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TransportabilityfromMultipleEnvironmentswithLimitedExperiments:Complet TransportabilityfromMultipleEnvironmentswithLimitedExperiments:Complet

TransportabilityfromMultipleEnvironmentswithLimitedExperiments:Complet - PDF document

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TransportabilityfromMultipleEnvironmentswithLimitedExperiments:Complet - PPT Presentation

EliasBareinboimComputerScienceUCLAebcsuclaeduJudeaPearlComputerScienceUCLAjudeacsuclaeduAbstractThispaperaddressestheproblemofmztransportabilitythatistransferringcausalknowledgecollectedinsev ID: 198443

EliasBareinboimComputerScienceUCLAeb@cs.ucla.eduJudeaPearlComputerScienceUCLAjudea@cs.ucla.eduAbstractThispaperaddressestheproblemofmz-transportability thatis transferringcausalknowledgecollectedinsev

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TransportabilityfromMultipleEnvironmentswithLimitedExperiments:CompletenessResults EliasBareinboimComputerScienceUCLAeb@cs.ucla.eduJudeaPearlComputerScienceUCLAjudea@cs.ucla.eduAbstractThispaperaddressestheproblemofmz-transportability,thatis,transferringcausalknowledgecollectedinseveralheterogeneousdomainstoatargetdomaininwhichonlypassiveobservationsandlimitedexperimentaldatacanbecollected.Thepaperrstestablishesanecessaryandsufcientconditionfordecidingthefeasibilityofmz-transportability,i.e.,whethercausaleffectsinthetargetdomainareestimablefromtheinformationavailable.Itfurtherprovesthatapreviouslyestablishedalgorithmforcomputingtransportformulaisinfactcomplete,thatis,failureofthealgorithmimpliesnon-existenceofatransportformula.Finally,thepapershowsthatthedo-calculusiscompleteforthemz-transportabilityclass.1MotivationTheissueofgeneralizingcausalknowledgeiscentralinscienticinferencessinceexperimentsareconducted,andconclusionsthatareobtainedinalaboratorysetting(i.e.,specicpopulation,do-main,study)aretransportedandappliedelsewhere,inanenvironmentthatdiffersinmanyaspectsfromthatofthelaboratory.Ifthetargetenvironmentisarbitrary,ordrasticallydifferentfromthestudyenvironment,nocausalrelationscanbelearnedandscienticprogresswillcometoastand-still.However,thefactthatscienticexperimentationcontinuestoprovideusefulinformationaboutourworldsuggeststhatcertainenvironmentssharecommoncharacteristicsandthat,owedtothesecommonalities,causalclaimswouldbevalidevenwhereexperimentshaveneverbeenperformed.Remarkably,theconditionsunderwhichthistypeofextrapolationcanbelegitimizedhavenotbeenformallyarticulateduntilveryrecently.Althoughtheproblemhasbeenextensivelydiscussedinstatistics,economics,andthehealthsciences,underrubricssuchas“externalvalidity”[1,2],“meta-analysis”[3],“quasi-experiments”[4],“heterogeneity”[5],thesediscussionsarelimitedtoverbalnarrativesintheformofheuristicguidelinesforexperimentalresearchers–noformaltreatmentoftheproblemhasbeenattemptedtoanswerthepracticalchallengeofgeneralizingcausalknowledgeacrossmultipleheterogeneousdomainswithdisparateexperimentaldataasposedinthispaper.Thelackofsoundmathematicalmachineryinsuchsettingsprecludesoneofthemaingoalsofmachinelearning(andbyandlargecomputerscience),whichisautomatingtheprocessofdiscovery.Theclassofproblemsofcausalgeneralizabilityiscalledtransportabilityandwasrstformallyarticulatedin[6].Weconsiderthemostgeneralinstanceoftransportabilityknowntodatethatistheproblemoftransportingexperimentalknowledgefromheterogeneoussettingstoacertainspe-cictarget.