( )J 292 543 :M f2_12 sf .75(P)A f0_6 sf 0 2 rm .307(p)A 0 -2 rm f0_12 sf .558 .056(, )J 311 543 :M -.165(then )A 335 543 :M .435(X)A f0_6 sf 0 2 rm .084(j)A 0 -2 rm f1_12 sf .151 .015( )J 350 543 :M .144 .014J 366 543 :M f0_12 sf .199(X)A f0_6 sf 0 2 rm .069(p)A 0 -2 rm f0_12 sf .069 .007( )J 382 543 :M -.167(in )A 395 543 :M .747(G)A f0_7 sf 0 3 rm .403(C)A 0 -3 rm f0_12 sf .471 .047(. )J 417 543 :M -.131(Since )A 447 543 :M .199(X)A f0_6 sf 0 2 rm .069(p)A 0 -2 rm f1_12 sf .069 .007( )J 463 543 :M .144 .014J 479 543 :M f0_12 sf .25(X)A f0_6 sf 0 2 rm (q)S 0 -2 rm 59 562 :M f0_12 sf -.027(in G)A f0_7 sf 0 3 rm (C)S 0 -3 rm f0_12 sf -.023(, it follows that X)A f0_6 sf 0 2 rm (j)S 0 -2 rm f0_12 sf -.025( is an ancestor )A 242 562 :M (of )S 256 562 :M .199(X)A f0_6 sf 0 2 rm .069(q)A 0 -2 rm f0_12 sf .069 .007( )J 272 562 :M -.167(in )A 285 562 :M .747(G)A f0_7 sf 0 3 rm .403(C)A 0 -3 rm f0_12 sf .471 .047(. )J 307 562 :M -.165(As )A 324 562 :M .354(G)A f0_7 sf 0 3 rm .191(C)A 0 -3 rm f3_12 sf .123 .012( )J 342 562 :M f0_12 sf (is )S 354 562 :M -.326(a )A 363 562 :M -.248(complete )A 409 562 :M -.181(ancestral )A 454 562 :M -.064(graph )A 485 562 :M -.668(it)A 59 579 :M .224 .022(then follows that X)J f0_6 sf 0 2 rm (j)S 0 -2 rm f1_12 sf .147 .015<20AE>J f0_12 sf .103(X)A f0_6 sf 0 2 rm (q)S 0 -2 rm f0_12 sf .114 .011( in G)J f0_7 sf 0 3 rm .055(C)A 0 -3 rm f0_12 sf .129 .013(, so X)J f0_6 sf 0 2 rm (j)S 0 -2 rm f0_12 sf ( )S f1_12 sf .125 .012J 256 579 :M f2_12 sf .75(P)A f0_6 sf 0 2 rm .307(q)A 0 -2 rm f0_12 sf .558 .056(. )J 275 579 :M -.128(Hence )A 309 579 :M .116(cov\()A f1_12 sf .115(e)A f0_6 sf 0 2 rm .066(q)A 0 -2 rm f0_12 sf .127(,X)A f0_6 sf 0 2 rm (j)S 0 -2 rm f0_12 sf .139 .014(\) )J 361 579 :M .211 .021(= )J 372 579 :M .833 .083(0, )J 386 579 :M (as )S 400 579 :M -.123(claimed. )A 444 579 :M -.164(It )A 455 579 :M (follows)S 59 597 :M .35 .035(that cov\()J f1_12 sf .083(e)A f0_7 sf 0 3 rm .055(q)A 0 -3 rm f0_12 sf (,)S f1_12 sf .083(e)A f0_7 sf 0 3 rm .055(p)A 0 -3 rm f0_12 sf .162 .016(\) = 0.)J 77 624 :M -.048(Finally, positive definiteness of )A f1_12 sf -.075(S)A f0_12 sf -.051( ensures that )A 300 624 :M -.245(each )A 325 624 :M f1_12 sf .104(e)A f0_7 sf 0 3 rm .069(k)A 0 -3 rm f0_12 sf .059 .006( )J 338 624 :M (has )S 358 624 :M -.124(positive )A 399 624 :M -.218(variance; )A 445 624 :M -.08(otherwise)A 59 645 :M -.059(X)A f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf -.034(would be a linear combination of previous X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.03('s and )A f1_12 sf (S)S f0_12 sf -.032( would not be positive definite. )A f1_12 sf <5C>S endp %%Page: 12 12 %%BeginPageSetup initializepage (peter; page: 12 of 18)setjob %%EndPageSetup gS 0 0 552 730 rC 461 698 30 27 rC 479 721 :M f0_12 sf (12)S gR gS 0 0 552 730 rC 59 58 :M f2_12 sf 1.122 .112(Lemma )J 104 58 :M .387(5:)A f0_12 sf .232 .023( )J 119 58 :M (In )S 133 58 :M -.326(a )A 142 58 :M -.331(MAG )A 173 58 :M 1.114 .111(G, )J 190 58 :M -.164(if )A 201 58 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 216 58 :M f1_12 sf .33A f0_12 sf .116 .012( )J 229 58 :M f2_12 sf .398(An)A f0_12 sf .328(\(X)A f0_7 sf 0 3 rm .101(i)A 0 -3 rm f0_12 sf .432 .043(\), )J 272 58 :M -.109(and )A 293 58 :M -.196(there )A 320 58 :M (is )S 332 58 :M (no )S 348 58 :M -.163(edge )A 374 58 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 390 58 :M f1_12 sf .126A f0_12 sf ( )S 407 58 :M .726(X)A f0_7 sf 0 3 rm .163(i)A 0 -3 rm f0_12 sf .457 .046(, )J 427 58 :M -.165(then )A 452 58 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 468 58 :M (is )S 481 58 :M (d-)S 59 79 :M .252 .025(separated from X)J f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .132 .013( given )J f2_12 sf .084(An)A f0_12 sf .069(\(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .059(\)\\{X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .064(,X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .096(}.)A 77 106 :M -.044(Proof. Suppose, on the contrary that there is a path U that )A 354 106 :M -.097(d-connects )A 409 106 :M -.082(some )A 438 106 :M -.219(member )A 480 106 :M .32(X)A f0_7 sf 0 3 rm (j)S 0 -3 rm 59 127 :M f1_12 sf S f0_12 sf ( )S f2_12 sf .044(An)A f0_12 sf .036(\(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .062 .006(\) to X)J f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .07 .007( given )J f2_12 sf .044(An)A f0_12 sf .036(\(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .031(\)\\{X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .034(,X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .091 .009(}. There are three cases: )J 352 127 :M -.219(either )A 382 127 :M -.196(there )A 409 127 :M (is )S 421 127 :M -.163(an )A 436 127 :M -.163(edge )A 462 127 :M .478(X)A f0_7 sf 0 3 rm .193(k)A 0 -3 rm f0_12 sf .166 .017( )J 479 127 :M f1_12 sf S 59 148 :M f0_12 sf .088(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .127 .013( on U, there is an edge X)J f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S f1_12 sf .12A f0_12 sf .098 .01( X)J f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf .124 .012( on U, or there is an edge X)J f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S f1_12 sf .126A f0_12 sf .098 .01( X)J f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf .12 .012( on U.)J 77 175 :M -.068(Suppose there is an edge )A 198 175 :M .478(X)A f0_7 sf 0 3 rm .193(k)A 0 -3 rm f0_12 sf .166 .017( )J 215 175 :M f1_12 sf .126A f0_12 sf ( )S 231 175 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 246 175 :M (on )S 262 175 :M 1.114 .111(U. )J 279 175 :M .478(X)A f0_7 sf 0 3 rm .193(k)A 0 -3 rm f0_12 sf .166 .017( )J 296 175 :M f1_12 sf 1.081A f0_12 sf .957A f0_7 sf 0 3 rm .319(j)A 0 -3 rm f0_12 sf .492 .049( )J 324 175 :M -.139(because )A 365 175 :M -.072(otherwise )A 415 175 :M -.196(there )A 442 175 :M (is )S 454 175 :M -.163(an )A 469 175 :M -.217(edge)A 59 196 :M .111(X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf ( )S f1_12 sf .151A f0_12 sf .124 .012( X)J f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .171 .017( in G. Hence X)J f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf .094 .009( is in )J 203 196 :M f2_12 sf .388(An)A f0_12 sf .321(\(X)A f0_7 sf 0 3 rm .099(i)A 0 -3 rm f0_12 sf .275(\)\\{X)A f0_7 sf 0 3 rm .099(j)A 0 -3 rm f0_12 sf .295(,X)A f0_7 sf 0 3 rm .099(j)A 0 -3 rm f0_12 sf .496 .05(}. )J 287 196 :M -.113(But )A 308 196 :M -.165(then )A 332 196 :M .478(X)A f0_7 sf 0 3 rm .193(k)A 0 -3 rm f0_12 sf .166 .017( )J 349 196 :M (is )S 361 196 :M -.111(not )A 380 196 :M -.326(a )A 389 196 :M -.206(collider )A 428 196 :M (on )S 444 196 :M 1.114 .111(U, )J 461 196 :M -.109(and )A 482 196 :M (U)S 59 217 :M .185 .019(does not d-connect X)J f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .087 .009( to X)J f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .109 .011( given )J f2_12 sf .069(An)A f0_12 sf .057(\(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .049(\)\\{X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .053(,X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .079(}.)A 77 244 :M .197 .02(Suppose )J 122 244 :M -.249(that )A 143 244 :M -.22(the )A 161 244 :M -.064(first )A 184 244 :M -.163(edge )A 210 244 :M (on )S 226 244 :M .306 .031(U )J 239 244 :M (is )S 251 244 :M -.163(an )A 266 244 :M -.163(edge )A 292 244 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 307 244 :M f1_12 sf .126A f0_12 sf ( )S 323 244 :M .876(X)A f0_7 sf 0 3 rm .354(k)A 0 -3 rm f0_12 sf .552 .055(. )J 345 244 :M -.164(It )A 357 244 :M (follows )S 398 244 :M -.249(that )A 420 244 :M -.219(either )A 451 244 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 467 244 :M (is )S 480 244 :M -.326(an)A 59 265 :M .115 .012(ancestor of X)J f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf (, )S 133 265 :M (or )S 147 265 :M -.196(there )A 174 265 :M (is )S 186 265 :M -.326(a )A 195 265 :M -.206(collider )A 234 265 :M (on )S 250 265 :M 1.114 .111(U. )J 267 265 :M -.139(Because )A 310 265 :M .306 .031(G )J 323 265 :M (is )S 335 265 :M -.122(acyclic, )A 375 265 :M -.109(and )A 396 265 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 411 265 :M (is )S 423 265 :M -.163(an )A 438 265 :M -.122(ancestor )A 481 265 :M (of)S 59 286 :M -.06(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.034(, X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.032( is not an ancestor of X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.032(. Suppose then that there is a collider X)A f0_7 sf 0 3 rm (l)S 0 -3 rm f0_12 sf ( )S 392 286 :M (on )S 408 286 :M 1.114 .111(U. )J 425 286 :M -.139(Because )A 468 286 :M .306 .031(U )J 481 286 :M (d-)S 59 307 :M .181 .018(connects X)J f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .083 .008( and X)J f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .082 .008( given )J f2_12 sf .052(An)A f0_12 sf .043(\(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .037(\)\\{X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .039(,X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .099 .01(}, X)J f0_7 sf 0 3 rm (l)S 0 -3 rm f0_12 sf .088 .009( is an ancestor )J 345 307 :M (of )S 359 307 :M f2_12 sf .388(An)A f0_12 sf .321(\(X)A f0_7 sf 0 3 rm .099(i)A 0 -3 rm f0_12 sf .275(\)\\{X)A f0_7 sf 0 3 rm .099(j)A 0 -3 rm f0_12 sf .295(,X)A f0_7 sf 0 3 rm .099(j)A 0 -3 rm f0_12 sf .496 .05(}, )J 443 307 :M -.109(and )A 464 307 :M -.245(hence)A 59 328 :M -.046(of X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.041(. It follows that G is cyclic, contrary to our assumption that G is a MAG.)A 77 355 :M -.026(Suppose that the first edge on U is X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S f1_12 sf -.068A