Causal Reasoning with Ancestral Graphs - Journal of Machine

38 Pages·2012·0.28 MB·English

JournalofMachineLearningResearch9(2008)1437-1474 Submitted6/07;Revised2/08;Published7/08
Causal Reasoning with Ancestral Graphs
JijiZhang [email protected]
DivisionoftheHumanitiesandSocialSciences
CaliforniaInstituteofTechnology
Pasadena,CA91106,USA
Editor:GregoryF.Cooper
Abstract
Causal reasoning is primarily concerned with what would happen to a system under external in-
terventions. Inparticular,weareofteninterestedinpredictingtheprobabilitydistributionofsome
randomvariablesthatwouldresultifsomeothervariableswereforcedtotakecertainvalues. One
prominentapproachtotacklingthisproblemisbasedoncausalBayesiannetworks,usingdirected
acyclicgraphsascausaldiagramstorelatepost-interventionprobabilitiestopre-interventionprob-
abilities that are estimable from observational data. However, such causal diagrams are seldom
fully testable given observational data. In consequence, many causal discovery algorithms based
ondata-miningcanonlyoutputanequivalenceclassofcausaldiagrams(ratherthanasingleone).
Thispaperisconcernedwithcausalreasoninggivenanequivalenceclassofcausaldiagrams,rep-
resentedbya(partial)ancestralgraph. Wepresenttwomainresults. ThefirstresultextendsPearl
(1995)’scelebrateddo-calculustothecontext ofancestralgraphs. Inthesecondresult, wefocus
onakeycomponentofPearl’scalculus—thepropertyofinvarianceunderinterventions,andgive
stronger graphical conditions for this property than those implied by the first result. The second
resultalsoimprovestheearlier,similarresultsduetoSpirtesetal. (1993).
Keywords: ancestralgraphs,causalBayesiannetwork,do-calculus,intervention
1. Introduction
Intellectual curiosity aside, an important reason for people to care about causality or causal expla-
nationistheneed—forexample,inpolicyassessmentordecisionmaking—topredictconsequences
of actions or interventions before actually carrying them out. Sometimes we can base that predic-
tion on similar past interventions or experiments, in which case the inference is but an instance of
the classical inductive generalization. Other times, however, we do not have access to sufficient
controlledexperimentalstudiesforvariousreasons,andcanonlymakepassiveobservationsbefore
interventionstakeplace. Underthelattercircumstances,weneedtoreasonfrompre-interventionor
observationaldatatoapost-interventionsetting.
AprominentmachineryforcausalreasoningofthiskindisknownascausalBayesiannetwork
(Spirtesetal.,1993;Pearl,2000),whichwewilldescribeinmoredetailinthenextsection. Inthis
framework,oncethecausalstructure—representedbyadirectedacyclicgraph(DAG)overasetof
attributes or random variables—is fully given, every query about post-intervention probability can
beansweredintermsofpre-interventionprobabilities. So,ifeveryvariableinthecausalstructureis
(passively)observed,theobservationaldatacanbeusedtoestimatethepost-interventionprobability
ofinterest.
(cid:13)c2008JijiZhang.
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Complications come in at least two ways. First, some variables in the causal DAG may be
unobserved, or worse, unobservable. So even if the causal DAG (with latent variables) is fully
known, we may not be able to predict certain intervention effects because we only have data from
themarginaldistributionovertheobservedvariablesinsteadofthejointdistributionoverallcausally
relevantvariables. Thequestionis whatpost-interventionprobabilityis oris notidentifiablegiven
a causal DAG with latent variables. Much of Pearl’s work (Pearl, 1995, 1998, 2000), and more
recentlyTianandPearl(2004)areparadigmaticattemptstoaddressthisproblem.