[6]introducedaformallanguageforencodingdifferencesandcommonalitiesbetweendomainsaccompaniedwithnecessaryorsufcientconditionsunderwhichtransportabilityofem-piricalndingsisfeasiblebetweentwodomains,asourceandatarget;then,theseconditionswereextendedforacompletecharacterizationfortransportabilityinonedomainwithunrestrictedexper-imentaldata[7,8].Subsequently,assumptionswererelaxedtoconsidersettingswhenonlylimitedexperimentsareavailableinthesourcedomain[9,10],furtherforwhenmultiplesourcedomains1 withunrestrictedexperimentalinformationareavailable[11,12],andthenformultipleheteroge-neoussourceswithlimitedanddistinctexperiments[13],whichwascalled“mz-transportability”.1Specically,themz-transportabilityproblemconcernswiththetransferofcausalknowledgefromaheterogeneouscollectionofsourcedomains=f1;:::;ngtoatargetdomain.Ineachdomaini2,experimentsoverasetofvariablesZicanbeperformed,andcausalknowledgegathered.In,potentiallydifferentfromi,onlypassiveobservationscanbecollected(thisconstraintwillbeweakened).TheproblemistoinferacausalrelationshipRinusingknowledgeobtainedin.Theproblemstudiedheregeneralizestheone-dimensionalversionoftransportabilitywithlimitedscopeandthemultipledimensionalwithunlimitedscopepreviouslystudied.Interestingly,whilecertaineffectsmightnotbeindividuallytransportabletothetargetdomainfromtheexperimentsinanyoftheavailablesources,combiningdifferentpiecesfromthevarioussourcesmayenabletheirestimation.Conversely,itisalsopossiblethateffectsarenotestimablefrommultipleexperimentsinindividualdomains,buttheyarefromexperimentsscatteredthroughoutdomains(discussedbelow).Thegoalofthispaperistoformallyunderstandtheconditionscausaleffectsinthetargetdo-mainare(non-parametrically)estimablefromtheavailabledata.Sufcientconditionsfor“mz-transportability”weregivenin[13],butthistreatmentfallsshortofprovidingguaranteeswhethertheseconditionsarealsonecessary,shouldbeaugmented,orevenreplacedbymoregeneralones.Thispaperestablishesthefollowingresults:Anecessaryandsufcientconditionfordecidingwhencausaleffectsinthetargetdomainareestimablefromboththestatisticalinformationavailableandthecausalinformationtransferredfromtheexperimentsinthedomains.Aproofthatthealgorithmproposedin[13]isinfactcompleteforcomputingthetransportformula,thatis,thestrategydevisedforcombiningtheempiricalevidencetosynthesizethetargetrelationcannotbeimprovedupon.Aproofthatthedo-calculusiscompleteforthemz-transportabilityclass.2BackgroundinTransportabilityInthissection,weconsiderothertransportabilityinstancesanddiscusstherelationshipwiththemz-transportabilitysetting.ConsiderFig.1(a)inwhichthenodeSrepresentsfactorsthatproducedifferencesbetweensourceandtargetpopulations.WeconductarandomizedtrialinLosAngeles(LA)andestimatethecausaleffectoftreatmentXonoutcomeYforeveryagegroupZ=z,denotedbyP(yjdo(x);z).WenowwishtogeneralizetheresultstothepopulationofNewYorkCity(NYC),butwendthedistributionP(x;y;z)inLAtobedifferentfromtheoneinNYC(callthelatterP(x;y;z)).Inparticular,theaverageageinNYCissignicantlyhigherthanthatinLA.HowarewetoestimatethecausaleffectofXonYinNYC,denotedR=P(yjdo(x))?23Theselectiondiagram–overlappingofthediagramsinLAandNYC–forthisexample(Fig.1(a))conveystheassumptionthattheonlydifferencebetweenthetwopopulationsarefactorsdeterminingagedistributions,shownasS!Z,whileage-speciceffectsP(yjdo(x);Z=z)areinvariantacrosspopulations.Difference-generatingfactorsarerepresentedbyaspecialsetofvariablescalledselectionvariablesS(orsimplyS-variables),whicharegraphicallydepictedassquarenodes().Fromthisassumption,theoverallcausaleffectinNYCcanbederivedasfollows:R=XzP(yjdo(x);z)P(z)=XzP(yjdo(x);z)P(z)(1)ThelastlineconstitutesatransportformulaforR;itcombinesexperimentalresultsobtainedinLA,P(yjdo(x);z),withobservationalaspectsofNYCpopulation,P(z),toobtainacausalclaim 1Traditionally,themachinelearningliteraturehasbeenconcernedaboutdiscrepanciesamongdomainsinthecontext,almostexclusively,ofpredictiveorclassicationtasksasopposedtolearningcausalorcounterfac-tualmeasures[14,15].Interestingly,recentworkonanticausallearningleveragesknowledgeaboutinvariancesoftheunderlyingdata-generatingstructureacrossdomains,movingtheliteraturetowardsmoregeneralmodal-itiesoflearning[16,17].2WewillusePx(yjz)interchangeablywithP(yjdo(x);z).3Weusethestructuralinterpretationofcausaldiagramsasdescribedin[18,pp.205](seealsoAppendix1).2 Figure1:(a)SelectiondiagramillustratingwhentransportabilityofR=P(yjdo(x))betweentwodomainsistriviallysolvedthroughsimplerecalibration.