f0_12 sf ( )S 274 355 :M .876(X)A f0_7 sf 0 3 rm .354(k)A 0 -3 rm f0_12 sf .552 .055(. )J 295 355 :M (If )S 307 355 :M -.196(there )A 334 355 :M (is )S 346 355 :M -.326(a )A 355 355 :M -.206(collider )A 394 355 :M (on )S 410 355 :M 1.114 .111(U, )J 427 355 :M -.165(then )A 451 355 :M .478(X)A f0_7 sf 0 3 rm .193(k)A 0 -3 rm f0_12 sf .166 .017( )J 468 355 :M (is )S 480 355 :M -.326(an)A 59 376 :M -.122(ancestor )A 103 376 :M (of )S 119 376 :M -.22(the )A 139 376 :M -.072(collider. )A 184 376 :M -.219(The )A 208 376 :M -.206(collider )A 249 376 :M (is )S 263 376 :M -.163(an )A 280 376 :M -.122(ancestor )A 325 376 :M (of )S 341 376 :M f2_12 sf .303(An)A f0_12 sf .25(\(X)A f0_7 sf 0 3 rm .077(i)A 0 -3 rm f0_12 sf .215(\)\\{X)A f0_7 sf 0 3 rm .077(j)A 0 -3 rm f0_12 sf .23(,X)A f0_7 sf 0 3 rm .077(j)A 0 -3 rm f0_12 sf .314 .031(} )J 423 376 :M -.139(because )A 466 376 :M .306 .031(U )J 481 376 :M (d-)S 59 397 :M -.052(connects X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.052( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.053( given )A 181 397 :M f2_12 sf .388(An)A f0_12 sf .321(\(X)A f0_7 sf 0 3 rm .099(i)A 0 -3 rm f0_12 sf .275(\)\\{X)A f0_7 sf 0 3 rm .099(i)A 0 -3 rm f0_12 sf .295(,X)A f0_7 sf 0 3 rm .099(j)A 0 -3 rm f0_12 sf .496 .05(}. )J 265 397 :M -.128(Hence )A 299 397 :M .478(X)A f0_7 sf 0 3 rm .193(k)A 0 -3 rm f0_12 sf .166 .017( )J 316 397 :M (is )S 328 397 :M -.163(an )A 343 397 :M -.122(ancestor )A 386 397 :M (of )S 400 397 :M f2_12 sf .388(An)A f0_12 sf .321(\(X)A f0_7 sf 0 3 rm .099(i)A 0 -3 rm f0_12 sf .275(\)\\{X)A f0_7 sf 0 3 rm .099(i)A 0 -3 rm f0_12 sf .295(,X)A f0_7 sf 0 3 rm .099(j)A 0 -3 rm f0_12 sf .496 .05(}. )J 484 397 :M -.327(It)A 59 418 :M -.055(follows that X)A f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf -.053( is an ancestor of X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.054(, contrary to the assumption that )A 381 418 :M .306 .031(G )J 394 418 :M (is )S 406 418 :M -.326(a )A 415 418 :M (MAG. )S 450 418 :M .056(Suppose)A 59 439 :M -.033(then that there is no collider on U. Hence X)A f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf -.032( is an ancestor of X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.035(. But X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.034( is by )A 431 439 :M -.033(hypothesis )A 486 439 :M (a)S 59 460 :M (member of )S f2_12 sf (An)S f0_12 sf (\(X)S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf (\). It follows that X)S f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf ( is an )S 266 460 :M -.122(ancestor )A 309 460 :M (of )S 323 460 :M .726(X)A f0_7 sf 0 3 rm .163(i)A 0 -3 rm f0_12 sf .457 .046(, )J 342 460 :M -.122(contrary )A 385 460 :M -.167(to )A 398 460 :M -.22(the )A 416 460 :M -.066(assumption )A 474 460 :M -.331(that)A 59 481 :M -.024(G is a MAG. )A f3_12 sf <5C>S 77 508 :M f2_12 sf 1.122 .112(Lemma )J 122 508 :M .387(6:)A f0_12 sf .232 .023( )J 137 508 :M (In )S 151 508 :M -.326(a )A 161 508 :M -.331(MAG )A 193 508 :M 1.114 .111(G, )J 211 508 :M -.164(if )A 223 508 :M -.163(an )A 239 508 :M -.164(undirected )A 293 508 :M -.165(path )A 318 508 :M .306 .031(U )J 332 508 :M -.167(in )A 346 508 :M .306 .031(G )J 360 508 :M -.097(d-connects )A 416 508 :M -.663(A )A 429 508 :M -.109(and )A 451 508 :M (B )S 464 508 :M -.165(given)A 59 529 :M f2_12 sf -.054(An)A f0_12 sf -.034(\(A\) )A f1_12 sf -.064A f0_12 sf ( )S f2_12 sf -.054(An)A f0_12 sf -.036(\(B\)\\{A,B} then U is an inducing path between A and B.)A 77 556 :M f2_12 sf 3.969 .397(Proof. )J 119 556 :M f0_12 sf (If )S 131 556 :M -.196(there )A 158 556 :M (is )S 170 556 :M -.326(a )A 179 556 :M -.165(path )A 203 556 :M .306 .031(U )J 216 556 :M -.249(that )A 237 556 :M -.097(d-connects )A 293 556 :M -.663(A )A 306 556 :M -.109(and )A 328 556 :M (B )S 341 556 :M -.132(given )A 372 556 :M f2_12 sf (An)S f0_12 sf (\(A\) )S 409 556 :M f1_12 sf -.161A f0_12 sf ( )S 423 556 :M f2_12 sf .135(An)A f0_12 sf .111(\(B\)\\{A,B})A 59 577 :M -.089(then every collider on U is an ancestor of a member of )A f2_12 sf -.141(An)A f0_12 sf -.121(\(A\) )A 354 577 :M f1_12 sf -.161A f0_12 sf ( )S 367 577 :M f2_12 sf .23(An)A f0_12 sf .848 .085(\(B\)\\{A,B}, )J 442 577 :M -.109(and )A 463 577 :M -.245(hence)A 59 598 :M -.071(an ancestor of A or B. Every vertex on U is an ancestor a collider on U or an )A 425 598 :M -.122(ancestor )A 468 598 :M (of )S 482 598 :M (A)S 59 619 :M -.089(or B; and hence every vertex on U )A 225 619 :M -.219(except )A 259 619 :M (for )S 277 619 :M -.22(the )A 295 619 :M -.073(endpoints )A 345 619 :M (is )S 357 619 :M -.167(in )A 370 619 :M f2_12 sf (An)S f0_12 sf (\(A\) )S 406 619 :M f1_12 sf -.161A f0_12 sf ( )S 419 619 :M f2_12 sf .24(An)A f0_12 sf .186(\(B\)\\{A,B}.)A 59 640 :M (If )S 71 640 :M .306 .031(U )J 84 640 :M -.097(d-connects )A 139 640 :M -.663(A )A 151 640 :M -.109(and )A 172 640 :M (B )S 184 640 :M -.132(given )A 214 640 :M f2_12 sf (An)S f0_12 sf (\(A\) )S 250 640 :M f1_12 sf -.161A f0_12 sf ( )S 263 640 :M f2_12 sf .23(An)A f0_12 sf .848 .085(\(B\)\\{A,B}, )J 338 640 :M -.165(then )A 362 640 :M -.129(every )A 392 640 :M -.163(vertex )A 425 640 :M -.249(that )A 447 640 :M (is )S 460 640 :M (on )S 477 640 :M 1.337(U,)A 59 661 :M -.069(except for the endpoints, is a collider. Hence U is an inducing path between A and B. )A 465 652 9 9 rC gS 1.286 1 scale 361.669 661 :M f1_10 sf <5C>S gR gR endp %%Page: 13 13 %%BeginPageSetup initializepage (peter; page: 13 of 18)setjob %%EndPageSetup gS 0 0 552 730 rC 461 698 30 27 rC 479 721 :M f0_12 sf (13)S gR gS 0 0 552 730 rC 59 58 :M f2_12 sf .822 .082(Theorem 5:)J f0_12 sf .304 .03( If G is a MAG, and )J f1_12 sf .196(S)A f0_12 sf .168 .017( is )J 245 58 :M -.326(a )A 254 58 :M -.124(positive )A 295 58 :M -.206(definite )A 334 58 :M -.196(covariance )A 388 58 :M -.22(matrix )A 422 58 :M (such )S 448 58 :M -.249(that )A 469 58 :M -.164(if )A 480 58 :M .32(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm 59 79 :M f0_12 sf -.109(and )A 82 79 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 99 79 :M -.215(are )A 119 79 :M -.117(d-separated )A 179 79 :M -.132(given )A 211 79 :M -.33(Z )A 224 79 :M -.167(in )A 239 79 :M 1.114 .111(G, )J 258 79 :M -.165(then )A 284 79 :M .102(cov\(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .099(,X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf (|)S f2_12 sf .136(Z)A f0_12 sf .108 .011(\) )J 351 79 :M .211 .021(= )J 364 79 :M .833 .083(0, )J 381 79 :M -.165(then )A 408 79 :M -.196(there )A 438 79 :M (is )S 453 79 :M -.326(a )A 465 79 :M -.263(linear)A 59 100 :M -.019(parameterization )A f1_12 sf (q)S f0_12 sf -.018( of G such that )A f1_12 sf (S)S f0_7 sf 0 3 rm (G\()S 0 -3 rm f1_7 sf 0 3 rm (q)S 0 -3 rm f0_7 sf 0 3 rm <29>S 0 -3 rm f0_12 sf -.017( = )A f1_12 sf (S)S f0_12 sf (.)S 77 128 :M -.065(Proof. By Lemma 3, there is a complete MAG G)A f0_7 sf 0 3 rm -.057(C)A 0 -3 rm f0_12 sf -.058( such that G is a )A 393 128 :M (subgraph )S 441 128 :M (of )S 455 128 :M .747(G)A f0_7 sf 0 3 rm .403(C)A 0 -3 rm f0_12 sf .471 .047(. )J 477 128 :M (By)S 59 147 :M -.034(Theorem 4, there is a parameterization )A f1_12 sf (q)S f0_12 sf -.035( of G)A f0_7 sf 0 3 rm (C)S 0 -3 rm f0_12 sf -.032( such that )A f1_12 sf -.051(S)A f0_7 sf 0 3 rm (G)S 0 -3 rm 0 5 rm (C)S 0 -5 rm 0 3 rm <28>S 0 -3 rm f1_7 sf 0 3 rm (q)S 0 -3 rm f0_7 sf 0 3 rm <29>S 0 -3 rm f0_12 sf -.03( = )A f1_12 sf -.036(S.)A f0_12 sf -.039( We will now show that)A 59 169 :M f1_12 sf -.06(q)A f0_12 sf -.045( assigns zeroes to every edge that is in G)A f0_7 sf 0 3 rm (C)S 0 -3 rm f0_12 sf -.051( but )A 285 169 :M -.111(not )A 304 169 :M -.167(in )A 317 169 :M 1.114 .111(G. )J 334 169 :M (First )S 360 169 :M -.081(consider )A 404 169 :M -.326(a )A 413 169 :M -.205(directed )A 454 169 :M -.163(edge )A 480 169 :M .32(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm 59 190 :M f1_12 sf -.071A f0_12 sf -.035( X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.026( that is in G)A f0_7 sf 0 3 rm (C)S 0 -3 rm f0_12 sf -.029( but not in G. By the method of )A 298 190 :M -.109(construction )A 360 190 :M (of )S 374 190 :M .747(G)A f0_7 sf 0 3 rm .403(C)A 0 -3 rm f0_12 sf .471 .047(, )J 396 190 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 411 190 :M (is )S 423 190 :M -.163(an )A 438 190 :M -.122(ancestor )A 481 190 :M (of)S 59 211 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 74 211 :M -.167(in )A 87 211 :M 1.114 .111(G. )J 104 211 :M -.139(Because )A 147 211 :M -.22(the )A 165 211 :M -.163(edge )A 191 211 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 206 211 :M f1_12 sf .126A f0_12 sf ( )S 222 211 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 237 211 :M (does )S 263 211 :M -.111(not )A 282 211 :M -.132(exist )A 308 211 :M -.167(in )A 321 211 :M 1.114 .111(G, )J 338 211 :M (by )S 354 211 :M -.33(Lemma )A 393 211 :M .833 .083(5, )J 408 211 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 424 211 :M (is )S 437 211 :M -.129(d-separated)A 59 232 :M -.08(from )A 90 232 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 109 232 :M -.132(given )A 143 232 :M f2_12 sf .303(An)A f0_12 sf .25(\(X)A f0_7 sf 0 3 rm .077(j)A 0 -3 rm f0_12 sf .215(\)\\{X)A f0_7 sf 0 3 rm .077(i)A 0 -3 rm f0_12 sf .23(,X)A f0_7 sf 0 3 rm .077(j)A 0 -3 rm f0_12 sf .314 .031(} )J 227 232 :M -.167(in )A 245 232 :M 1.114 .111(G. )J 267 232 :M -.128(Hence )A 306 232 :M .172(cov)A f1_7 sf 0 3 rm .123(S)A 0 -3 rm f0_12 sf .188(\(X)A f0_7 sf 0 3 rm .058(i)A 0 -3 rm f0_12 sf .174(,X)A f0_7 sf 0 3 rm .058(j)A 0 -3 rm f0_12 sf .071(|)A f2_12 sf .228(An)A f0_12 sf .188(\(X)A f0_7 sf 0 3 rm .058(j)A 0 -3 rm f0_12 sf .162(\)\\{X)A f0_7 sf 0 3 rm .058(i)A 0 -3 rm f0_12 sf .174(,X)A f0_7 sf 0 3 rm .058(j)A 0 -3 rm f0_12 sf .316 .032(}\) )J 448 232 :M .211 .021(= )J 464 232 :M (0 )S 479 232 :M (by)S 59 254 :M .067(hypothesis.)A 77 280 :M -.139(Because )A 120 280 :M .354(G)A f0_7 sf 0 3 rm .191(C)A 0 -3 rm f0_12 sf .123 .012( )J 138 280 :M (is )S 150 280 :M -.326(a )A 159 280 :M (MAG, )S 194 280 :M f1_12 sf -.11(e)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf ( )S 205 280 :M (is )S 217 280 :M -.163(uncorrelated )A 280 280 :M -.083(with )A 306 280 :M -.22(the )A 325 280 :M (errors )S 358 280 :M (of )S 373 280 :M -.109(any )A 395 280 :M -.122(ancestor )A 439 280 :M (of )S 454 280 :M .726(X)A f0_7 sf 0 3 rm .163(j)A 0 -3 rm f0_12 sf .457 .046(, )J 474 280 :M -.163(and)A 59 301 :M -.077(hence uncorrelated with any ancestor of X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.088(. Hence )A 301 301 :M -.167(in )A 314 301 :M f1_12 sf .506(q)A f0_12 sf .242 .024( )J 325 301 :M -.22(the )A 343 301 :M -.163(coefficients )A 401 301 :M (of )S 415 301 :M -.22(the )A 433 301 :M -.071(ancestors )A 481 301 :M (of)S 59 323 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 75 323 :M -.167(in )A 90 323 :M -.22(the )A 110 323 :M -.165(equation )A 156 323 :M (for )S 176 323 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 193 323 :M -.215(are )A 213 323 :M -.197(equal )A 244 323 :M -.167(to )A 259 323 :M -.22(the )A 279 323 :M -.235(partial )A 314 323 :M -.031(regression )A 369 323 :M -.163(coefficients )A 429 323 :M (of )S 445 323 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 462 323 :M (on )S 480 323 :M -.168(its)A 59 344 :M -.06(ancestors. But when X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.053( is regressed on its ancestors, the partial regression coefficient )A 466 344 :M (of )S 480 344 :M .32(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm 59 363 :M f0_12 sf .103 .01(in the equation for X)J f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .093 .009(, is equal to zero when cov)J f0_7 sf 0 3 rm (G)S 0 -3 rm 0 5 rm (C)S 0 -5 rm 0 3 rm <28>S 0 -3 rm f1_7 sf 0 3 rm (q)S 0 -3 rm f0_7 sf 0 3 rm <29>S 0 -3 rm f0_12 sf .076 .008( \(X)J f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .037(,X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf (|)S f2_12 sf .048(An)A f0_12 sf .04(\(X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .034(\)\\{X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .037(,X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .059 .006(}\) = 0. )J 446 363 :M .224 .022(Hence, )J 484 363 :M -.327(if)A 59 385 :M (there is no edge X)S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S f1_12 sf S f0_12 sf ( X)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf ( in G, Xi and Xj )S 258 385 :M -.215(are )A 276 385 :M -.117(d-separated )A 334 385 :M -.132(given )A 364 385 :M f2_12 sf .303(An)A f0_12 sf .25(\(X)A f0_7 sf 0 3 rm .077(j)A 0 -3 rm f0_12 sf .215(\)\\{X)A f0_7 sf 0 3 rm .077(i)A 0 -3 rm f0_12 sf .23(,X)A f0_7 sf 0 3 rm .077(j)A 0 -3 rm f0_12 sf .314 .031(} )J 444 385 :M -.167(in )A 457 385 :M 1.114 .111(G, )J 474 385 :M -.163(and)A 59 405 :M (by )S 81 405 :M -.033(hypothesis )A 143 405 :M .172(cov)A f1_7 sf 0 3 rm .123(S)A 0 -3 rm f0_12 sf .188(\(X)A f0_7 sf 0 3 rm .058(i)A 0 -3 rm f0_12 sf .174(,X)A f0_7 sf 0 3 rm .058(j)A 0 -3 rm f0_12 sf .071(|)A f2_12 sf .228(An)A f0_12 sf .188(\(X)A f0_7 sf 0 3 rm .058(j)A 0 -3 rm f0_12 sf .162(\)\\{X)A f0_7 sf 0 3 rm .058(i)A 0 -3 rm f0_12 sf .174(,X)A f0_7 sf 0 3 rm .058(j)A 0 -3 rm f0_12 sf .316 .032(}\) )J 287 405 :M .211 .021(= )J 305 405 :M .833 .083(0. )J 326 405 :M .16(cov)A f0_7 sf 0 3 rm .14(G)A 0 -3 rm 0 5 rm .129(C)A 0 -5 rm 0 3 rm .065<28>A 0 -3 rm f1_7 sf 0 3 rm .101(q)A 0 -3 rm f0_7 sf 0 3 rm .065<29>A 0 -3 rm f0_12 sf .175(\(X)A f0_7 sf 0 3 rm .054(i)A 0 -3 rm f0_12 sf .162(,X)A f0_7 sf 0 3 rm .054(j)A 0 -3 rm f0_12 sf .067(|)A f2_12 sf .213(An)A f0_12 sf .175(\(X)A f0_7 sf 0 3 rm .054(j)A 0 -3 rm f0_12 sf .151(\)\\{X)A f0_7 sf 0 3 rm .054(i)A 0 -3 rm f0_12 sf .162(,X)A f0_7 sf 0 3 rm .054(j)A 0 -3 rm f0_12 sf .295 .029(}\) )J 484 405 :M (=)S 59 427 :M .162(cov)A f1_7 sf 0 3 rm .116(S)A 0 -3 rm f0_12 sf .177(\(X)A f0_7 sf 0 3 rm .054(i)A 0 -3 rm f0_12 sf .163(,X)A f0_7 sf 0 3 rm .054(j)A 0 -3 rm f0_12 sf .067(|)A f2_12 sf .215(An)A f0_12 sf .177(\(X)A f0_7 sf 0 3 rm .054(j)A 0 -3 rm f0_12 sf .152(\)\\{X)A f0_7 sf 0 3 rm .054(i)A 0 -3 rm f0_12 sf .163(,X)A f0_7 sf 0 3 rm .054(j)A 0 -3 rm f0_12 sf .336 .034(}\) = 0. Hence in )J f1_12 sf .175(q)A f0_12 sf .153 .015(, )J 278 427 :M -.22(the )A 296 427 :M -.235(partial )A 329 427 :M -.031(regression )A 382 427 :M -.208(coefficient )A 435 427 :M (of )S 449 427 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 464 427 :M -.167(in )A 477 427 :M -.33(the)A 59 449 :M -.165(equation )A 103 449 :M (for )S 121 449 :M .726(X)A f0_7 sf 0 3 rm .163(j)A 0 -3 rm f0_12 sf .457 .046(, )J 140 449 :M (is )S 152 449 :M -.197(equal )A 181 449 :M -.167(to )A 194 449 :M .236 .024(zero. )J 222 449 :M -.164(It )A 233 449 :M (follows )S 274 449 :M -.249(that )A 296 449 :M (we )S 315 449 :M -.217(can )A 336 449 :M -.163(remove )A 376 449 :M -.08(from )A 404 449 :M .354(G)A f0_7 sf 0 3 rm .191(C)A 0 -3 rm f0_12 sf .123 .012( )J 423 449 :M -.129(every )A 454 449 :M -.234(directed)A 59 467 :M -.009(edge that is in G)A f0_7 sf 0 3 rm (C)S 0 -3 rm f0_12 sf -.009( but not in G. Call this graph G)A f0_7 sf 0 3 rm (C)S 0 -3 rm 0 6 rm (0)S 0 -6 rm f0_12 sf (.)S 77 494 :M .273 .027(We now show that G)J f0_7 sf 0 3 rm .073(C)A 0 -3 rm 0 6 rm .055(0)A 0 -6 rm f0_12 sf .195 .02( is a MAG. G)J f0_7 sf 0 3 rm .073(C)A 0 -3 rm 0 6 rm .055(0)A 0 -6 rm f0_12 sf .176 .018(. G)J f0_7 sf 0 3 rm .073(C)A 0 -3 rm 0 6 rm .055(0)A 0 -6 rm f0_12 sf .143 .014( has )J 309 494 :M -.22(the )A 327 494 :M -.163(same )A 355 494 :M -.122(ancestor )A 398 494 :M -.146(relations )A 442 494 :M (as )S 456 494 :M .354(G)A f0_7 sf 0 3 rm .191(C)A 0 -3 rm f0_12 sf .123 .012( )J 474 494 :M -.163(and)A 59 515 :M .077 .008(G. If A*)J f1_12 sf .064A f0_12 sf .051 .005( B in G)J f0_7 sf 0 3 rm (C)S 0 -3 rm 0 6 rm (0)S 0 -6 rm f0_12 sf .082 .008(, then A*)J f1_12 sf .064A f0_12 sf .051 .005( B in G)J f0_7 sf 0 3 rm (C. )S 0 -3 rm f0_12 sf .146 .015(Because G)J f0_7 sf 0 3 rm (C)S 0 -3 rm f0_12 sf .058 .006( is a MAG, B )J 380 515 :M (is )S 392 515 :M -.111(not )A 411 515 :M -.163(an )A 426 515 :M -.122(ancestor )A 469 515 :M (of )S 483 515 :M (A)S 59 536 :M (in G)S f0_7 sf 0 3 rm (C)S 0 -3 rm f0_12 sf .014 .001(. Hence B is not an ancestor of A in G)J f0_7 sf 0 3 rm (C)S 0 -3 rm 0 6 rm (0)S 0 -6 rm f0_12 sf (.)S 77 563 :M .197 .02(Suppose )J 123 563 :M -.249(that )A 145 563 :M -.196(there )A 173 563 :M (is )S 186 563 :M -.163(an )A 202 563 :M -.124(inducing )A 248 563 :M -.165(path )A 273 563 :M -.139(between )A 317 563 :M -.663(A )A 330 563 :M -.109(and )A 352 563 :M (B )S 365 563 :M -.167(in )A 379 563 :M .823(G)A f0_7 sf 0 3 rm .443(C)A 0 -3 rm 0 6 rm .332(0)A 0 -6 rm f0_12 sf .518 .052(. )J 406 563 :M -.139(Because )A 450 563 :M .51(G)A f0_7 sf 0 3 rm .275(C)A 0 -3 rm 0 6 rm .206(0)A 0 -6 rm f0_12 sf .177 .018( )J 473 563 :M (is )S 486 563 :M (a)S 59 587 :M -.03(subgraph of G)A f0_7 sf 0 3 rm (C)S 0 -3 rm f0_12 sf -.028(, there is an inducing path between A and B in G)A f0_7 sf 0 3 rm (C)S 0 -3 rm f0_12 sf -.024(. If )A 387 587 :M -.663(A )A 399 587 :M (is )S 411 587 :M -.163(an )A 426 587 :M -.122(ancestor )A 469 587 :M (of )S 483 587 :M (B)S 59 608 :M -.167(in )A 72 608 :M .747(G)A f0_7 sf 0 3 rm .403(C)A 0 -3 rm f0_12 sf .471 .047(, )J 94 608 :M -.165(then )A 118 608 :M -.139(because )A 160 608 :M -.129(every )A 191 608 :M -.163(vertex )A 225 608 :M (on )S 242 608 :M -.22(the )A 261 608 :M -.124(inducing )A 307 608 :M -.165(path )A 332 608 :M (is )S 345 608 :M -.163(an )A 361 608 :M -.122(ancestor )A 405 608 :M (of )S 420 608 :M -.663(A )A 433 608 :M (or )S 448 608 :M .83 .083(B, )J 465 608 :M -.161(every)A 59 629 :M -.122(vertex on the inducing path )A 191 629 :M (is )S 203 629 :M -.163(an )A 218 629 :M -.122(ancestor )A 261 629 :M (of )S 275 629 :M .83 .083(B. )J 291 629 :M -.113(But )A 312 629 :M -.22(the )A 330 629 :M -.057(predecessor )A 390 629 :M (of )S 404 629 :M (B )S 416 629 :M (on )S 432 629 :M -.22(the )A 450 629 :M -.142(inducing)A 59 650 :M -.099(path is a collider on the inducing path, and hence is )A 304 650 :M -.111(not )A 323 650 :M -.163(an )A 338 650 :M -.122(ancestor )A 381 650 :M (of )S 395 650 :M .83 .083(B. )J 411 650 :M -.164(It )A 422 650 :M (follows )S 462 650 :M -.249(that )A 483 650 :M (A)S 59 671 :M -.072(is not an ancestor of B. Similarly, B is not an ancestor of A. Hence the edge between A and)A endp %%Page: 14 14 %%BeginPageSetup initializepage (peter; page: 14 of 18)setjob %%EndPageSetup gS 0 0 552 730 rC 461 698 30 27 rC 479 721 :M f0_12 sf (14)S gR gS 0 0 552 730 rC 59 56 :M f0_12 sf (B )S 71 56 :M -.167(in )A 84 56 :M .354(G)A f0_7 sf 0 3 rm .191(C)A 0 -3 rm f0_12 sf .123 .012( )J 102 56 :M (is )S 114 56 :M -.663(A )A 126 56 :M f1_12 sf .4A f0_12 sf .096 .01( )J 143 56 :M .83 .083(B. )J 159 56 :M -.164(It )A 170 56 :M (follows )S 210 56 :M -.249(that )A 231 56 :M -.22(the )A 249 56 :M -.163(edge )A 275 56 :M -.663(A )A 287 56 :M f1_12 sf .4A f0_12 sf .096 .01( )J 304 56 :M .83 .083(B. )J 320 56 :M (is )S 332 56 :M -.167(in )A 345 56 :M .823(G)A f0_7 sf 0 3 rm .443(C)A 0 -3 rm 0 6 rm .332(0)A 0 -6 rm f0_12 sf .518 .052(. )J 371 56 :M (By )S 389 56 :M -.187(Theorem )A 435 56 :M .833 .083(1, )J 450 56 :M .51(G)A f0_7 sf 0 3 rm .275(C)A 0 -3 rm 0 6 rm .206(0)A 0 -6 rm f0_12 sf .177 .018( )J 473 56 :M (is )S 486 56 :M (a)S 59 80 :M (MAG.)S 77 106 :M -.099(Let )A f1_12 sf -.13(q)A f0_7 sf 0 3 rm -.073(0)A 0 -3 rm f0_12 sf -.098( be the same as )A f1_12 sf -.13(q)A f0_12 