Second, the causal structure is seldom, if ever, fully known. In the situation we are concerned
withinthispaper,wherenosubstantialbackgroundknowledgeorcontrolledstudyisavailable,we
have to relyuponobservationaldatato informus aboutcausalstructure. Thefamiliarcurseis that
very rarely can observational data determine a unique causal structure, and many causal discovery
algorithms in the literature output an equivalence class of causal structures based on observational
data (Spirtes et al., 1993; Meek, 1995a; Spirtes et al., 1999; Chickering, 2002).1 Different causal
structures in the class may or may not give the same answer to a query about post-intervention
probability. Forasimpleillustration,considertwocausalBayesiannetworks(seeSection2below),
X !Y ! Z and X Y ! Z, over three variables X;Y and Z. The two causal structures are
indistinguishable (without strong parametric assumptions) by observational data. Suppose we are
interested in the post-intervention probability distribution ofY given that X is manipulated to take
some fixed value x. The structure X !Y !Z entails that the post-intervention distribution ofY
is identical to the pre-intervention distribution of Y conditional on X = x, whereas the structure
X Y !Z entails that the post-intervention distribution ofY is identical to the pre-intervention
marginaldistributionofY. Sothetwostructuresgivedifferentanswerstothisparticularquery. By
contrast, if we are interested in the post-intervention distribution of Z under an intervention onY,
thetwostructuresgivethesameanswer.
The matter becomes formidably involved when both complications are present. Suppose we
observeasetofrandomvariablesO,butforallweknow,theunderlyingcausalstructuremayinvolve
extralatentvariables. Wewillnotworryabouttheestimationofthepre-interventiondistributionof
Ointhispaper,sowemaywellassumeforsimplicitythatthepre-interventiondistributionofOis
known. Butweareinterestedinqueriesaboutpost-interventionprobability,suchastheprobability
of Y conditional on Z that would result under an intervention on X (where X;Y;Z (cid:18) O). The
questioniswhetherandhowwecananswersuchqueriesfromthegivenpre-interventiondistribution
ofO.
Thisproblemisnaturallydividedintotwoparts. Thefirstpartiswhatsomecausaldiscoveryal-
gorithmsattempttoachieve,namely,tolearnsomethingaboutthecausalstructure—usuallyfeatures
sharedbyallcausalstructuresinanequivalenceclass—fromthepre-interventiondistributionofO.
The second part is to figure out, given the learned causal information, whether a post-intervention
probabilityisidentifiableintermsofpre-interventionprobabilities.
Thispaperprovidessomeresultsconcerningthesecondpart,assumingtheavailablecausalin-
formation is summarized in a (partial) ancestral graph. Ancestral graphical models (Richardson
and Spirtes, 2002, 2003) have proved to be an elegant and useful surrogate for DAG models with
latentvariables(moredetailsfollowinSection3),nottheleastbecauseprovablycorrectalgorithms
are available for learning an equivalence class of ancestral graphs represented by a partial ances-
tral graph from the pre-intervention distribution of the observed variables—in particular, from the
1.Therecentworkonlinearnon-Gaussianstructuralequationmodels(Shimizuetal.,2006)isanexception.However,
wedonotmakeparametricassumptionsinthispaper.
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conditionalindependenceanddependencerelationsimpliedbythedistribution(Spirtesetal.,1999;
Zhang,forthcoming).
We have two main results. First, we extend the do-calculus of Pearl (1995) to the context of
ancestralgraphs(Section4),sothattheresultingcalculusisbasedonanequivalenceclassofcausal
DAGswithlatentvariablesratherthanasingleone. Second,wefocusonakeycomponentofPearl’s
calculus—thepropertyofinvarianceunderinterventionsstudiedbySpirtesetal. (1993), andgive
stronger graphical conditions for this property than those implied by the first result (Section 5).
OurresultimprovesupontheSpirtes-Glymour-Scheinesconditionsforinvarianceformulatedwith
respecttotheso-calledinducingpathgraphs,whoserelationshipwithancestralgraphsisdiscussed
inAppendixA.
2. CausalBayesianNetwork
ABayesiannetworkforasetofrandomvariablesVconsistsofapairhG;Pi,whereG isadirected
acyclicgraph(DAG)withVasthesetofvertices,andPisthejointprobabilityfunctionofV,such
thatPfactorizesaccordingtoG asfollows:
(cid:213)
P(V)= P(YjPa (Y))
G
Y2V
where Pa (Y) denotes the set of parents ofY in G. In a causal Bayesian network, the DAG G is
G
interpreted causally, as a representation of the causal structure over V. That is, for X;Y 2V, an
arrowfromX toY (X !Y)inG meansthatX hasadirectcausalinfluenceonY relativetoV. We
refertoacausallyinterpretedDAGasacausalDAG.Thepostulatethatthe(pre-intervention)joint
distributionPfactorizesaccordingtothecausalDAGG isknownasthecausalMarkovcondition.