(b)Thesmallestdiagraminwhichacausalrelationisnottransportable.(c,d)SelectiondiagramsillustratingtheimpossibilityofestimatingRthroughindividualtransportabilityfromaandbevenwhenZ=fZ1;Z2g.IfexperimentsoverfZ2gisavailableinaandoverfZ1ginb,Ristransportable.(e,f)Selectiondiagramsillustratingoppositephenomenon–transportabilitythroughmultipledomainsisnotfeasible,butifZ=fZ1;Z2ginonedomainis.TheselectionvariablesSaredepictedassquarenodes().P(yjdo(x))aboutNYC.Inthistrivialexample,thetransportformulaamountstoasimplere-calibration(orre-weighting)oftheage-speciceffectstoaccountforthenewagedistribution.Ingeneral,however,amoreinvolvedmixtureofexperimentalandobservationalndingswouldbenecessarytoobtainanunbiasedestimateofthetargetrelationR.Incertaincasesthereisnowaytosynthesizeatransportformula,forinstance,Fig.1(b)depictsthesmallestexampleinwhichtransportabilityisnotfeasible(evenwithXrandomized).Ourgoalistocharacterizethesecases.Inrealworldapplications,itmayhappenthatonlyalimitedamountofexperimentalinformationcanbegatheredatthesourceenvironment.Thequestionariseswhetheraninvestigatorinpossessionofalimitedsetofexperimentswouldstillbeabletoestimatethedesiredeffectsatthetargetdomain.Toillustratesomeofthesubtleissuesthatmz-transportabilityentails,considerFig.1(c,d)whichconcernsthetransportofexperimentalresultsfromtwosources(fa;bg)toinfertheeffectofXonYin,R=P(yjdo(x)).Inthesediagrams,Xmayrepresentthetreatment(e.g.,cholesterollevel),Z1representsapre-treatmentvariable(e.g.,diet),Z2representsanintermediatevariable(e.g.,biomarker),andYrepresentstheoutcome(e.g.,heartfailure).AssumethatexperimentalstudiesrandomizingfZ2gcanbeconductedindomainaandfZ1gindomainb.AsimpleanalysiscanshowthatRcannotbetransportedfromeithersourcealone(evenwhenexperimentsareavailableoverbothvariables)[9].Still,combiningexperimentsfrombothsourcesallowsonetodeterminetheeffectinthetargetthroughthefollowingtransportformula[13]:P(yjdo(x))=Xz2P(b)(z2jx;do(Z1))P(a)(yjdo(z2))(2)ThistransportformulaisamixtureoftheexperimentalresultoverfZ1gfromb,P(b)(z2jx;do(Z1)),withtheresultoftheexperimentoverfZ2gina,P(a)(yjdo(z2)),andconsti-tuteaconsistentestimandofthetargetrelationin.FurtherconsiderFig.1(e,f)whichillustratestheoppositephenomenon.Inthiscase,ifexperimentsoverfZ2gareavailableindomainaandoverfZ1ginb,Risnottransportable.However,iffZ1;Z2gareavailableinthesamedomain,saya,RistransportableandequalsP(a)(yjx;do(Z1;Z2)),independentlyofthevaluesofZ1andZ2.Theseintriguingresultsentailtwofundamentalissuesthatwillbeansweredthroughoutthispaper.First,whetherthedo-calculusiscompleterelativetosuchproblems,thatis,whetheritwouldalwaysndatransportformulawheneversuchexists.Second,assumingthatthereexistsasequenceofapplicationsofdo-calculusthatachievesthereductionrequiredbymz-transportability,tondsuchasequencemaybecomputationalintractable,soanefcientwayisneededforobtainingsuchformula.3AGraphicalConditionformz-transportabilityThebasicsemanticalframeworkinouranalysisrestsonstructuralcausalmodelsasdenedin[18,pp.205],alsocalleddata-generatingmodels.Inthestructuralcausalframework[18,Ch.7],actionsaremodicationsoffunctionalrelationships,andeachactiondo(x)onacausalmodelMproducesanewmodelMx=hU;V;Fx;P(U)i,whereVisthesetofobservablevariables,Uisthesetofunobservablevariables,andFxisobtainedafterreplacingfX2FforeveryX2Xwithanewfunctionthatoutputsaconstantvaluexgivenbydo(x).Wefollowtheconventionsgivenin[18].Wedenotevariablesbycapitallettersandtheirrealizedvaluesbysmallletters.Similarly,setsofvariableswillbedenotedbyboldcapitalletters,sets3 