A Linear “Microscope” for Interventions and Counterfactuals Judea Pearl.pdf

该【A Linear “Microscope” for Interventions and Counterfactuals Judea Pearl 】是由【金钏】上传分享，文档一共【15】页，该文档可以免费在线阅读，需要了解更多关于【A Linear “Microscope” for Interventions and Counterfactuals Judea Pearl 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。:..;20170003Causal,CasualandCuriousJudeaPearlALinear?Microscope?-2017-0003Abstract:Thisnoteillustrates,usingsimpleexamples,howcausalquestionsofnon-trivialcharactercanberepresented,,linearanalysisallowsforswiftassessmentofhowvariousfeaturesofthemodelimpacttheques-,representationandidenti?cationofcounterfactualexpressions,robustnesstomodelmisspeci?cation,:causalinference,structuralequationmodels,counterfactuals,generalization,robustness1IntroductionTwoyearsago,Iwroteapaperentitled“LinearModels:AUseful‘Microscope’forCausalAnalysis”[1]inwhichlinearstructuralequationsmodels(SEM),wereusedas“microscopes”,linearSEMsenableustoderiveclose-,alsocalled“es,”whichoftenpresentaformidablechallengetonon-,wewilldemonstrate,usingsimpleexamples,howconceptsandissuesinmoderncounterfactualanalysiscanbeunderstoodandana-:Causaleffectidenti?cation,mediation,themediationfallacy,unit-speci?ceffects,theeffectoftreatmentonthetreated(ETT),generalizationacrosspopulations,[1].Section3introducesd-separationandthegraphicalde?nitionsofinterventionsandcounterfactuals,andprovidesthebasictoolsfortheidenti??cproblemsofcausalandcounterfactualnature,includingmediation,sequentialidenti?cation,robustness,,regression,andcorrelationWestartwiththestandardde??nedas1ThissectionistakenfromPearl[1]andcanbeskippedbyreadersfamiliarwithmultipleregression,pathdiagramsandWright’srulesofpathtracing.*Correspondingauthor:puterScienceDepartment,UniversityofCalifornia,LosAngeles,CA90095-1596,USAE-mail:******@|2/13/1812:45AM:..:ALinear?Microscope?forInterventionsandCounterfactuals32=E[X–E(X)]2xandmeasuresthedegreetowhichXdeviatesfromitsmeanE(X).ThecovarianceofXandYisde?nedas3xy=E[(X–E(X))(Y–E(Y))],woothermeasuresofassociation:(1)theregressioncoef?-cient"yxand(2)thecorrelationcoef?:3xy1xy=(1)3x3y3xy3y"yx=2=1xy.(2)3x3xWenotethat1xy=1yxisdimensionlessandiscon?nedtotheunitinterval;0≤1xy≤?cient,"yx,representstheslopeoftheleastsquareerrorlineinthepredictionofYgivenX?"yx=E(Y|X=x).??cient"yx?zwhichisgivenby?"yx?z=E(Y|X=x,Z=z).?xInwords,"yx?zistheslopeoftheregressionlineofYonXwhenweconsideronlycasesforwhichZ=?cient1xy?zcanbede?nedbynormalizing"yx?z:1xy?z="?z3y?[2]permitsustoexpress1xy?zrecursivelyintermsofpair-wiseregressioncoef?,thisreductionreads:1yx–1yz1xz1yx?z=1.(3)[(1–12)(1–12)]2yzxzAccordingly,wecanalsoexpress"yx?zand3yx?zintermsofpair-wiserelationships,whichgives:3yx?z=3xx–3xz2/3z23yy–3yz2/3z21yx?z(4)23yz3zx3yx?z=3x["yx–"yz"zx]3yx–2(5)3z"yx–"yz"zx323yx–3yz3zx3y1yx–1yz?1zx"==z=.(6)yx?z222222321–"zx3x/3z3x3z–3xzx1–1xzUnauthenticatedDownloadDate|2/13/1812:45AM:..:ALinear?Microscope?forInterventionsandCounterfactuals3NotethatnoneoftheseconditionalassociationsdependsonthelevelzatwhichweconditionvariableZ;thisisoneofthefeaturesthatmakeslinearanalysiseasytomanageand,atthesametime,(SEM)isasystemoflinearequationsamongasetVofvariables,,thevariableonitslefthandsideiscalledthedependentvariable,,theequationbelowY=!X+"Z+UY(7)declaresYasthedependentvariable,XandZasexplanatoryvariables,andUYasan“error”or“disturbance”term,representingallfactorsomittedfromVthat,,.,hevalueofY,natureconsultsthevalueofvariablesX,ZandUYand,binationineq.(7),.(7)non-symmetrical,sincethevaluesofXandZarenotdeterminedbyinvertingeq.(7)butbyotherequations,forexample,X=#Z+UX(8)Z=UZ.(9)Thedirectionalityofthisassignmentprocessiscapturedbyapath-diagram,inwhichthenodesrepresentvariables,andthearrowsrepresentthe(potentially)non-zerocoef?(a)representstheSEMequationsof(7)–(9)andtheassumptionofzerocorrelationsbetweentheUvariables,3UX,UY=3UX,UZ=3UZ,UY=(b)ontheotherhandrepresentseqs.(7)–(9)togetherwiththeassumption3UX,UZ=3UZ,UY=0while3UX,UY=(a)(b)Figure1:Pathdiagramscapturinglthedirectionalityoftheassignmentprocessofeqs.(7)?