What about interventions? For simplicity, let us focus on what Pearl (2000) calls atomic
interventions—interventions that fix the values of the target variables—though the results in Sec-
tion 5 also apply to more general types of interventions (such as interventions that confer a non-
degenerate probability distribution on the target variables). In the framework of causal Bayesian
network, an intervention on X(cid:18)V is supposed to be effective in the sense that the value of X is
completelydeterminedbytheintervention,andlocalinthesensethattheconditionaldistributions
of other variables (variables not in X) given their respective parents in the causal DAG are not af-
fected by the intervention. Graphically, such an intervention amounts to erasing all arrows into X
in the causal DAG (because variables in X do not depend on their original parents any more), but
otherwisekeepingthegraphasitis. Callthismodifiedgraphthepost-interventioncausalgraph.
Based on this understanding of interventions, the following postulate has been proposed by
severalauthorsinvariousforms(Robins,1986;Spirtesetal.,1993;Pearl,2000):
Intervention Principle Given a causal DAG G over V and a (pre-intervention) joint distri-
butionP thatfactorizesaccordingto G, thepost-interventiondistributionP (V)—thatis,
X:=x
thejointdistributionofVafterX(cid:18)Varemanipulatedtovaluesxbyanintervention—takes
asimilar,truncatedformoffactorization,asfollows:
(cid:213) P(YjPa (Y)) forvaluesofVconsistentwithX=x;
P (V)= Y2VnX G
X:=x (cid:26) 0 otherwise:
Note that in the case of a null intervention (when X=Ø), the intervention principle implies the
factorizationofthepre-interventiondistributionPaccordingtoG,whichisjustthecausalMarkov
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condition. SotheinterventionprinciplegeneralizesthecausalMarkovcondition: itassumesthatthe
post-intervention distribution also satisfies the causal Markov condition with the post-intervention
causalgraph.
Bytheinterventionprinciple,oncethecausalDAGisgiven,thepost-interventionjointdistribu-
tion can be calculated in terms of pre-intervention probabilities.2 So if every variable is observed,
andhencethosepre-interventionprobabilitiescanbeestimated,anypost-interventionprobabilityis
estimableaswell.
It is time to recall the two complications mentioned in the last section. First, the intervention
principleisonlyplausiblewhenthegivensetofvariablesiscausallysufficient. Hereiswhatcausal
sufficiencymeans. GivenasetofvariablesV,andtwovariablesA;B2V,avariableC (notneces-
sarilyincludedinV)iscalledacommondirectcauseofAandBrelativetoVifChasadirectcausal
influenceon A and also a direct causalinfluenceon B relative to V[fCg. V is said to be causally
sufficientif for every pair of variablesV ;V 2V, every commondirect causeofV andV relative
1 2 1 2
to V is also a member of V. It is well known that the causal Markov condition tends to fail for a
causally insufficient set of variables (Spirtes et al., 1993), and even more so with the intervention
principle. Butinmostrealsituations,thereisnoreasontoassumethatthesetofobservedvariables
iscausallysufficient,sothecausalBayesiannetworkmaywellinvolvelatentvariables.
Second, the causal DAG is not fully learnable with observational, pre-intervention data. The
causaldiscoveryalgorithmsintheliterature—someofwhichareprovablycorrectinthelargesam-
ple limit assuming the causal Markov condition and its converse, causal Faithfulness condition—
typically return an equivalence class of DAGs that imply the same conditional independence rela-
tionsamongtheobservedvariables(accordingtotheMarkovcondition),withsomecausalfeatures
incommonthatconstitutethelearnedcausalinformation. Givensuchlimitedcausalinformation,a
post-interventionprobabilitymayormaynotbeuniquelyidentifiable.
Takingbothcomplicationsintoaccount,theinterestingquestionisthis: whatcausalreasoningis
warrantedgiventhecausalinformationlearnablebyalgorithmsthatdonotassumecausalsufficiency
forthesetofobservedvariables,suchastheFCIalgorithmpresentedinSpirtesetal.(1999)? Before
weexplorethequestion,letusmakeitalittlemoreprecisewiththeformalismofancestralgraphs.