ofrealizedvaluesbyboldsmallletters.Weusethetypicalgraph-theoreticterminologywiththecorrespondingabbreviationsDe(Y)G,Pa(Y)G,andAn(Y)G,whichwilldenoterespectivelythesetofobservabledescendants,parents,andancestorsofthenodesetYinG.AgraphGYwilldenotetheinducedsubgraphGcontainingnodesinYandallarrowsbetweensuchnodes.Finally,G XZ standsfortheedgesubgraphofGwhereallarrowsincomingintoXandallarrowsoutgoingfromZareremoved.Keytotheanalysisoftransportabilityisthenotionofidentiability[18,pp.77],whichexpressestherequirementthatcausaleffectsarecomputablefromacombinationofnon-experimentaldataPandassumptionsembodiedinacausaldiagramG.Causalmodelsandtheirinduceddiagramsareassociatedwithoneparticulardomain(i.e.,setting,population,environment),andthisrepresentationisextendedintransportabilitytocapturepropertiesoftwodomainssimultaneously.Thisispossibleifweassumethatthestructuralequationssharethesamesetofarguments,thoughthefunctionalformsoftheequationsmayvaryarbitrarily[7].4Denition1(SelectionDiagrams).LethM;Mibeapairofstructuralcausalmodelsrelativetodomainsh;i,sharingadiagramG.hM;MiissaidtoinduceaselectiondiagramDifDisconstructedasfollows:everyedgeinGisalsoanedgeinD;DcontainsanextraedgeSi!Viwhenevertheremightexistadiscrepancyfi6=fiorP(Ui)6=P(Ui)betweenMandM.Inwords,theS-variableslocatethemechanismswherestructuraldiscrepanciesbetweenthetwodo-mainsaresuspectedtotakeplace.5Armedwiththeconceptofidentiabilityandselectiondiagrams,mz-transportabilityofcausaleffectscanbedenedasfollows[13]:Denition2(mz-Transportability).LetD=fD(1);:::;D(n)gbeacollectionofselectiondiagramsrelativetosourcedomains=f1;:::;ng,andtargetdomain,respectively,andZi(andZ)bethevariablesinwhichexperimentscanbeconductedindomaini(and).LethPi;Iizibethepairofobservationalandinterventionaldistributionsofi,whereIiz=SZ0ZiPi(vjdo(z0)),andinananalogousmanner,hP;Izibetheobservationalandinterventionaldistributionsof.ThecausaleffectR=Px(y)issaidtobemz-transportablefromtoinDifPx(y)isuniquelycomputablefromSi=1;:::;nhPi;Iizi[hP;IziinanymodelthatinducesD.Whilethisdenitionmightappearconvoluted,itisnothingmorethanaformalizationofthestate-ment“RneedtobeuniquelycomputablefromtheinformationsetISalone.”Naturally,whenIShasmanycomponents(multipleobservationalandinterventionaldistributions),itbecomeslengthy.ThisrequirementofcomputabilityfromhP;IziandhPi;Iizifromallsourceshasasyntacticimageinthedo-calculus,whichiscapturedbythefollowingsufcientcondition:Theorem1([13]).LetD=fD(1);:::;D(n)gbeacollectionofselectiondiagramsrelativetosourcedomains=f1;:::;ng,andtargetdomain,respectively,andSirepresentsthecollectionofS-variablesintheselectiondiagramD(i).LetfhPi;IizigandhP;Iziberespectivelythepairsofobservationalandinterventionaldistributionsinthesourcesandtarget.TheeffectR=P(yjdo(x))ismz-transportablefromtoinDiftheexpressionP(yjdo(x);S1;:::;Sn)isreducible,usingtherulesofthedo-calculus,toanexpressioninwhich(1)do-operatorsthatapplytosubsetsofIizhavenoSi-variablesor(2)do-operatorsapplyonlytosubsetsofIz.ItisnotdifculttoseethatinFig.1(c,d)(andalsoinFig.1(e,f))asequenceofapplicationsoftherulesofdo-calculusindeedreachesthereductionrequiredbythetheoremandyieldsatransportformulaasshowninSection2.Itisnotobvious,however,whethersuchsequenceexistsinFig.2(a,b)whenexperimentsoverfXgareavailableinaandfZginb,andifitdoesnotexist,itisalsonotclearwhetherthiswouldimplytheinabilitytotransport.ItturnsoutthatinthisspecicexamplethereisnotsuchsequenceandthetargetrelationRisnottransportable,whichmeansthatthereexisttwomodelsthatareequallycompatiblewiththedata(i.e.,bothcouldgeneratethesamedataset)whileeachmodelentailsadifferentanswerfortheeffectR(violatingtheuniquenessrequirementofDef.2).6Todemonstratethisfactformally,weshowtheexistenceoftwostructural 