sf -.097(, except that all of the zero coefficients corresponding to directed)A 59 125 :M -.017(edges in G)A f0_7 sf 0 3 rm (C)S 0 -3 rm f0_12 sf -.016( but not in G have been removed. )A f1_12 sf (q)S f0_7 sf 0 3 rm (0)S 0 -3 rm f0_12 sf -.015( is a parameterization of G)A f0_7 sf 0 3 rm (C)S 0 -3 rm 0 6 rm (0)S 0 -6 rm f0_12 sf -.015(, and )A f1_12 sf (S)S f0_7 sf 0 3 rm (G)S 0 -3 rm 0 5 rm (C)S 0 -5 rm 0 3 rm <28>S 0 -3 rm f1_7 sf 0 3 rm (q)S 0 -3 rm 0 5 rm (0)S 0 -5 rm f0_7 sf 0 3 rm <29>S 0 -3 rm f0_12 sf ( )S 484 125 :M (=)S 59 147 :M f1_12 sf .444(S)A f0_7 sf 0 3 rm .316(G)A 0 -3 rm 0 5 rm .292(C)A 0 -5 rm 0 3 rm .146<28>A 0 -3 rm f1_7 sf 0 3 rm .228(q)A 0 -3 rm f0_7 sf 0 3 rm .146<29>A 0 -3 rm f0_12 sf .347 .035( = )J f1_12 sf .444(S)A f0_12 sf (.)S 77 176 :M -.046(Now consider the double-headed arrows in G)A f0_7 sf 0 3 rm (C)S 0 -3 rm f0_12 sf -.043( but not in G. Arrange these )A 434 176 :M -.064(pairs )A 461 176 :M (on )S 477 176 :M -.33(the)A 59 197 :M -.073(following )A 110 197 :M -.062(order )A 140 197 :M -.499(O: )A 156 197 :M .422({X)A f0_7 sf 0 3 rm .114(i)A 0 -3 rm f0_12 sf .341(,X)A f0_7 sf 0 3 rm .114(j)A 0 -3 rm f0_12 sf .466 .047(} )J 199 197 :M .211 .021(< )J 211 197 :M .364({X)A f0_7 sf 0 3 rm .177(k)A 0 -3 rm f0_12 sf .294(,X)A f0_7 sf 0 3 rm .275(m)A 0 -3 rm f0_12 sf .402 .04(} )J 259 197 :M -.164(if )A 271 197 :M -.039(max\(i,j\) )A 314 197 :M .211 .021(< )J 326 197 :M .536 .054(max\(k,m\). )J 382 197 :M -.083(This )A 409 197 :M -.081(ordering )A 455 197 :M (has )S 477 197 :M -.33(the)A 59 217 :M -.073(following )A 109 217 :M -.109(property: )A 156 217 :M (If )S 168 217 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 183 217 :M f1_12 sf .4A f0_12 sf .096 .01( )J 200 217 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 215 217 :M -.121(precedes )A 260 217 :M .478(X)A f0_7 sf 0 3 rm .193(k)A 0 -3 rm f0_12 sf .166 .017( )J 277 217 :M f1_12 sf .4A f0_12 sf .096 .01( )J 294 217 :M -.055(X)A f0_7 sf 0 3 rm (m)S 0 -3 rm f0_12 sf ( )S 312 217 :M -.167(in )A 325 217 :M -.22(the )A 343 217 :M -.062(order )A 372 217 :M -.165(then )A 396 217 :M .478(X)A f0_7 sf 0 3 rm .193(k)A 0 -3 rm f0_12 sf .166 .017( )J 413 217 :M f1_12 sf .4A f0_12 sf .096 .01( )J 430 217 :M -.055(X)A f0_7 sf 0 3 rm (m)S 0 -3 rm f0_12 sf ( )S 449 217 :M (does )S 476 217 :M -.167(not)A 59 239 :M -.03(occur on any inducing path between X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.031( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.028( in G)A f0_7 sf 0 3 rm (C)S 0 -3 rm f0_12 sf -.027(. Suppose, for a reductio, that this )A 472 239 :M .172(was)A 59 260 :M -.111(not )A 78 260 :M -.22(the )A 96 260 :M .236 .024(case. )J 124 260 :M -.139(Because )A 167 260 :M .478(X)A f0_7 sf 0 3 rm .193(k)A 0 -3 rm f0_12 sf .166 .017( )J 184 260 :M -.109(and )A 205 260 :M -.055(X)A f0_7 sf 0 3 rm (m)S 0 -3 rm f0_12 sf ( )S 223 260 :M -.129(occur )A 253 260 :M (on )S 269 260 :M -.163(an )A 284 260 :M -.124(inducing )A 329 260 :M -.165(path )A 353 260 :M -.139(between )A 397 260 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 413 260 :M -.109(and )A 435 260 :M .726(X)A f0_7 sf 0 3 rm .163(j)A 0 -3 rm f0_12 sf .457 .046(, )J 455 260 :M -.245(each )A 481 260 :M (of)S 59 281 :M -.249(them )A 86 281 :M (is )S 98 281 :M -.163(an )A 113 281 :M -.122(ancestor )A 156 281 :M (of )S 170 281 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 185 281 :M (or )S 199 281 :M .726(X)A f0_7 sf 0 3 rm .163(j)A 0 -3 rm f0_12 sf .457 .046(. )J 218 281 :M .197 .02(Suppose )J 264 281 :M -.095(without )A 305 281 :M .237 .024(loss )J 329 281 :M (of )S 344 281 :M -.197(generality )A 395 281 :M -.249(that )A 417 281 :M (k )S 428 281 :M .211 .021(> )J 440 281 :M .555 .055(m. )J 458 281 :M (By )S 477 281 :M -.33(the)A 59 302 :M -.039(ordering of the variable pairs, k > max\(i,j\). By the )A 301 302 :M -.081(ordering )A 345 302 :M (of )S 359 302 :M -.22(the )A 377 302 :M -.166(individual )A 428 302 :M -.031(variables, )A 478 302 :M .596(X)A f0_7 sf 0 3 rm (k)S 0 -3 rm 59 323 :M f0_12 sf -.034(is not an ancestor of X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.035( or X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.033(. This is a contradiction.)A 77 347 :M -.094(Let G)A f0_7 sf 0 3 rm -.08(C)A 0 -3 rm 0 6 rm -.06(n)A 0 -6 rm f0_12 sf -.085( be the graph resulting from removing the )A 313 347 :M -.064(first )A 336 347 :M (n )S 346 347 :M -.125(double-headed )A 419 347 :M .223 .022(arrows )J 456 347 :M -.249(that )A 477 347 :M -.323(are)A 59 368 :M .116 .012(in G)J f0_7 sf 0 3 rm (C)S 0 -3 rm 0 6 rm (0)S 0 -6 rm f0_12 sf .079 .008( but not in G, in )J 169 368 :M -.22(the )A 187 368 :M -.062(order )A 216 368 :M -.167(in )A 229 368 :M -.065(which )A 262 368 :M -.165(they )A 286 368 :M -.129(occur )A 316 368 :M -.167(in )A 329 368 :M .281 .028(O, )J 345 368 :M -.109(and )A 366 368 :M f1_12 sf .612(q)A f0_7 sf 0 3 rm .343(n)A 0 -3 rm f0_12 sf .294 .029( )J 381 368 :M -.163(be )A 396 368 :M -.22(the )A 414 368 :M -.241(parameterization)A 59 389 :M (of )S 74 389 :M .51(G)A f0_7 sf 0 3 rm .275(C)A 0 -3 rm 0 6 rm .206(n)A 0 -6 rm f0_12 sf .177 .018( )J 97 389 :M -.249(that )A 119 389 :M -.046(results )A 156 389 :M -.08(from )A 185 389 :M -.123(removing )A 236 389 :M -.08(from )A 265 389 :M f1_12 sf .506(q)A f0_12 sf .242 .024( )J 278 389 :M -.22(the )A 298 389 :M -.163(parameters )A 355 389 :M -.049(corresponding )A 429 389 :M -.167(to )A 444 389 :M -.22(the )A 464 389 :M -.08(edges)A 59 410 :M -.14(removed )A 104 410 :M -.08(from )A 131 410 :M .823(G)A f0_7 sf 0 3 rm .443(C)A 0 -3 rm 0 6 rm .332(0)A 0 -6 rm f0_12 sf .518 .052(. )J 157 410 :M -.219(The )A 179 410 :M (proof )S 209 410 :M -.249(that )A 230 410 :M .51(G)A f0_7 sf 0 3 rm .275(C)A 0 -3 rm 0 6 rm .206(n)A 0 -6 rm f0_12 sf .177 .018( )J 252 410 :M (is )S 264 410 :M -.326(a )A 274 410 :M -.331(MAG )A 306 410 :M (is )S 319 410 :M (by )S 336 410 :M -.148(induction )A 385 410 :M (on )S 402 410 :M -.22(the )A 421 410 :M -.22(the )A 440 410 :M -.109(number )A 481 410 :M (of)S 59 431 :M -.125(double-headed )A 132 431 :M .223 .022(arrows )J 169 431 :M -.14(removed )A 214 431 :M -.08(from )A 241 431 :M .823(G)A f0_7 sf 0 3 rm .443(C)A 0 -3 rm 0 6 rm .332(0)A 0 -6 rm f0_12 sf .518 .052(. )J 268 431 :M .258 .026(For )J 290 431 :M -.161(zero )A 315 431 :M -.125(double-headed )A 389 431 :M .223 .022(arrows )J 427 431 :M (removed, )S 477 431 :M (we)S 59 452 :M -.163(have )A 86 452 :M -.187(already )A 125 452 :M .447 .045(shown )J 162 452 :M -.249(that )A 184 452 :M .51(G)A f0_7 sf 0 3 rm .275(C)A 0 -3 rm 0 6 rm .206(0)A 0 -6 rm f0_12 sf .177 .018( )J 207 452 :M (is )S 220 452 :M -.326(a )A 230 452 :M (MAG, )S 267 452 :M -.109(and )A 290 452 :M f1_12 sf .378(S)A f0_7 sf 0 3 rm .269(G)A 0 -3 rm 0 5 rm .249(C)A 0 -5 rm 0 6 rm .186(n)A 0 -6 rm 0 3 rm .124<28>A 0 -3 rm f1_7 sf 0 3 rm .194(q)A 0 -3 rm f0_7 sf 0 5 rm .186(n)A 0 -5 rm 0 3 rm .124<29>A 0 -3 rm f0_12 sf .16 .016( )J 330 452 :M .211 .021(= )J 343 452 :M f1_12 sf .22(S)A f0_7 sf 0 3 rm .156(G)A 0 -3 rm 0 5 rm .144(C)A 0 -5 rm 0 3 rm .072<28>A 0 -3 rm f1_7 sf 0 3 rm .113(q)A 0 -3 rm f0_7 sf 0 3 rm .072<29>A 0 -3 rm f0_12 sf .093 .009( )J 375 452 :M .211 .021(= )J 388 452 :M f1_12 sf 1.029(S)A f0_12 sf .79 .079(. )J 406 452 :M -.33(Let )A 427 452 :M -.22(the )A 447 452 :M -.166(induction)A 59 473 :M -.022(hypothesis be that G)A f0_7 sf 0 3 rm (C)S 0 -3 rm 0 6 rm (n)S 0 -6 rm f0_12 sf ( )S 169 473 :M (is )S 181 473 :M -.326(a )A 190 473 :M -.331(MAG )A 221 473 :M -.109(and )A 242 473 :M f1_12 sf .308(S)A f0_7 sf 0 3 rm .219(G)A 0 -3 rm 0 5 rm .202(C)A 0 -5 rm 0 3 rm .101<28>A 0 -3 rm f1_7 sf 0 3 rm .158(q)A 0 -3 rm 0 5 rm .152(0)A 0 -5 rm f0_7 sf 0 3 rm .101<29>A 0 -3 rm f0_12 sf .13 .013( )J 276 473 :M .211 .021(= )J 287 473 :M f1_12 sf .22(S)A f0_7 sf 0 3 rm .156(G)A 0 -3 rm 0 5 rm .144(C)A 0 -5 rm 0 3 rm .072<28>A 0 -3 rm f1_7 sf 0 3 rm .113(q)A 0 -3 rm f0_7 sf 0 3 rm .072<29>A 0 -3 rm f0_12 sf .093 .009( )J 317 473 :M .211 .021(= )J 328 473 :M f1_12 sf 1.029(S)A f0_12 sf .79 .079(. )J 344 473 :M -.33(Let )A 363 473 :M -.22(the )A 381 473 :M -.163(edge )A 407 473 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 422 473 :M f1_12 sf .4A f0_12 sf .096 .01( )J 439 473 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 454 473 :M -.163(be )A 469 473 :M -.217(edge)A 59 494 :M (n+1 in the ordering O. Suppose that in )S f1_12 sf (q)S f0_7 sf 0 3 rm (n)S 0 -3 rm f0_12 sf (, cov\()S f1_12 sf (e)S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf (,)S f1_12 sf (e)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf (\) = c. We will now show that )S 444 494 :M .546(G)A f0_7 sf 0 3 rm .294(C)A 0 -3 rm 0 6 rm .23(n+1)A 0 -6 rm f0_12 sf .189 .019( )J 474 494 :M (is )S 486 494 :M (a)S 59 518 :M (MAG.)S 77 542 :M .281 .028(No )J 97 542 :M -.125(double-headed )A 171 542 :M (arrow )S 204 542 :M -.249(that )A 226 542 :M (is )S 239 542 :M -.167(in )A 253 542 :M .51(G)A f0_7 sf 0 3 rm .275(C)A 0 -3 rm 0 6 rm .206(n)A 0 -6 rm f0_12 sf .177 .018( )J 276 542 :M -.111(but )A 296 542 :M -.111(not )A 317 542 :M -.167(in )A 332 542 :M .306 .031(G )J 347 542 :M -.091(appears )A 389 542 :M (on )S 407 542 :M -.163(an )A 424 542 :M -.124(inducing )A 471 542 :M -.22(path)A 59 563 :M .056 .006(between X)J f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .025 .003( and X)J f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf ( in G)S f0_7 sf 0 3 rm (C)S 0 -3 rm 0 6 rm (n)S 0 -6 rm f0_12 sf .034 .003(, because )J 227 563 :M -.22(the )A 245 563 :M -.083(only )A 270 563 :M -.064(edges )A 301 563 :M -.249(that )A 322 563 :M -.331(lie )A 337 563 :M (on )S 353 563 :M -.163(an )A 368 563 :M -.124(inducing )A 413 563 :M -.165(path )A 437 563 :M -.139(between )A 