(9)|2/13/1812:45AM:..:ALinear?Microscope?forInterventionsandCounterfactualsThecoef?cients!,",and#arecalledpathcoef?cients,,!standsforthechangeinYinducedbyraisingXoneunit,,includingtheerrorvariables;apropertycalled“effecthomogeneity.”Sinceerrors(.,UX,UY,YZ)capturevariationsamongindividualunits(.,subjects,samples,orsituations),effecthomogen-eityamountstoclaimingthatallunitsreactequallytoanytreatment,?spath-tracingrulesIn1921,icistSewallWrightdevelopedaningeniousmethodbywhichthecovariance3xyofanytwovariablescanbedeterminedswiftly,bymereinspectionofthediagram[3].Wright’smethodconsistsofequatingthe(standardized3)covariance3xy=1xybetweenanypairofvariableswithasumofproductsofpathcoef?cientsanderrorcovariancesalongalld--connectedifitdoesnottraverseanycollider(.,head-to-headarrows,asinX→Y←Z).Forexample,inFigure1(a),thestandardizedcovariance3xyisobtainedbysumming!withtheproduct"#,thusyielding3xy=!+"#,whileinFigure1(b)weget:3xy=!+"#+,weget3xz=#sincethepathX→Y←Zisnotd-,namely,-standardizedvariablesthemethodneedstobemodi?edslightly,multiplyingtheproductassociatedwithapathpbythevarianceofthevariablethatactsasthe“root”,forFigure1(a)wehave3xy=32!+32"#,sinceXservesastherootforpathX→YandZservesastherootforxzX←Z→(b),however,weget3xy=32!+32"#+CXYwherethedoublearrowUX?-wisecorrelationssummarizedineqs.(4)–(6),binedwithWright’spath-,putethepartialregressioncoef?cient"yx?z,westartwithastandardizedmodelwhereallvariancesareunity(hence3xy=1xy="xy),andapplyeq.(6)with3x=3z=1toget:(3yx–3yz3zx)"yx?z=2(10)(1–3xz)Atthispoint,eachpair-putedfromthediagramthroughpath-tracingand,substitutedineq.(10),yieldsanexpressionforthepartialregressioncoef?cient"yx?!amoreformalde?nitionusingbothinterventionalinterpretation:!=?E[(Y|do(x),do(z))]?xandacounterfactualinterpretation!=?Yxz(u).?x3Standardizedparametersrefertosystemsinwhich(withoutlossofgenerality)allvariablesarenormalizedtohavezeromeanandunitvariance,whichsigni?cantlysimpli?|2/13/1812:45AM:..:ALinear?Microscope?forInterventionsandCounterfactuals5Towitness,thepair-wisecovariancesforFigure1(a)are:3yx=!+"#(11)3xz=#(12)3yz="+!#(13)Substitutingineq.(10),weget"yx?z=[(!+"#)–("+#!)#]/(1–#2)=!(1–#2)/(1–#2)=!(14)Indeed,weknowthat,foraconfounding-freemodellikeFigure1(a)thedirecteffect!isidenti?ableandgivenbythepartialregressioncoef?cient"xy?(b)yields:"yx?z=!+CXYleaving!non-identi?=Z1,Z2,...,Zkofregressorsthepartialcorrelation1yx?z1,z2,...,putedbyapplyingeq.(3),whenthenumberofregressorsislarge,,however,canbereadilyidenti?,whichisessentialfortheanalysisofinter-ventions,isfacilitatedthroughagraphicalcriterioncalledd-separation[4].Inotherwords,thecriterionpermitsustoglanceatthediagramanddeterminewhenasetofvariablesZ=Z1,Z2,...,Zkrenderstheequality1yx?z=-separationistoassociatezerocorrelationwithseparation;namely,theequality1yx?z=0wouldbevalidwheneverthesetZ“separates”??nition1(d-Separation)→B→CoraforkA←B→CsuchthatthemiddlenodeBisinZ(.,Bisconditionedon),→B←CsuchthatthecollisionnodeBisnotinZ,,thenXandYared-separated,conditionalonZ,andthenthepartialcorrelationcoef?cient1yx?zvanishes[5].Armedwiththeabilitytoreadvanishingpartials,|2/13/1812:45AM:..:ALinear?Microscope?=[Y|do(x)][Y|do(x)]andtheparametersofanygivenmodelcanreadilybeobtainedbyexplicatinghowaninterventionmodi?esthedata-(x)overridesallpreexistingcausesofXand,hence,transformsthegraphGintoamodi?edgraphGXinwhichallarrowsenteringXareeliminated,asshowninFigure2(b).GGGXXZ1Z1Z1Z2Z2Z2W1W1W1Z3W2Z3W2Z3W2XX=xXW3W3W3YYY(a)(b)(c)Figure2:Illustratingthegraphicalreadingofinterventions.(a)Theoriginalgraph.(b)Themodi?edgraphGXrepresentingtheinterventiondo(x).(c)Themodi?edgraphGXinwhichseparatingXfromYrepresentsnon-,theinterventionalexpectationE[Y|do(x)]isgivenbytheconditionalexpectationE(Y|X=x)evalu-atedinthemodi?’sRuletothismodel,weobtainawellknownresultinpathanalysis:E[Y|do(x)]=4x,where4standsforthesumofproductsofpathcoef?,theaveragecausaleffect(ACE)ofXonY,de?nedbythedifferenceACE=E[Y|do(x+1)]–E[Y|do(x)](15)isaconstant,independentofx,andisgivenby4,Intheearlydaysofpathanalysis,totaleffectswereestimatedby?rstestimatingallpathcoef?-separationcriterionofputationsigni?(Identi?cationoftotaleffects)ThetotaleffectofXonYcanbeidenti?edingraphGwheneverasetZofobservedvariablesexists,non-desc