4Asdiscussedinthereference,theassumptionofnostructuralchangesbetweendomainscanberelaxed,butsomestructuralassumptionsregardingthediscrepanciesbetweendomainsmuststillhold(e.g.,acyclicity).5Transportabilityassumesthatenoughstructuralknowledgeaboutbothdomainsisknowninordertosub-stantiatetheproductionoftheirrespectivecausaldiagrams.Intheabsenceofsuchknowledge,causaldiscoveryalgorithmsmightbeusedtoinferthediagramsfromdata[19,18].6Thisisusuallyanindicationthatthecurrentstateofscienticknowledgeabouttheproblem(encodedintheformofaselectiondiagram)doesnotconstrainttheobserveddistributionsinsuchawaythatananswerisentailedindependentlyofthedetailsofthefunctionsandprobabilityovertheexogenous.4 Figure2:(a,b)SelectiondiagramsinwhichisnotpossibletotransportR=P(yjdo(x))withexperimentsoverfXginaandfZginb.(c,d)Exampleofdiagramsinwhichsomepathsneedtobeextendedforsatisfyingthedenitionofmz-shedge.modelsM1andM2suchthatthefollowingequalitiesandinequalitybetweendistributionshold,8�����&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;:P(a)M1(X;Z;Y)=P(a)M2(X;Z;Y);P(b)M1(X;Z;Y)=P(b)M2(X;Z;Y);P(a)M1(Z;Yjdo(X))=P(a)M2(Z;Yjdo(X));P(b)M1(X;Yjdo(Z))=P(b)M2(X;Yjdo(Z));PM1(X;Z;Y)=PM2(X;Z;Y);(3)forallvaluesofX,Z,andY,andPM1(Yjdo(X))6=PM2(Yjdo(X));(4)forsomevalueofXandY.LetusassumethatallvariablesinU[Varebinary.LetU1;U22UbethecommoncausesofXandYandZandY,respectively;letU3;U42UbetherandomdisturbancesexclusivetoZandY,respectively,andU5;U62UbeextrarandomdisturbancesexclusivetoY.LetSaandSbindexthemodelinthefollowingway:thetupleshSa=1;Sb=0i,hSa=0;Sb=1i,hSa=0;Sb=0irepresentdomainsa,b,and,respectively.Denethetwomodelsasfollows:M1=8&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;:X=U1Z=U2(U3^Sa)Y=((XZU1U2(U4^Sb))^U5)+(:U5^U6)M2=8&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;:X=U1Z=U2(U3^Sa)Y=((ZU2(U4^Sb))^U5)(:U5^U6)whererepresentstheexclusiveorfunction.BothmodelsagreeinrespecttoP(U),whichisdenedasP(Ui)=1=2,i=1;:::;6.ItisnotdifculttoevaluatethesemodelsandnotethattheconstraintsgiveninEqs.(3)and(4)areindeedsatised(includingpositivity),theresultfollows.7GiventhatourgoalistodemonstratetheconverseofTheorem1,wecollectdifferentexamplesofnon-transportability,asthepreviousone,andtrytomakesensewhetherthereisapatterninsuchcasesandhowtogeneralizethemtowardsacompletecharacterizationofmz-transportability.Onesyntacticsubtaskofmz-transportabilityistodeterminewhethercertaineffectsareidentiableinsomesourcedomainswhereinterventionaldataisavailable.Therearetwofundamentalresultsdevelopedforidentiabilitythatwillberelevantformz-transportabilityaswell.First,weshouldconsiderconfoundedcomponents(orc-components),whichweredenedin[20]andstandforaclusterofvariablesconnectedthroughbidirectededges(whicharenotseparablethroughtheob-servablesinthesystem).Onekeyresultisthateachcausalgraph(andsubgraphs)inducesanuniqueC-componentdecomposition([20,Lemma11]).Thisdecompositionwasindeedinstrumentalforaseriesofconditionsforordinaryidentication[21]andtheinabilitytorecursivelydecomposeacertaingraphwaslaterusedtoprovecompleteness.Denition3(C-component).LetGbeacausaldiagramsuchthatasubsetofitsbidirectedarcsformsaspanningtreeoverallverticesinG.ThenGisaC-component(confoundedcomponent).Subsequently,[22]proposedanextensionofC-componentscalledC-forests,essentiallyenforcingthateachC-componenthastobeaspanningforestandclosedunderancestralrelations[20]. 