480 563 :M .32(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm 59 584 :M f0_12 sf -.109(and )A 80 584 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 95 584 :M -.167(in )A 108 584 :M .51(G)A f0_7 sf 0 3 rm .275(C)A 0 -3 rm 0 6 rm .206(n)A 0 -6 rm f0_12 sf .177 .018( )J 130 584 :M -.109(and )A 151 584 :M -.249(that )A 172 584 :M -.215(are )A 190 584 :M -.111(not )A 209 584 :M -.167(in )A 222 584 :M 1.114 .111(G, )J 239 584 :M -.129(occur )A 269 584 :M -.064(prior )A 296 584 :M -.167(to )A 309 584 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 324 584 :M f1_12 sf .4A f0_12 sf .096 .01( )J 341 584 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 356 584 :M -.167(in )A 370 584 :M -.22(the )A 389 584 :M .186 .019(ordering, )J 438 584 :M -.109(and )A 460 584 :M (by )S 477 584 :M -.33(the)A 59 605 :M -.109(induction hypothesis have already been removed from )A 318 605 :M .823(G)A f0_7 sf 0 3 rm .443(C)A 0 -3 rm 0 6 rm .332(n)A 0 -6 rm f0_12 sf .518 .052(. )J 344 605 :M -.13(Every )A 376 605 :M -.205(directed )A 417 605 :M -.163(edge )A 443 605 :M -.249(that )A 464 605 :M -.066(exists)A 59 626 :M -.043(in G)A f0_7 sf 0 3 rm (C)S 0 -3 rm 0 6 rm (n)S 0 -6 rm f0_12 sf -.038( also exists in G. Hence if there is an inducing )A 311 626 :M -.165(path )A 335 626 :M -.139(between )A 378 626 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 393 626 :M -.109(and )A 414 626 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 429 626 :M -.167(in )A 442 626 :M .823(G)A f0_7 sf 0 3 rm .443(C)A 0 -3 rm 0 6 rm .332(n)A 0 -6 rm f0_12 sf .518 .052(, )J 468 626 :M -.245(there)A 59 650 :M -.053(is an inducing path between X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.057( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.054( in G. But because G is a MAG, and there )A 441 650 :M (is )S 453 650 :M (no )S 469 650 :M -.217(edge)A 59 671 :M -.017(between X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.015( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.013( in G, there is no )A 230 671 :M -.124(inducing )A 275 671 :M -.165(path )A 299 671 :M -.139(between )A 342 671 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 357 671 :M -.109(and )A 378 671 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 393 671 :M -.167(in )A 406 671 :M 1.114 .111(G. )J 423 671 :M -.164(It )A 434 671 :M (follows )S 474 671 :M -.331(that)A endp %%Page: 15 15 %%BeginPageSetup initializepage (peter; page: 15 of 18)setjob %%EndPageSetup gS 0 0 552 730 rC 461 698 30 27 rC 479 721 :M f0_12 sf (15)S gR gS 0 0 552 730 rC 59 56 :M f0_12 sf -.023(there is no inducing path between X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.024( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.022( in G)A f0_7 sf 0 3 rm (C)S 0 -3 rm 0 6 rm (n)S 0 -6 rm f0_12 sf -.022(, other than the edge X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S f1_12 sf -.057A f0_12 sf -.027( X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.022(. It follows)A 59 77 :M -.027(that in G)A f0_7 sf 0 3 rm (C)S 0 -3 rm 0 6 rm -.021(n+1)A 0 -6 rm f0_12 sf -.027(, if there is no edge between X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.031( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.027(, then there )A 355 77 :M (is )S 367 77 :M (no )S 383 77 :M -.124(inducing )A 428 77 :M -.165(path )A 452 77 :M -.163(between)A 59 101 :M .266(X)A f0_7 sf 0 3 rm .06(i)A 0 -3 rm f0_12 sf .377 .038( and X)J f0_7 sf 0 3 rm .06(j)A 0 -3 rm f0_12 sf (.)S 77 125 :M (G)S f0_7 sf 0 3 rm (C)S 0 -3 rm 0 6 rm (n+1)S 0 -6 rm f0_12 sf .047 .005( has the same ancestor relations as G)J f0_7 sf 0 3 rm (C)S 0 -3 rm f0_12 sf .034 .003( and G. If A*)J f1_12 sf S f0_12 sf .027 .003( B in G)J f0_7 sf 0 3 rm (Cn+1)S 0 -3 rm f0_12 sf .033 .003(, then )J 440 125 :M -.125(A*)A f1_12 sf -.203A f0_12 sf ( )S 470 125 :M (B )S 482 125 :M -.334(in)A 59 149 :M (G)S f0_7 sf 0 3 rm (C. )S 0 -3 rm f0_12 sf -.026(Because G)A f0_7 sf 0 3 rm (C)S 0 -3 rm f0_12 sf -.022( is a MAG, B is not an ancestor of )A 299 149 :M -.663(A )A 311 149 :M -.167(in )A 324 149 :M .747(G)A f0_7 sf 0 3 rm .403(C)A 0 -3 rm f0_12 sf .471 .047(. )J 346 149 :M -.128(Hence )A 380 149 :M (B )S 392 149 :M (is )S 404 149 :M -.111(not )A 423 149 :M -.163(an )A 438 149 :M -.122(ancestor )A 481 149 :M (of)S 59 167 :M (A in G)S f0_7 sf 0 3 rm (C)S 0 -3 rm 0 6 rm (n+1)S 0 -6 rm f0_12 sf .01 .001(. It follows from Theorem 1 that G)J f0_7 sf 0 3 rm (C)S 0 -3 rm 0 6 rm (n+1)S 0 -6 rm f0_12 sf ( is a MAG.)S 77 194 :M .033 .003(If there is no edge between X)J f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .026 .003( and X)J f0_7 sf 0 3 rm (j )S 0 -3 rm f0_12 sf (in G)S f0_7 sf 0 3 rm (C)S 0 -3 rm 0 6 rm (n+1)S 0 -6 rm f0_12 sf (, by )S 315 194 :M -.187(Theorem )A 361 194 :M (1 )S 371 194 :M -.196(there )A 398 194 :M (is )S 410 194 :M (no )S 426 194 :M -.124(inducing )A 471 194 :M -.22(path)A 59 215 :M -.139(between )A 103 215 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 119 215 :M -.109(and )A 141 215 :M .45(X)A f0_7 sf 0 3 rm .174 .017(j )J 0 -3 rm 156 215 :M f0_12 sf -.167(in )A 170 215 :M .778(G)A f0_7 sf 0 3 rm .419(C)A 0 -3 rm 0 6 rm .328(n+1)A 0 -6 rm f0_12 sf .49 .049(. )J 205 215 :M (By )S 224 215 :M -.33(Lemma )A 264 215 :M .833 .083(6, )J 279 215 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 295 215 :M -.109(and )A 317 215 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 333 215 :M -.215(are )A 352 215 :M -.117(d-separated )A 411 215 :M -.132(given )A 442 215 :M f2_12 sf .223(An)A f0_12 sf .184(\(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .184 .018(\) )J 482 215 :M f1_12 sf S 59 236 :M f2_12 sf .388(An)A f0_12 sf .321(\(X)A f0_7 sf 0 3 rm .099(j)A 0 -3 rm f0_12 sf .275(\)\\{X)A f0_7 sf 0 3 rm .099(i)A 0 -3 rm f0_12 sf .295(,X)A f0_7 sf 0 3 rm .099(j)A 0 -3 rm f0_12 sf .496 .05(}. )J 148 236 :M -.128(Hence )A 187 236 :M .168(cov)A f0_7 sf 0 3 rm .147(G)A 0 -3 rm 0 5 rm .136(C)A 0 -5 rm 0 6 rm .106(n+1)A 0 -6 rm 0 3 rm .068<28>A 0 -3 rm f1_7 sf 0 3 rm .106(q)A 0 -3 rm f0_7 sf 0 5 rm .106(n+1)A 0 -5 rm 0 3 rm .068<29>A 0 -3 rm f0_12 sf .087 .009( )J 255 236 :M .231(\(X)A f0_7 sf 0 3 rm .071(i)A 0 -3 rm f0_12 sf .213(,X)A f0_7 sf 0 3 rm .071(j)A 0 -3 rm f0_12 sf .088(|)A f2_12 sf .28(An)A f0_12 sf .231(\(X)A f0_7 sf 0 3 rm .071(i)A 0 -3 rm f0_12 sf .232 .023(\) )J 331 236 :M f1_12 sf -.161A f0_12 sf ( )S 350 236 :M f2_12 sf .288(An)A f0_12 sf .238(\(X)A f0_7 sf 0 3 rm .073(j)A 0 -3 rm f0_12 sf .205(\)\\{X)A f0_7 sf 0 3 rm .073(i)A 0 -3 rm f0_12 sf .219(,X)A f0_7 sf 0 3 rm .073(j)A 0 -3 rm f0_12 sf .4 .04(}\) )J 440 236 :M .211 .021(= )J 457 236 :M .833 .083(0. )J 477 236 :M (By)S 59 258 :M .32 .032(hypothesis, )J 127 258 :M .118(cov)A f0_7 sf 0 3 rm .104(G)A 0 -3 rm 0 5 rm .096(C)A 0 -5 rm 0 3 rm <28>S 0 -3 rm f1_7 sf 0 3 rm .075(q)A 0 -3 rm f0_7 sf 0 3 rm <29>S 0 -3 rm f0_12 sf .13(\(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .12(,X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf (|)S f2_12 sf .157(An)A f0_12 sf .13(\(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .13 .013(\) )J 242 258 :M f1_12 sf -.161A f0_12 sf ( )S 265 258 :M f2_12 sf .288(An)A f0_12 sf .238(\(X)A f0_7 sf 0 3 rm .073(j)A 0 -3 rm f0_12 sf .205(\)\\{X)A f0_7 sf 0 3 rm .073(i)A 0 -3 rm f0_12 sf .219(,X)A f0_7 sf 0 3 rm .073(j)A 0 -3 rm f0_12 sf .4 .04(}\) )J 359 258 :M .211 .021(= )J 380 258 :M .129(cov)A f1_7 sf 0 3 rm .093(S)A 0 -3 rm f0_12 sf .142(\(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .131(,X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .054(|)A f2_12 sf .172(An)A f0_12 sf .142(\(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .142 .014(\) )J 482 258 :M f1_12 sf S 59 280 :M f2_12 sf .37(An)A f0_12 sf .306(\(X)A f0_7 sf 0 3 rm .094(j)A 0 -3 rm f0_12 sf .263(\)\\{X)A f0_7 sf 0 3 rm .094(i)A 0 -3 rm f0_12 sf .282(,X)A f0_7 sf 0 3 rm .094(j)A 0 -3 rm f0_12 sf .585 .059(}\). )J 148 280 :M -.139(Because )A 192 280 :M -.22(the )A 211 280 :M -.163(edge )A 238 280 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 254 280 :M f1_12 sf .4A f0_12 sf .096 .01( )J 272 280 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 289 280 :M (does )S 317 280 :M -.111(not )A 338 280 :M -.129(occur )A 370 280 :M -.167(in )A 385 280 :M 1.114 .111(G, )J 404 280 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 421 280 :M -.109(and )A 444 280 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 461 280 :M -.215(are )A 481 280 :M (d-)S 59 301 :M -.144(separated )A 110 301 :M -.132(given )A 144 301 :M f2_12 sf .223(An)A f0_12 sf .184(\(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .184 .018(\) )J 187 301 :M f1_12 sf -.161A f0_12 sf ( )S 204 301 :M f2_12 sf .303(An)A f0_12 sf .25(\(X)A f0_7 sf 0 3 rm .077(j)A 0 -3 rm f0_12 sf .215(\)\\{X)A f0_7 sf 0 3 rm .077(i)A 0 -3 rm f0_12 sf .23(,X)A f0_7 sf 0 3 rm .077(j)A 0 -3 rm f0_12 sf .314 .031(} )J 288 301 :M -.167(in )A 305 301 :M 1.114 .111(G, )J 326 301 :M -.109(and )A 351 301 :M -.196(hence )A 386 301 :M .129(cov)A f1_7 sf 0 3 rm .093(S)A 0 -3 rm f0_12 sf .142(\(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .131(,X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .054(|)A f2_12 sf .172(An)A f0_12 sf .142(\(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .142 .014(\) )J 482 301 :M f1_12 sf S 59 323 :M f2_12 sf .164(An)A f0_12 sf .136(\(X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .116(\)\\{X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .125(,X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .316 .032(}\) = 0. By Lemma 4,)J 166 345 :M .078 .008(0 = cov)J f0_7 sf 0 3 rm (G)S 0 -3 rm 0 5 rm (C)S 0 -5 rm 0 6 rm (n)S 0 -6 rm 0 3 rm <28>S 0 -3 rm f1_7 sf 0 3 rm (q)S 0 -3 rm f0_7 sf 0 5 