3. AncestralGraphicalModels
Ancestral graphical models are motivated by the need to represent data generating processes that
mayinvolvelatentconfoundersand/orselectionbias,3 withoutexplicitlymodellingtheunobserved
variables (Richardson and Spirtes, 2002). We do not deal with selection bias in this paper, so we
useonlypartofthemachinery.
A(directed)mixedgraphisavertex-edgegraphthatmaycontaintwokindsofedges: directed
edges(!)andbi-directededges($). Betweenanytwoverticesthereisatmostoneedge. Thetwo
ends of an edge we call marks. Obviously there are two kinds of marks: arrowhead (>) and tail
((cid:0)). The marks of a bi-directededge are both arrowheads, and a directed edge has one arrowhead
2.Atechnicalissueisthatsomeconditionalprobabilitiesmaybeundefinedinthepre-interventiondistribution. Inthis
paperweignorethatissuebyassumingthatthepre-interventiondistributionisstrictlypositive. Otherwisewejust
needtoaddtheproviso“whenalltheconditionalprobabilitiesinvolvedaredefined”toallourresults.
3.Roughlyspeaking,thereisselectionbiasiftheprobabilityofaunitbeingsampleddependsoncertainpropertiesof
theunit.Thekindofselectionbiasthatisespeciallytroublesomeforcausalinferenceiswhentwoormoreproperties
ofinterestaffecttheprobabilityofbeingsampled,givingriseto“misleading”associationsinthesample.
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A B A B
C D C D
(a) (b)
Figure1: (a)anancestralgraphthatisnotmaximal;(b)amaximalancestralgraph.
and one tail. We say an edge is into (or out of) a vertex if the mark of the edge at the vertex is an
arrowhead(ortail).
Two vertices are said to be adjacentin a graph if there is an edge (of any kind) betweenthem.
Given a mixed graph G and two adjacent vertices X,Y therein, X is called a parent ofY andY a
child of X if X !Y is in G; X is called a spouse ofY (andY a spouse of X) if X $Y is in G. A
pathinG isasequenceofdistinctverticeshV ;:::;V isuchthatforall0(cid:20)i(cid:20)n(cid:0)1,V andV are
0 n i i+1
adjacentinG. AdirectedpathfromV toV inG isasequenceofdistinctverticeshV ;:::;V isuch
0 n 0 n
thatforall0(cid:20)i(cid:20)n(cid:0)1,V isaparentofV inG. X iscalledanancestorofY andY adescendant
i i+1
ofX ifX =Y orthereisadirectedpathfromX toY. WeusePa ;Ch ;Sp ;An ;De todenote
G G G G G
the set of parents, children, spouses, ancestors, and descendants of a vertex in G, respectively. A
directedcycleoccursinG whenY !X isinG andX 2An (Y). Analmostdirectedcycleoccurs
G
whenY $X isinG andX 2An (Y).4
G
Given a path p=hV ;:::;V i with n>1,V (1(cid:20)i(cid:20)n(cid:0)1) is a collider on p if the two edges
0 n i
incidenttoV arebothintoV,thatis,haveanarrowheadatV;otherwiseitiscalledanoncollideron
i i i
p. InFigure1(a),forexample,BisacollideronthepathhA;B;Di,butisanon-collideronthepath
hC;B;Di. Acolliderpathisapathonwhicheveryvertexexceptfortheendpointsisacollider. For
example, in Figure 1(a), the path hC;A;B;Di is a collider path because both A and B are colliders
on the path. Let L be any subset of vertices in the graph. An inducing path relative to L is a path
onwhicheveryvertexnotinL(exceptfortheendpoints)isacollideronthepathandeverycollider
isanancestorofanendpointofthepath. Forexample,anysingle-edgepathistriviallyaninducing
path relative to any set of vertices (because the definition does not constrain the endpoints of the
path). InFigure1(a),thepathhC;B;DiisaninducingpathrelativetofBg,butnotaninducingpath
relativetotheemptyset(becauseBisnotacollider). However,thepathhC;A;B;Diisaninducing
path relative to the empty set, because both A and B are colliders on the path, A is an ancestor of
D, and B is an ancestor ofC. To simplify terminology, we will henceforth refer to inducing paths
relativetotheemptysetsimplyasinducingpaths.5
Definition1(MAG) Amixedgraphiscalledamaximalancestralgraph(MAG)if
i. thegraphdoesnotcontainanydirectedoralmostdirectedcycles(ancestral);and
4.The terminologyof “almostdirectedcycle”is motivated by the fact that removingthe arrowheadatY onY $X
resultsinadirectedcycle.