7Toamoresophisticatedargumentonhowtoevaluatethesemodels,seeproofsinappendix3.5 YXZ(c)XY(b) Denition4(C-forest).LetGbeacausaldiagramwhereYisthemaximalrootset.ThenGisaY-rootedC-forestifGisaC-componentandallobservablenodeshaveatmostonechild.Forconcreteness,considerFig.1(c)andnotethatthereexistsaC-forestovernodesfZ1;X;Z2gandrootedinfZ2g.ThereexistsanotherC-forestovernodesfZ1;X;Z2;YgrootedinfYg.ItisalsothecasethatfZ2gandfYgarethemselvestrivialC-forests.WhenwehaveapairofC-forestsasfZ1;X;Z2gandfZ2gorfZ1;X;Z2;YgandfYg–i.e.,therootsetdoesnotintersectthetreatmentvariables;thesestructuresarecalledhedgesandidentiabilitywasshowntobeinfeasiblewheneverahedgeexists[22].Clearly,despitetheexistenceofhedgesinFig.1(c,d),theeffectsofinterestwereshowntobemz-transportable.Thisexampleisanindicationthathedgesdonotcaptureinanimmediatewaythestructureneededforcharacterizingmz-transportability–i.e.,agraphmightbeahedge(orhaveahedgeasanedgesub–graph)butthetargetquantitymightstillbemz-transportable.Basedontheseobservations,weproposethefollowingdenitionthatmayleadtotheboundariesoftheclassofmz-transportablerelations:Denition5(mz-shedge).LetD=(D(1);:::;D(n))beacollectionofselectiondiagramsrel-ativetosourcedomains=(1;:::;n)andtargetdomain,respectively,SirepresentsthecollectionofS-variablesintheselectiondiagramD(i),andletD()bethecausaldiagramof.LetfhPi;Iizigbethecollectionofpairsofobservationalandinterventionaldistributionsoffig,whereIiz=SZ0ZiPi(vjdo(z0)),andinananalogousmanner,hP;Izibetheobservationalandinterventionaldistributionsof,forZithesetofexperimentalvariablesini.ConsiderapairofR-rootedC-forestsF=hF;F0isuchthatF0F,F0\X=;,F\X6=;,andRAn(Y)G X(calledahedge[22]).WesaythattheinducedcollectionofpairsofR-rootedC-forestsovereachdiagram,hF();F(1);:::;F(n)i,isanmz-shedgeforPx(y)relativetoexperiments(Iz;I1z;:::;Inz)iftheyareallhedgesandoneofthefollowingconditionsholdforeachdomaini,i=f;1;:::;ng:1.ThereexistsatleastonevariableofSipointingtotheinduceddiagramF0(i),or2.(F(i)nF0(i))\Ziisanemptyset,or3.ThecollectionofpairsofC-forestsinducedoverdiagrams,hF();F(1);:::;F(i)nZi;:::;F(n)i,isalsoanmz-shedgerelativeto(Iz;I1z;:::;Iiznzi;:::;Inz),whereZi=(F(i)nF0(i))\Zi.Furthermore,wecallmz-shedgethemz-shedgeinwhichthereexistonedirectedpathfromRn(R\De(X)F)to(R\De(X)F)notpassingthroughX(seealsoappendix3).Thedenitionofmz-shedgemightappearinvolved,butitisnothingmorethanthearticulationofthecomputabilityrequirementofDef.2(andimplicitlythesyntacticgoalofThm.1)inamoreexplicitgraphicalfashion.Specically,foracertainfactorQineededforthecomputationoftheeffectQ=P(yjdo(x)),inatleastonedomain,(i)itshouldbeenforcedthattheS-nodesareseparablefromtheinducingrootsetofthecomponentinwhichQibelongs,andfurther,(ii)theexperimentsavailableinthisdomainaresufcientforsolvingQi.Forinstance,assumingwewanttocomputeQ=P(yjdo(x))inFig.1(c,d),Qcanbedecomposedintotwofactors,Q1=Pz1;x(z2)andQ2=Pz1;x;z2(y).ItisthecasethatforfactorQ1,(i)holdstrueinband(ii)theexperimentsavailableoverZ1areenoughtoguaranteethecomputabilityofthisfactor(similaranalysisappliestoQ2)–i.e.,thereisnomz-shedgeandQiscomputablefromtheavailabledata.Def.5alsoasksfortheexplicitexistenceofapathfromthenodesintherootsetRn(R\De(X)F)to(R\De(X)F),asimpleexamplecanhelptoillustratethisrequirement.ConsiderFig.2(c)andthegoalofcomputingQ=P(yjdo(x))withoutextraexperimentalinformation.Thereex-istsahedgeforQinducedoverfX;Z;YgwithoutthenodeW(notethatfWgisac-componentitself)andtheinducedgraphGfX;Z;Ygindeedleadstoacounter-exampleforthecomputabilityofP(z;yjdo(x)).Usingthissubgraphalone,however,itwouldnotbepossibletoconstructacounter-exampleforthemarginaleffectP(yjdo(x)).DespitethefactthatP(z;yjdo(x))isnotcomputablefromP(x;z;y),thequantityP(yjdo(x))isidentiableinGfX;Z;Yg,andsoanystructuralmodelcompatiblewiththissubgraphwillgeneratethesamevalueunderthemarginalizationoverZfromP(z;yjdo(x)).Also,itmighthappenthattherootsetRmustbeaugmented(Fig.2(d)),soweprefertoaddthisrequirementexplicitlytothedenition.