rm (n)S 0 -5 rm 0 3 rm <29>S 0 -3 rm f0_12 sf .173 .017(\(Xi,Xj|An\(Xi\) )J f1_12 sf .054A f0_12 sf .148 .015( An\(Xj\)\\{Xi,Xj}\) =)J 146 370 :M .151(cov)A f0_7 sf 0 3 rm .133(G)A 0 -3 rm 0 5 rm .122(C)A 0 -5 rm 0 6 rm .096(n+1)A 0 -6 rm 0 3 rm .061<28>A 0 -3 rm f1_7 sf 0 3 rm .096(q)A 0 -3 rm f0_7 sf 0 5 rm .096(n+1)A 0 -5 rm 0 3 rm .061<29>A 0 -3 rm f0_12 sf .166(\(X)A f0_7 sf 0 3 rm .051(i)A 0 -3 rm f0_12 sf .153(,X)A f0_7 sf 0 3 rm .051(j)A 0 -3 rm f0_12 sf .063(|)A f2_12 sf .201(An)A f0_12 sf .166(\(X)A f0_7 sf 0 3 rm .051(i)A 0 -3 rm f0_12 sf .153 .015(\) )J f1_12 sf .242A f0_12 sf .071 .007( )J f2_12 sf .201(An)A f0_12 sf .166(\(X)A f0_7 sf 0 3 rm .051(j)A 0 -3 rm f0_12 sf .143(\)\\{X)A f0_7 sf 0 3 rm .051(i)A 0 -3 rm f0_12 sf .153(,X)A f0_7 sf 0 3 rm .051(j)A 0 -3 rm f0_12 sf .232 .023(}\) + c = 0 + c)J 59 395 :M .293 .029(It follows that c = 0, and hence )J f1_12 sf .155(S)A f0_7 sf 0 3 rm .11(G)A 0 -3 rm 0 5 rm .102(C)A 0 -5 rm 0 6 rm .08(n+1)A 0 -6 rm 0 3 rm .051<28>A 0 -3 rm f1_7 sf 0 3 rm .079(q)A 0 -3 rm f0_7 sf 0 5 rm .08(n+1)A 0 -5 rm 0 3 rm .074 .007(\) )J 0 -3 rm f0_12 sf .177 .018(= )J f1_12 sf .155(S)A f0_7 sf 0 3 rm .123 .012( G)J 0 -3 rm 0 5 rm .102(C)A 0 -5 rm 0 6 rm .076(n)A 0 -6 rm 0 3 rm .051<28>A 0 -3 rm f1_7 sf 0 3 rm .079(q)A 0 -3 rm f0_7 sf 0 5 rm .076(n)A 0 -5 rm 0 3 rm .074 .007(\) )J 0 -3 rm f1_12 sf .28 .028(= S)J f0_12 sf .109 .011(. )J f3_12 sf <5C>S 59 424 :M f2_12 sf 1.122 .112(Lemma )J 105 424 :M .387(7:)A f0_12 sf .232 .023( )J 121 424 :M -.219(The )A 145 424 :M .231(n)A f0_7 sf 0 -5 rm .105(th)A 0 5 rm f0_12 sf .115 .012( )J 163 424 :M -.197(derivative )A 215 424 :M (of )S 231 424 :M -.326(a )A 242 424 :M -.206(rational )A 283 424 :M -.123(function )A 328 424 :M .101(f)A f0_7 sf 0 3 rm .089(1)A 0 -3 rm f0_12 sf .121(\(X\)/f)A f0_7 sf 0 3 rm .089(2)A 0 -3 rm f0_12 sf .382 .038(\(X\) )J 387 424 :M -.05(\(where )A 426 424 :M .193(f)A f0_7 sf 0 3 rm .169(1)A 0 -3 rm f0_12 sf .145 .014( )J 440 424 :M -.109(and )A 463 424 :M .193(f)A f0_7 sf 0 3 rm .169(2)A 0 -3 rm f0_12 sf .145 .014( )J 477 424 :M -.323(are)A 59 445 :M -.065(polynomials\) whose denominator is nowhere 0 on its )A 314 445 :M -.166(domain )A 353 445 :M (is )S 365 445 :M -.326(a )A 374 445 :M -.206(rational )A 413 445 :M (function, )S 460 445 :M .086(whose)A 59 466 :M -.055(denominator is a positive integral power of f)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.076(\(X\).)A 77 493 :M -.075(Proof. Consider the first derivative. The derivative is equal )A 359 493 :M -.167(to )A 372 493 :M .099(\(f)A f0_7 sf 0 3 rm .087(1)A 0 -3 rm f0_12 sf .419 .042J 409 493 :M (* )S 419 493 :M .126(f)A f0_7 sf 0 3 rm .111(2)A 0 -3 rm f0_12 sf .477 .048(\(X\) )J 448 493 :M (- )S 456 493 :M .126(f)A f0_7 sf 0 3 rm .111(1)A 0 -3 rm f0_12 sf .477 .048(\(X\) )J 485 493 :M (*)S 59 514 :M .087(f)A f0_7 sf 0 3 rm .076(2)A 0 -3 rm f0_12 sf .1(\325\(X\)\)/f)A f0_7 sf 0 3 rm .076(2)A 0 -3 rm f0_12 sf .121(\(X\))A f0_7 sf 0 -5 rm .076(2)A 0 5 rm f0_12 sf .318 .032(, where f)J f0_7 sf 0 3 rm .076(1)A 0 -3 rm f0_12 sf .253 .025(\325 and f)J f0_7 sf 0 3 rm .076(2)A 0 -3 rm f0_12 sf .139 .014J 214 514 :M -.215(are )A 232 514 :M -.082(also )A 255 514 :M -.027(polynomials. )A 321 514 :M -.128(Hence )A 355 514 :M -.22(the )A 373 514 :M -.197(derivative )A 423 514 :M (is )S 435 514 :M -.206(rational )A 474 514 :M -.163(and)A 59 535 :M -.079(has a denominator that is a positive integral function of f)A f0_7 sf 0 3 rm -.06(2)A 0 -3 rm f0_12 sf -.112(\(X\).)A 77 562 :M .197 .02(Suppose )J 123 562 :M -.22(the )A 142 562 :M .231(n)A f0_7 sf 0 -5 rm .105(th)A 0 5 rm f0_12 sf .115 .012( )J 159 562 :M -.197(derivative )A 210 562 :M (is )S 223 562 :M -.326(a )A 233 562 :M -.206(rational )A 273 562 :M (function, )S 321 562 :M -.109(and )A 344 562 :M -.22(the )A 364 562 :M -.15(denominator )A 429 562 :M (is )S 443 562 :M -.326(a )A 454 562 :M -.142(positive)A 59 583 :M .383 .038(power of f)J f0_7 sf 0 3 rm .08(2)A 0 -3 rm f0_12 sf .31 .031(\(X\), i.e. the )J 174 583 :M .231(n)A f0_7 sf 0 -5 rm .105(th)A 0 5 rm f0_12 sf .115 .012( )J 190 583 :M -.197(derivative )A 240 583 :M -.109(equals )A 274 583 :M .123(f)A f0_7 sf 0 3 rm .107(3)A 0 -3 rm f0_12 sf .147(\(X\)/f)A f0_7 sf 0 3 rm .107(2)A 0 -3 rm f0_12 sf .17(\(X\))A f0_7 sf 0 -5 rm .167(m)A 0 5 rm f0_12 sf .167 .017(, )J 340 583 :M -.062(where )A 373 583 :M .126(f)A f0_7 sf 0 3 rm .111(3)A 0 -3 rm f0_12 sf .477 .048(\(X\) )J 402 583 :M (is )S 414 583 :M -.326(a )A 423 583 :M -.06(polynomial. )A 484 583 :M -.327(It)A 59 604 :M -.056(follows that the n+1)A f0_7 sf 0 -5 rm -.027(st)A 0 5 rm f0_12 sf -.052( derivative is a rational function \(f)A f0_7 sf 0 3 rm (3)S 0 -3 rm f0_12 sf -.054(\325\(X\) * f)A f0_7 sf 0 3 rm (2)S 0 -3 rm 0 -5 rm (n)S 0 5 rm f0_12 sf -.065(\(X\) - m )A 407 604 :M (* )S 417 604 :M .167(f)A f0_7 sf 0 3 rm .147(2)A 0 -3 rm f0_12 sf .233(\(X\))A f0_7 sf 0 -5 rm .13(n-1)A 0 5 rm f0_12 sf .126 .013( )J 456 604 :M (* )S 466 604 :M .142(f)A f0_7 sf 0 3 rm .124(1)A 0 -3 rm f0_12 sf .296(\(X\))A 59 625 :M (* )S 69 625 :M .127(f)A f0_7 sf 0 3 rm .112(2)A 0 -3 rm f0_12 sf .146(\325\(X\)\)/f)A f0_7 sf 0 3 rm .112(2)A 0 -3 rm f0_12 sf .177(\(X\))A f0_7 sf 0 -5 rm .137(m+1)A 0 5 rm f0_12 sf .174 .017(, )J 151 625 :M -.062(where )A 184 625 :M .142(f)A f0_7 sf 0 3 rm .124(1)A 0 -3 rm f0_12 sf .225 .023J 200 625 :M -.109(and )A 221 625 :M .142(f)A f0_7 sf 0 3 rm .124(3)A 0 -3 rm f0_12 sf .225 .023J 237 625 :M -.215(are )A 255 625 :M -.082(also )A 278 625 :M -.166(polynomial )A 335 625 :M .173 .017(functions. )J 387 625 :M -.219(The )A 409 625 :M -.15(denominator )A 473 625 :M (is )S 486 625 :M (a)S 59 646 :M -.027(positive power of f)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.027(\(X\), and hence is nowhere 0 in its domain. )A f3_12 sf <5C>S endp %%Page: 16 16 %%BeginPageSetup initializepage (peter; page: 16 of 18)setjob %%EndPageSetup gS 0 0 552 730 rC 461 698 30 27 rC 479 721 :M f0_12 sf (16)S gR gS 0 0 552 730 rC 77 58 :M f0_12 sf -.33(Let )A 96 58 :M -.22(the )A 114 58 :M -.109(set )A 131 58 :M (of )S 146 58 :M -.109(values )A 181 58 :M (of )S 196 58 :M -.188(natural )A 233 58 :M -.163(parameters )A 289 58 :M (of )S 304 58 :M -.326(a )A 314 58 :M -.165(full )A 335 58 :M -.139(regular )A 373 58 :M -.18(exponential )A 432 58 :M -.22(family )A 467 58 :M f2_12 sf .917(S)A f0_12 sf .412 .041( )J 480 58 :M -.326(be)A 59 79 :M .171 .017(denoted by )J f2_12 sf .082(N)A f0_12 sf .047 .005(, )J f2_12 sf .063(S)A f2_7 sf 0 3 rm (0)S 0 -3 rm f0_12 sf .11 .011( be a subfamily of )J f2_12 sf .063(S)A f0_12 sf .047 .005(, )J f2_12 sf .082(N)A f2_7 sf 0 3 rm (0)S 0 -3 rm f0_12 sf .113 .011( be the set of values of the )J 390 79 :M -.188(natural )A 426 79 :M -.163(parameters )A 481 79 :M (of)S 59 100 :M f2_12 sf 1.167(S)A f2_7 sf 0 3 rm .612(0)A 0 -3 rm f0_12 sf .954 .095(, )J 80 100 :M -.109(and )A 102 100 :M -.164(if )A 114 100 :M f2_12 sf .993(U)A f0_12 sf .344 .034( )J 129 100 :M (is )S 142 100 :M -.163(an )A 158 100 :M -.082(open )A 186 100 :M -.054(neighborhood )A 257 100 :M -.167(in )A 271 100 :M f2_12 sf 1.381(N)A f0_12 sf .869 .087(, )J 290 100 :M f2_12 sf .578(S)A f2_7 sf 0 -5 rm .438(U)A 0 5 rm f0_12 sf .26 .026( )J 308 100 :M -.163(be )A 324 100 :M -.22(the )A 343 100 :M -.109(set )A 361 100 :M (of )S 376 100 :M -.077(distributions )A 441 100 :M -.167(in )A 456 100 :M f2_12 sf .917(S)A f0_12 sf .412 .041( )J 470 100 :M -.11(with)A 59 121 :M .094 .009(parameters in )J f2_12 sf (U)S f0_12 sf (.)S 59 148 :M f2_12 sf 1.973 .197(Theorem )J 113 148 :M .387(6:)A f0_12 sf .232 .023( )J 128 148 :M -.219(The )A 150 148 :M -.22(family )A 184 148 :M (of )S 198 148 :M -.077(distributions )A 261 148 :M -.117(represented )A 319 148 :M (by )S 335 148 :M -.326(a )A 344 148 :M -.219(linear )A 374 148 :M -.331(MAG )A 405 148 :M -.667(M )A 419 148 :M -.08(over )A 444 148 :M -.326(a )A 453 148 :M -.109(set )A 470 148 :M (of )S 485 148 :M (k)S 59 169 :M -.145(variables )A 107 169 :M (is )S 121 169 :M -.326(a )A 132 169 :M -.236(locally )A 169 169 :M -.227(parameterized )A 240 169 :M -.108(curved )A 279 169 :M -.18(exponential )A 340 169 :M -.22(family )A 377 169 :M (of )S 394 169 :M -.111(dimension )A 450 169 :M -.197(equal )A 482 169 :M -.334(to)A 59 190 :M -.105(k\(k+1\)/2 minus the number of pairs of variables in M that are not adjacent to each other.)A 77 217 :M .839 .084(Proof. )J 112 217 :M -.183(According )A 165 217 :M -.167(to )A 178 217 :M -.187(Theorem )A 224 217 :M 1.333 .133(4.2.1 )J 254 217 :M -.167(in )A 267 217 :M .483 .048(Kass )J 295 217 :M -.109(and )A 316 217 :M .342 .034(Vos\(1997\), )J 375 217 :M -.326(a )A 384 217 :M -.11(subfamily )A 435 217 :M f2_12 sf .926(S)A f2_7 sf 0 3 rm .486(0)A 0 -3 rm f0_12 sf .417 .042( )J 451 217 :M (of )S 465 217 :M -.163(an )A 481 217 :M (n-)S 59 238 :M -.15(dimensional )A 121 238 :M -.139(regular )A 159 238 :M -.18(exponential )A 218 238 :M -.22(family )A 253 238 :M f2_12 sf .917(S)A f0_12 sf .412 .041( )J 266 238 :M (is )S 280 238 :M -.326(a )A 291 238 :M -.236(locally )A 328 238 :M -.227(parameterized )A 399 238 :M -.108(curved )A 437 238 :M -.198(exponential)A 59 259 :M -.22(family )A 93 259 :M -.164(if )A 104 259 :M (for )S 122 259 :M -.245(each )A 147 259 :M f1_12 sf .14(h)A f1_7 sf 0 3 rm .068(0)A 0 -3 rm f0_12 sf .058 .006( )J 162 259 :M -.167(in )A 175 259 :M f2_12 sf 1.049(N)A f2_7 sf 0 3 rm .424(0)A 0 -3 rm f0_12 sf .363 .036( )J 194 259 :M -.196(there )A 222 259 :M (is )S 235 259 :M -.163(an )A 251 259 :M -.082(open )A 279 259 :M -.054(neighborhood )A 350 259 :M f2_12 sf .993(U)A f0_12 sf .344 .034( )J 365 259 :M -.167(in )A 379 259 :M f2_12 sf .993(N)A f0_12 sf .344 .034( )J 394 259 :M -.165(containing )A 448 259 :M f1_12 sf .14(h)A f1_7 sf 0 3 rm .068(0)A 0 -3 rm f0_12 sf .058 .006( )J 464 259 :M -.109(and )A 486 259 :M (a)S 59 280 :M (diffeomorphism h: )S f2_12 sf (U)S f0_12 sf ( )S f1_12 sf S f0_12 sf ( )S f2_12 sf (R)S f2_7 sf 0 -5 rm (k)S 0 5 rm f0_12 sf ( )S f1_12 sf S f0_12 sf ( )S f2_12 sf (R)S f2_7 sf 0 -5 rm (n-k)S 0 5 rm f0_12 sf ( such that )S 271 268 15 17 rC 286 285 :M psb currentpoint pse 271 268 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 480 div 544 3 -1 roll exch div scale currentpoint translate 64 60 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Bold f1 (S) -16 324 sh 224 ns (0) 205 422 sh (U) 206 152 sh end MTsave restore pse gR gS 0 0 552 730 rC 286 280 :M f0_12 sf .412 .041(= {P)J f1_7 sf 0 3 rm .11(h)A 0 -3 rm f0_12 sf .166 .017( in )J f2_12 sf .173(S)A f2_7 sf 0 -5 rm .131(U)A 0 5 rm f2_12 sf .151 .015(: )J f0_12 sf .13(h\()A f1_12 sf .188(h)A f0_12 sf .215 .022(\) = \()J f1_12 sf .171(b)A f0_12 sf .078(,)A f1_12 sf .214(y)A f0_12 sf .273 .027(\) and )J f1_12 sf .214(y)A f0_12 sf .286 .029( = 0}.)J 77 305 :M -.183(According )A 131 305 :M -.167(to )A 145 305 :M -.187(Theorem )A 192 305 :M .833 .083(4, )J 207 305 :M -.22(the )A 226 305 :M -.077(distributions )A 290 305 :M -.117(represented )A 349 305 :M (by )S 366 305 :M -.326(a )A 377 305 :M -.132(given )A 409 305 :M -.331(MAG )A 442 305 :M -.667(M )A 458 305 :M -.217(can )A 480 305 :M -.326(be)A 59 322 :M -.227(parameterized )A 130 322 :M -.167(in )A 145 322 :M -.22(the )A 165 322 :M -.073(following )A 217 322 :M .722 .072(way. )J 247 322 :M .258 .026(For )J 270 322 :M -.326(a )A 281 322 :M -.132(given )A 313 322 :M -.196(covariance )A 369 322 :M -.22(matrix )A 406 322 :M f1_12 sf .632(S)A f0_12 sf .267 .027( )J 421 322 :M -.132(among )A 460 322 :M -.22(the )A 481 322 :M f2_12 sf (X)S 59 341 :M f0_12 sf .119 .012(variables, regress X)J f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf .05 .005( on the set )J f2_12 sf (P)S f2_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf .059 .006( := {X)J f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .036 .004( | X)J f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .044 .004( )J cF f1_12 sf .004A sf .044 .004( X)J f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf .061 .006( and X)J f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .066 .007( is an ancestor of X)J f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf .092 .009(}. Let)J 244 363 :M ( )S 247 350 76 29 rC 323 379 :M psb currentpoint pse 247 350 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2432 div 928 3 -1 roll exch div scale currentpoint translate 64 50 translate 69 258 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (\303) show -4 366 moveto 384 /Times-Roman f1 (X) show 299 462 moveto 224 ns (k) show 551 366 moveto 384 /Symbol f1 (=) show 1485 366 moveto 384 /Symbol f1 (a) show 1755 462 moveto 224 /Times-Roman f1 (kj) show 1942 366 moveto 384 ns (X) show 2257 462 moveto 224 ns (j) show 869 739 moveto 224 /Times-Roman f1 ( ) show 925 739 moveto 224 /Times-Roman f1 (X) show 1123 796 moveto 160 /Times-Roman f1 (j) show 1212 739 moveto 192 /Symbol f1 (\316) show 1335 739 moveto 224 /Times-Bold f1 (P) show 1499 796 moveto 160 /Times-Roman f1 (k) show 1036 453 moveto 576 /Symbol f1 (\345) show end pse gR gS 0 0 552 730 rC 59 402 :M f0_12 sf -.013(be the linear predictor of X)A f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf ( on )S f2_12 sf (P)S f2_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf -.011(, or 0 if )A f2_12 sf (P)S f2_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf -.014( is empty. Now let )A 362 389 67 17 rC 429 406 :M psb currentpoint pse 362 389 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2144 div 544 3 -1 roll exch div scale currentpoint translate 64 50 translate -8 366 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Symbol f1 (e) show 187 462 moveto 224 /Times-Roman f1 (k) show 332 366 moveto 384 /Times-Roman f1 (:) show 467 366 moveto 384 /Symbol f1 (=) show 781 366 moveto 384 /Times-Roman f1 (X) show 1084 462 moveto 224 ns (k) show 1320 366 moveto 384 /Symbol f1 (-) show 1690 258 moveto 384 /Times-Roman f1 (\303) show 1617 366 moveto 384 /Times-Roman f1 (X) show 1920 462 moveto 224 ns (k) show end pse gR gS 0 0 552 730 rC 429 402 :M f0_12 sf .175 .017(. The )J f1_12 sf .124(a)A f0_7 sf 0 3 rm .045(kj)A 0 -3 rm f0_12 sf .257 .026( and)J 59 423 :M -.114(the non-zero covariances among the )A f1_12 sf -.119(e)A f0_7 sf 0 3 rm -.079(k)A 0 -3 rm f0_12 sf -.11( parameterize a MAG. Call this set of parameters )A f2_12 sf -.256(M)A f0_12 sf (.)S 59 450 :M -.073(First we will show that there is a diffeomorphism )A 296 450 :M -.08(from )A 323 450 :M -.22(the )A 341 450 :M -.109(set )A 358 450 :M (of )S 372 450 :M -.196(covariance )A 426 450 :M -.205(matrices )A 469 450 :M f3_12 sf .632(S)A f0_12 sf .267 .027( )J 481 450 :M (of)S 59 471 :M -.085(the normal distribution \(with zero means\) over k variables to )A f2_12 sf -.203(M)A f0_12 sf -.103(. By )A 383 471 :M -.109(Corollary )A 432 471 :M .259 .026(A.3 )J 454 471 :M -.167(in )A 467 471 :M .226(Kass)A 59 492 :M -.109(and )A 81 492 :M .478 .048(Vos, )J 109 492 :M -.334(it )A 120 492 :M -.038(suffices )A 162 492 :M -.167(to )A 176 492 :M .479 .048(show )J 207 492 :M -.249(that )A 229 492 :M -.196(there )A 257 492 :M (is )S 270 492 :M -.326(a )A 280 492 :M -.056(smooth )A 320 492 :M -.097(one-to-one )A 376 492 :M -.123(function )A 420 492 :M -.08(from )A 448 492 :M f3_12 sf .632(S)A f0_12 sf .267 .027( )J 461 492 :M -.167(to )A 476 492 :M f2_12 sf .533(M)A f0_12 sf (,)S 59 513 :M (whose inverse is also smooth.)S 59 541 :M -.102(From Theorem 4 it follows that)A 77 550 400 75 rC 477 625 :M psb currentpoint pse 77 550 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 12800 div 2400 3 -1 roll exch div scale currentpoint translate 64 44 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (cov\() 3520 366 sh (,) 4550 366 sh (\)) 5035 366 sh (=) 5245 366 sh (cov\(X) 5553 366 sh (-) 6754 366 sh (X) 6956 366 sh (X) 7539 366 sh (-) 8060 366 sh (X) 8262 366 sh (X) 2599 1108 sh (X) 3182 1108 sh (X) 4831 1108 sh (X) 5414 1108 sh (X) 7063 1108 sh (X) 7646 1108 sh (\)) 8106 1108 sh (X) 9297 1108 sh (X) 9880 1108 sh (X) 687 1830 sh (X) 1270 1830 sh (cov\(X) 3271 1830 sh (X) 4535 1830 sh (\)) 4943 1830 sh (cov\(X) 6470 1830 sh (X) 7735 1830 sh (\)) 8148 1830 sh (X) 11621 1830 sh 224 ns (p) 4407 462 sh (q) 4873 462 sh (p) 6537 462 sh (p) 7259 462 sh (q) 7837 462 sh (q) 8560 462 sh (p) 2902 1204 sh (q) 3480 1204 sh (p) 5134 1204 sh (q) 5712 1204 sh (p) 7366 1204 sh (q) 7944 1204 sh (p) 9600 1204 sh (q) 10178 1204 sh (p) 990 1926 sh (q) 1568 1926 sh (pi) 3078 1926 sh (p) 4255 1926 sh (i) 4834 1926 sh (X) 2230 2203 sh (qj) 6288 1926 sh (q) 7449 1926 sh (j) 8050 1926 sh (X) 5443 2203 sh (pi) 10227 1926 sh (X) 9377 2203 sh (X) 8650 2203 sh (qj) 10679 1926 sh 160 ns (i) 2417 2260 sh (p) 2788 2259 sh (j) 5641 2260 sh (q) 6001 2259 sh (j) 9575 2260 sh (q) 9935 2259 sh (i) 8837 2260 sh (p) 9208 2259 sh 384 /Symbol f1 (e) 4212 366 sh (e) 4683 366 sh (a) 2808 1830 sh (a) 6023 1830 sh (a) 9957 1830 sh (a) 10414 1830 sh /mt_vec StandardEncoding 256 array copy def /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 mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\303) 7029 258 sh (,) 7402 366 sh (\303) 8335 258 sh (\)) 8722 366 sh (cov\() 1903 1108 sh (,) 3045 1108 sh (\)) 3642 1108 sh (cov\() 4135 1108 sh (,) 5277 1108 sh (\303) 5487 1000 sh (\)) 5874 1108 sh (cov\() 6367 1108 sh (\303) 7136 1000 sh (,) 7509 1108 sh (cov\() 8601 1108 sh (\303) 9370 1000 sh (,) 9743 1108 sh (\303) 9953 1000 sh (\)) 10340 1108 sh (cov\() -9 1830 sh (,) 1133 1830 sh (\)) 1730 1830 sh (,) 4398 1830 sh (,) 7598 1830 sh (cov\() 10925 1830 sh 384 /Symbol f1 (=) 8939 366 sh (-) 3843 1108 sh (-) 6075 1108 sh (+) 8308 1108 sh (=) 10557 1108 sh (-) 1931 1830 sh (-) 5144 1830 sh (+) 8350 1830 sh 224 ns (\316) 2512 2203 sh (\316) 5728 2203 sh (\316) 9662 2203 sh (\316) 8932 2203 sh 576 ns (\345) 2359 1917 sh (\345) 5574 1917 sh (\345) 9508 1917 sh (\345) 8779 1917 sh 224 /Times-Bold f1 (P) 2652 2203 sh (P) 5868 2203 sh (P) 9802 2203 sh (P) 9072 2203 sh 224 /MT_Times-Roman f1 (i) 11920 1926 sh (i) 11920 1926 sh (j) 12468 1926 sh 384 ns (X) 12153 1830 sh 384 /MT_Times-Roman f1 (,) 12016 1830 sh (\)) 12566 1830 sh end MTsave restore pse gR gS 0 0 552 730 rC 59 648 :M f0_12 sf -.246(Each )A 87 648 :M (of )S 102 648 :M -.22(the )A 121 648 :M f1_12 sf .466(a)A f0_7 sf 0 3 rm .167(kj)A 0 -3 rm f0_12 sf .184 .018( )J 140 648 :M (is )S 153 648 :M -.326(a )A 163 648 :M -.031(regression )A 218 648 :M -.108(coefficient, )A 277 648 :M -.109(and )A 300 648 :M -.196(hence )A 333 648 :M -.326(a )A 344 648 :M -.206(rational )A 385 648 :M -.123(function )A 430 648 :M (of )S 446 648 :M f3_12 sf .632(S)A f0_12 sf .267 .027( )J 460 648 :M -.249(that )A 483 648 :M (is)S 59 669 :M -.095(everywhere )A 120 669 :M -.14(defined )A 161 669 :M (on )S 179 669 :M -.112(its )A 196 669 :M -.166(domain )A 237 669 :M -.121(\(because )A 284 669 :M -.129(every )A 316 669 :M -.196(covariance )A 372 669 :M -.22(matrix )A 408 669 :M -.167(in )A 424 669 :M f3_12 sf .632(S)A f0_12 sf .267 .027( )J 439 669 :M (is )S 454 669 :M -.142(positive)A endp %%Page: 17 17 %%BeginPageSetup initializepage (peter; page: 17 of 18)setjob %%EndPageSetup gS 0 0 552 730 rC 461 698 30 27 rC 479 721 :M f0_12 sf (17)S gR gS 0 0 552 730 rC 59 58 :M f0_12 sf -.025(definite\). cov\()A f1_12 sf (e)S f0_7 sf 0 3 rm (p)S 0 -3 rm f0_12 sf (,)S f1_12 sf (e)S f0_7 sf 0 3 rm (q)S 0 -3 rm f0_12 sf -.025(\) is a rational function of the )A f1_12 sf (a)S f0_7 sf 0 3 rm (kj)S 0 -3 rm f0_12 sf -.026( and )A f3_12 sf (S)S f1_12 sf ( )S 333 58 :M f0_12 sf -.249(that )A 354 58 :M (is )S 366 58 :M -.095(everywhere )A 425 58 :M -.14(defined )A 464 58 :M (on )S 480 58 :M -.168(its)A 59 79 :M -.107(domain \(because every covariance matrix in )A f3_12 sf -.152(S)A f0_12 sf -.095( is positive definite\). )A 377 79 :M -.139(Because )A 420 79 :M f2_12 sf -.258(M)A f0_12 sf ( )S 435 79 :M (is )S 447 79 :M -.326(a )A 456 79 :M -.235(rational)A 59 100 :M -.123(function )A 102 100 :M (of )S 116 100 :M f3_12 sf .632(S)A f1_12 sf .267 .027( )J 128 100 :M f0_12 sf -.249(that )A 149 100 :M (is )S 161 100 :M -.095(everywhere )A 220 100 :M -.14(defined )A 260 100 :M (on )S 277 100 :M -.112(its )A 293 100 :M (domain, )S 337 100 :M (by )S 354 100 :M -.33(Lemma )A 394 100 :M (7 )S 405 100 :M -.196(there )A 433 100 :M (is )S 446 100 :M -.326(a )A 456 100 :M -.067(smooth)A 59 121 :M (function from )S f3_12 sf (S)S f0_12 sf ( to )S f2_12 sf (M)S f0_12 sf (.)S 77 140 :M -.085(It was also shown in Theorem 4 that each variable X)A f0_7 sf 0 3 rm -.06(k)A 0 -3 rm f0_12 sf -.085( could be written as)A 227 149 113 28 rC 340 177 :M psb currentpoint pse 227 149 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3616 div 896 3 -1 roll exch div scale currentpoint translate 64 38 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (X) -4 346 sh (X) 2386 346 sh 224 ns (k) 299 442 sh (kj) 2199 442 sh (j) 2701 442 sh (k) 3362 442 sh (X) 867 719 sh (\(X) 2036 719 sh (\)) 2436 719 sh 160 ns (j) 1065 776 sh (k) 2300 776 sh 384 /Symbol f1 (=) 551 346 sh (+) 2873 346 sh 224 ns (\316) 1152 719 sh 576 ns (\345) 1480 433 sh 384 /Symbol f1 (a) 1929 346 sh (e) 3167 346 sh 224 /Times-Bold f1 (Parents) 1292 719 sh end MTsave restore pse gR gS 0 0 552 730 rC 59 200 :M f0_12 sf -.128(Hence )A 94 200 :M -.22(the )A 113 200 :M -.123(function )A 157 200 :M -.142(mapping )A 203 200 :M f3_12 sf .632(S)A f0_12 sf .267 .027( )J 216 200 :M -.167(to )A 230 200 :M f2_12 sf -.258(M)A f0_12 sf ( )S 246 200 :M (has )S 267 200 :M -.163(an )A 283 200 :M -.092(inverse )A 323 200 :M -.109(and )A 346 200 :M (is )S 360 200 :M (one-to-one. )S 421 200 :M (In )S 437 200 :M -.036(addition, )A 485 200 :M -.668(it)A 59 221 :M -.096(follows that there is a reduced form for the )A 264 221 :M f2_12 sf .993(X)A f0_12 sf .344 .034( )J 278 221 :M -.031(variables, )A 328 221 :M .957 .096(i.e. )J 348 221 :M -.165(they )A 372 221 :M -.215(are )A 390 221 :M -.326(a )A 399 221 :M -.206(rational )A 438 221 :M -.123(function )A 481 221 :M (of)S 59 242 :M -.059(the values of the )A f1_12 sf -.097(a)A f0_7 sf 0 3 rm -.035(kj)A 0 -3 rm f0_12 sf -.062( parameters and the )A f1_12 sf -.068(e)A f0_12 sf -.061( variables. Hence )A f3_12 sf -.091(S)A f0_12 sf -.06( is )A 360 242 :M -.326(a )A 369 242 :M -.206(rational )A 408 242 :M -.123(function )A 451 242 :M (of )S 465 242 :M f2_12 sf .441(M)A f0_12 sf .212 .021(. )J 484 242 :M -.327(It)A 59 263 :M -.051(follows that there is a smooth function from )A f2_12 sf -.122(M)A f0_12 sf -.041( to )A f3_12 sf -.076(S)A f0_12 sf (.)S 77 290 :M -.053(Hence there is a diffeomorphism from )A f3_12 sf -.074(S)A f0_12 sf -.04( to )A f2_12 sf -.119(M)A f0_12 sf (.)S 77 317 :M -.006(There is also a diffeomorphism from )A f2_12 sf (N)S f0_12 sf ( to )S f3_12 sf (S)S f0_12 sf -.006( \(Kass and Vos, )A 367 317 :M .833 .083(p. )J 381 317 :M .671 .067(101\). )J 411 317 :M -.219(The )A 433 317 :M -.133(composition)A 59 338 :M -.041(of two diffeomophisms is a diffeomorphism \(Kass )A 303 338 :M -.109(and )A 324 338 :M .478 .048(Vos, )J 351 338 :M .833 .083(p. )J 365 338 :M .671 .067(101\), )J 395 338 :M -.109(and )A 416 338 :M -.196(hence )A 447 338 :M -.196(there )A 474 338 :M (is )S 486 338 :M (a)S 59 359 :M (diffeomorphism from )S f2_12 sf (N)S f0_12 sf ( to )S f2_12 sf (M)S f0_12 sf (.)S 77 386 :M -.246(Each )A 105 386 :M -.22(family )A 140 386 :M (of )S 155 386 :M -.077(distributions )A 219 386 :M -.117(represented )A 278 386 :M (by )S 295 386 :M -.326(a )A 305 386 :M -.331(MAG )A 337 386 :M -.217(can )A 358 386 :M -.163(be )A 375 386 :M -.226(characterized )A 442 386 :M (by )S 460 386 :M -.166(setting)A 59 407 :M -.125(some subset of the parameters of a complete MAG )A 301 407 :M -.197(equal )A 330 407 :M -.167(to )A 343 407 :M .236 .024(zero. )J 371 407 :M -.164(It )A 382 407 :M (follows )S 422 407 :M -.08(from )A 449 407 :M -.218(Theorem)A 59 428 :M -.089(4.2.1 that the distributions represented by a MAG are a curved exponential family.)A 77 455 :M -.119(Since the dimensionality of the full space of k normal variables with )A 401 455 :M -.161(zero )A 425 455 :M -.247(mean )A 454 455 :M (is )S 466 455 :M -.247(equal)A 59 476 :M -.167(to )A 72 476 :M .48 .048(k\(k+1\)/2, )J 122 476 :M -.22(the )A 140 476 :M -.166(dimensionality )A 213 476 :M (of )S 227 476 :M -.326(a )A 236 476 :M -.235(compete )A 279 476 :M -.331(MAG )A 310 476 :M (is )S 322 476 :M .48 .048(k\(k+1\)/2. )J 373 476 :M -.33(Let )A 393 476 :M -.667(M )A 408 476 :M -.163(be )A 424 476 :M -.163(an )A 440 476 :M -.257(incomplete)A 59 497 :M (MAG. )S 94 497 :M (By )S 113 497 :M -.33(Lemma )A 153 497 :M .833 .083(3, )J 168 497 :M -.667(M )A 183 497 :M (has )S 204 497 :M -.326(a )A 214 497 :M -.248(complete )A 261 497 :M -.11(extension )A 311 497 :M .261 .026(M\325, )J 334 497 :M -.109(and )A 356 497 :M -.22(the )A 375 497 :M -.166(dimensionality )A 449 497 :M (of )S 464 497 :M -.33<4DD520>A 483 497 :M (is)S 59 518 :M -.104(k\(k+1\)/2. Each parameter in M\325 that is set to )A 272 518 :M -.161(zero )A 296 518 :M -.08(\(one )A 321 518 :M (of )S 335 518 :M -.22(the )A 353 518 :M f1_12 sf .79(a)A f0_7 sf 0 3 rm .284(kj)A 0 -3 rm f0_12 sf .569 .057(, )J 375 518 :M (or )S 389 518 :M -.326(a )A 398 518 :M -.196(covariance )A 452 518 :M -.163(between)A 59 539 :M -.065(two error terms )A f1_12 sf -.071(e)A f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf -.063( and )A f1_12 sf -.071(e)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.063(\) corresponds to a pair of variables )A 342 539 :M -.167(in )A 355 539 :M -.667(M )A 369 539 :M -.249(that )A 390 539 :M -.215(are )A 408 539 :M -.111(not )A 427 539 :M -.108(adjacent. )A 473 539 :M -.328(The)A 59 560 :M -.102(number of parameters in M is equal to k\(k+1\)/2 )A 287 560 :M -.067(minus )A 320 560 :M -.22(the )A 338 560 :M -.109(number )A 378 560 :M (of )S 392 560 :M -.163(parameters )A 447 560 :M -.167(in )A 460 560 :M -.33<4DD520>A 478 560 :M -.164(set)A 59 581 :M -.009(to zero, i.e. k\(k+1\)/2 minus the number of )A 264 581 :M -.064(pairs )A 291 581 :M (of )S 305 581 :M -.145(variables )A 351 581 :M -.167(in )A 364 581 :M -.33<4DD520>A 382 581 :M -.249(that )A 403 581 :M -.215(are )A 421 581 :M -.111(not )A 440 581 :M -.247(adjacent )A 482 581 :M -.334(to)A 59 602 :M -.069(each other. )A f3_12 sf <5C>S endp %%Page: 18 18 %%BeginPageSetup initializepage (peter; page: 18 of 18)setjob %%EndPageSetup gS 0 0 552 730 rC 461 698 30 27 rC 479 721 :M f0_12 sf (18)S gR gS 0 0 552 730 rC 258 58 :M f0_12 sf -.143(References)A 59 103 :M 1.118 .112(Kass, )J 91 103 :M .83 .083(R. )J 107 103 :M -.109(and )A 128 103 :M .478 .048(Vos, )J 155 103 :M 1.107 .111(P. )J 170 103 :M (\(1997\) )S 206 103 :M f4_12 sf -.27(Geometrical )A 268 103 :M -.151(Foundations )A 332 103 :M .555 .055(of )J 347 103 :M .343 .034(Asymptotic )J 406 103 :M -.029(Inference)A f0_12 sf (, )S 459 103 :M -.064(Wiley,)A 77 124 :M .337(NY.)A 59 151 :M .375 .037(Spirtes, )J 100 151 :M 1.792 .179(P., )J 120 151 :M .375 .038(Glymour, )J 172 151 :M 1.535 .154(C., )J 193 151 :M -.109(and )A 215 151 :M .184 .018(Scheines, )J 266 151 :M .83 .083(R. )J 283 151 :M (\(1993\) )S 320 151 :M f4_12 sf -.134(Causation, )A 376 151 :M -.15(Prediction, )A 433 151 :M -.333(and )A 455 151 :M -.049(Search)A f0_12 sf (,)S 77 172 :M -.05(Springer-Verlag Lecture Notes in Statistics 81, N.Y.)A 59 199 :M .739 .074(Spirtes, P., Richardson, T., )J 197 199 :M -.064(Meek, )A 231 199 :M 1.535 .154(C., )J 251 199 :M .184 .018(Scheines, )J 301 199 :M .83 .083(R. )J 317 199 :M -.109(and )A 338 199 :M .375 .038(Glymour, )J 389 199 :M .83 .083(C. )J 405 199 :M (\(1997\) )S 441 199 :M (\322Using )S 479 199 :M -.656(D-)A 77 220 :M -.131(separation )A 131 220 :M -.167(to )A 146 220 :M -.257(Calculate )A 195 220 :M -.162(Zero )A 223 220 :M -.188(Partial )A 259 220 :M -.109(Correlations )A 323 220 :M -.167(in )A 338 220 :M -.218(Linear )A 374 220 :M -.166(Models )A 415 220 :M -.083(with )A 442 220 :M -.182(Correlated)A 77 241 :M -.052(Errors\323, Technical Report Ò»±¾µÀÎÞÂë-72-Phil.)A 59 268 :M .375 .037(Spirtes, )J 100 268 :M 1.792 .179(P., )J 119 268 :M -.109(and )A 140 268 :M .166 .017(Richardson, )J 202 268 :M .558 .056(T. )J 217 268 :M .596 .06(\(1996\). )J 257 268 :M -.663(A )A 270 268 :M -.133(Polynomial )A 329 268 :M -.331(Time )A 358 268 :M -.221(Algorithm )A 411 268 :M .258 .026(For )J 433 268 :M -.264(Determining)A 77 289 :M -.195(DAG Equivalence in the )A 195 289 :M -.079(Presence )A 241 289 :M (of )S 255 289 :M -.275(Latent )A 288 289 :M -.219(Variables )A 336 289 :M -.109(and )A 357 289 :M -.183(Selection )A 404 289 :M .446 .045(Bias, )J 433 289 :M -.064(Proceedings)A 77 310 :M -.127(of the 6th International Workshop on Artificial Intelligence and Statistics.)A 59 337 :M -.025(Verma, T. and Pearl, J. \(1990\). Equivalence and synthesis of causal models )A 423 337 :M -.167(in )A 436 337 :M .673 .067(Proc. )J 466 337 :M -.085(Sixth)A 77 358 :M -.13(Conference )A 135 358 :M (on )S 151 358 :M -.149(Uncertainty )A 210 358 :M -.167(in )A 223 358 :M .264 .026(AI. )J 244 358 :M -.15(Association )A 304 358 :M (for )S 323 358 :M -.149(Uncertainty )A 383 358 :M -.167(in )A 397 358 :M .264 .026(AI, )J 418 358 :M 1.12 .112(Inc., )J 446 358 :M -.237(Mountain)A 77 379 :M .193 .019(View, CA.)J endp %%Trailer end %%EOF