(Therearemoreinvolvedscenariosthat6 PROCEDURETRmz(y;x;P;I;S;W;D)INPUT:x;y:valueassignments;P:localdistributionrelativetodomainS(S=0indexes)andactiveexperimentsI;W:weightingscheme;D:backboneofselectiondiagram;Si:selectionnodesini(S0=;relativeto);[Thefollowingsetanddistributionsaregloballydened:Zi;P;P(i)Zi.]OUTPUT:Px(y)intermsofP;PZ;P(i)ZiorFAIL(D;C0).1ifx=;,returnPVnYP.2ifVnAn(Y)D6=;,returnTRmz(y;x\An(Y)D;PVnAn(Y)DP;I;S;W;DAn(Y)).3setW=(VnX)nAn(Y)D X.ifW6=;,returnTRmz(y;x[w;P;I;S;W;D).4ifC(DnX)=fC0;C1;:::;Ckg,returnPVnfY;XgQiTRmz(ci;vnci;P;I;S;W;D).5ifC(DnX)=fC0g,6ifC(D)6=fDg,7ifC02C(D),returnQijVi2C0PVnV(i)DP=PVnV(i�1)DP.8if(9C0)C0C02C(D),forfijVi2C0g,seti=i[v(i�1)DnC0.returnTRmz(y;x\C0,QijVi2C0P(VijV(i�1)D\C0;i);I;S;W;C0).9else,10ifI=;,fori=0;:::;jDj,if�(Si??YjX)D(i) X^(Zi\X6=;),Ei=TRmz(y;xnzi;P;Zi\X;i;W;DnfZi\Xg).11ifjEj�0,returnPjEji=1w(j)iEi.12else,FAIL(D;C0).Figure3:Modiedversionofidenticationalgorithmcapableofrecognizingmz-transportability.weprefertoomitforthesakeofpresentation.)AfteraddingthedirectedpathfromZtoYthatpassesthroughW,wecanconstructthefollowingcounter-exampleforQ:M1=8���&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;:X=U1Z=U1U2W=((ZU3)_B)(B^(1Z))Y=((XWU2)^A)(A_(1XWU2));M2=8&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;:X=U1Z=U2W=((ZU3)_B)(B^(1Z))Y=((WU2)^A)(A_(1WU2));withP(Ui)=1=2;8i,P(A)=P(B)=1=2.Itisnotimmediatetoshowthatthetwomodelsproducethedesiredproperty.RefertoAppendix2foraformalproofofthisstatement.Giventhatthedenitionofmz-shedgeisjustiedandwell-understood,wecannowstatetheconnectionbetweenhedgesandmz-shedgesmoredirectly(theproofcanbefoundinAppendix3):Theorem2.IfthereisahedgeforPx(y)inGandnoexperimentaldataisavailable(i.e.,Iz=fg),thereexistsanmz-shedgeforPx(y)inG.Wheneveronedomainisconsideredandnoexperimentaldataisavailable,thisresultstatesthatamz-shedgecanalwaysbeconstructedfromahedge,whichimpliesthatwecanoperatewithmz-shedgesfromnowon(theconverseholdsforZ=fg).Finally,wecanconcentrateonthemostgeneralcaseofmz-shedgeswithexperimentaldatainmultipledomainsasstatedinthesequel:Theorem3.LetD=fD(1);:::;D(n)gbeacollectionofselectiondiagramsrelativetosourcedomains=f1;:::;ng,andtargetdomain,respectively,andfIizg,fori=f;1;:::;ngdenedappropriately.Ifthereisanmz-shedgefortheeffectR=Px(y)relativetoexperiments(Iz;I1z;:::;Inz)inD,Risnotmz-transportablefromtoinD.Thisisapowerfulresultthatstatesthattheexistenceofamz-shedgeprecludesmz-transportability.(Theproofofthisstatementissomewhatinvolved,seethesupplementarymaterialformoredetails.)Forconcreteness,letusconsidertheselectiondiagramsD=(D(a);D(b))relativetodomainsaandbinFig.2(a,b).Ourgoalistomz-transportQ=P(yjdo(x))withexperimentsoverfXginaandfZginb.Itisthecasethatthereexistsanmz-shedgerelativetothegivenexperiments.Towitness,rstnotethatF0=fY;ZgandF=F0[fXg,andalsothatthereexistsaselectionvariableSpointingtoF0inbothdomains–therstconditionofDef.5issatised.Thisisatrivialgraphwith3variablesthatcanbesolvedbyinspection,butitissomewhatinvolvedtoefcientlyevaluatetheconditionsofthedenitioninmoreintricatestructures,whichmotivatesthesearchforaprocedureforrecognizingmz-shedgesthatcanbecoupledwiththeprevioustheorem.7 4CompleteAlgorithmformz-transportabilityThereexistsanextensiveliteratureconcernedwiththeproblemofcomputabilityofcausalrelationsfromacombinationofassumptionsanddata[21,22,7,13].Inthissection,webuildontheworksthattreatthisproblembygraphicalmeans,andweconcentrateparticularlyinthealgorithmcalledTRmzconstructedin[13](seeFig.3)thatfollowedsomeoftheresultsin[21,22,7].ThealgorithmTRmztakesasinputacollectionofselectiondiagramswiththecorrespondingex-perimentaldatafromthecorrespondingdomains,anditreturnsatransportformulawheneveritisabletoproduceone.Themainideaofthealgorithmistoleveragethec-componentfactorization[20]andrecursivelydecomposethetargetrelationintomanageablepieces(line4),soastotrytosolveeachofthemseparately.Wheneverthisstandardevaluationfailsinthetargetdomain(line6),TRmztriestousetheexperimentalinformationavailablefromthetargetandsourcedomains(line10).(ForaconcreteviewofhowTRmzworks,seetherunningexamplein[13,pp.7].)Inasystematicfashion,thealgorithmbasicallyimplementsthedeclarativeconditiondelineatedinTheorem1.TRmzwasshowntobesound[13,Thm.3],butthereisnotheoreticalguaranteeonwhetherfailureinndingatransportformulaimpliesitsnon-existenceandperhaps,thecompletelackoftransportability.Thisguaranteeispreciselywhatwestateinthesequel.Theorem4.AssumeTRmzfailstotransporttheeffectPx(y)(exitswithfailureexecutingline12).ThenthereexistsX0X,Y0Y,suchthatthegraphpairD;C0returnedbythefailconditionofTRmzcontainsasedgesubgraphsC-forestsF,F'thatspanamz-shedgeforPx0(y0).Proof.LetDbethesubgraphlocaltothecallinwhichTRmzfailed,andRbetherootsetofD.ItispossibletoremovesomedirectedarrowsfromDwhilepreservingRasroot,whichresultinaR-rootedc-forestF.SincebyconstructionF0=F\C0isclosedunderdescendentsandonlydirectedarrowswereremoved,bothF;F0areC-forests.AlsobyconstructionRAn(Y)G XtogetherwiththefactthatXandYfromtherecursivecallareclearlysubsetsoftheoriginalinput.Beforefailure,TRmzevaluatedfalseconsecutivelyatlines6,10,and11,anditisnotdifculttoseethatanS-nodepointstoF0ortherespectiveexperimentswerenotabletobreakthelocalhedge(lines10and11).Itremainstobeshowedthatthismz-shedgecanbestretchedtogenerateamz-shedge,butnowthesameconstructiongiveninThm.2canbeapplied(seealsosupplementarymaterial). Finally,wearereadytostatethecompletenessofthealgorithmandthegraphicalcondition.Theorem5(completeness).TRmziscomplete.Corollary1(mz-shedgecharacterization).Px(y)ismz-transportablefromtoinDifandonlyifthereisnotmz-shedgeforPx0(y0)inDforanyX0XandY0Y.Furthermore,weshowbelowthatthedo-calculusiscompleteforestablishingmz-transportability,whichmeansthatfailureintheexhaustiveapplicationofitsrulesimpliesthenon-existenceofamappingfromtheavailabledatatothetargetrelation(i.e.,thereisnomz-transportformula),inde-pendentlyofthemethodusedtoobtainsuchmapping.Corollary2(do-calculuscharacterization).Therulesofdo-calculustogetherwithstandardproba-bilitymanipulationsarecompleteforestablishingmz-transportabilityofcausaleffects.5ConclusionsInthispaper,weprovidedacompletecharacterizationintheformofagraphicalconditionforde-cidingmz-transportability.Wefurthershowedthattheprocedureintroducedin[1]forcomputingthetransportformulaiscomplete,whichmeansthatthesetoftransportableinstancesidentiedbythealgorithmcannotbebroadenedwithoutstrengtheningtheassumptions.Finally,weshowedthatthedo-calculusiscompleteforthisclassofproblems,whichmeansthatndingaproofstrategyinthislanguagesufcestosolvetheproblem.Thenon-parametriccharacterizationestablishedinthispapergivesrisetoanewsetofresearchquestions.Whileouranalysisaimedatachievingunbiasedtransportunderasymptoticconditions,additionalconsiderationsneedtobetakenintoaccountwhendealingwithnitesamples.Specically,whensamplesizesvarysignicantlyacrossstudies,statis-ticalpowerconsiderationsneedtobeinvokedalongwithbiasconsiderations.Furthermore,whennotransportformulaexists,approximationtechniquesmustberesortedto,forexample,replacingtherequirementofnon-parametricanalysiswithassumptionsaboutlinearityormonotonicityofcer-tainrelationshipsinthedomains.Thenonparametriccharacterizationprovidedinthispapershouldserveasaguidelineforsuchapproximationschemes.8 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