5.TheyarecalledprimitiveinducingpathsbyRichardsonandSpirtes(2002).
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ii. thereisnoinducingpathbetweenanytwonon-adjacentvertices(maximal).
The first condition is obviously an extension of the defining condition for DAGs. It follows
that in an ancestral graph an arrowhead, whether on a directed edge or a bi-directed edge, implies
non-ancestorship. Thesecondconditionisatechnicalone,buttheoriginalmotivationisthefamiliar
pairwise Markov property of DAGs: if two vertices are not adjacent, then they are d-separated by
some set of other vertices. The notion of d-separation carries over to mixed graphs in a straight-
forward way, as we will see shortly. But in general an ancestral graph does not need to satisfy the
pairwiseMarkovproperty,orwhatiscalledmaximalityhere. Asufficientandnecessarycondition
for maximality turns out to be precisely the second clause in the above definition, as proved by
RichardsonandSpirtes(2002). SoalthoughthegraphinFigure1(a)isancestral,itisnotmaximal
becausethereisaninducingpathbetweenCandD(i.e.,hC;A;B;Di),butCandDarenotadjacent.
However,eachnon-maximalancestralgraphhasauniquesupergraphthatisancestralandmaximal.
Forexample,Figure1(b)istheuniqueMAGthatisalsoasupergraphofFigure1(a);theformerhas
anextrabi-directededgebetweenCandD.
It is worth noting that both conditions in Definition 1 are obviously met by a DAG. Hence,
syntacticallyaDAGisalsoaMAG,onewithoutbi-directededges.
Animportantnotionindirectedgraphicalmodelsisthatofd-separation,whichcapturesexactly
the conditional independence relations entailed by a DAG according to the Markov condition. It
is straightforward to extend the notion to mixed graphs, which, following Richardson and Spirtes
(2002),wecallm-separation.
Definition2(m-separation) In a mixed graph, a path p between vertices X and Y is active (or
m-connecting)relativetoa(possiblyempty)setofverticesZ(X;Y 2= Z)if
i. everynon-collideron pisnotamemberofZ;
ii. everycollideron pisanancestorofsomememberofZ.
X andY aresaidtobem-separatedbyZifthereisnoactivepathbetweenX andY relativeto
Z.
Two disjoint sets of variables X and Y are m-separated by Z if every variable in X is m-
separatedfromeveryvariableinYbyZ.
InDAGs,obviously,m-separationreducestod-separation. The(global)Markovpropertyofances-
tralgraphicalmodelsisdefinedbym-separation.
AnicepropertyofMAGsisthattheycanrepresentthemarginalindependencemodelsofDAGs
in the following sense: given any DAG G over V=O[L—where O denotes the set of observed
variables,andLdenotesthesetoflatentvariables—thereisaMAGoverOalonesuchthatforany
disjointX;Y;Z(cid:18)O,XandYared-separatedbyZinG (andhenceentailedbyG tobeindependent
conditional on Z) if and only if they are m-separated by Z in the MAG (and hence entailed by the
MAGtobeindependentconditionalonZ). ThefollowingconstructiongivesussuchaMAG:
Input: aDAGG overhO;Li
Output: aMAGM overO
G
1. for each pair of variables A;B 2 O, A and B are adjacent in M if and only if there is an
G
inducingpathbetweenthemrelativetoLinG;
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2. for each pair of adjacent variables A;B in M , orient the edge as A!B in M if A is an
G G
ancestorofBinG;orientitasA BinM ifBisanancestorofAinG;orientitasA$B
G
inM otherwise.
G
ItcanbeshownthatM isindeedaMAGandrepresentsthemarginalindependencemodeloverO
G
(RichardsonandSpirtes,2002; alsoseeLemma20below). Moreimportantly, M alsoretainsthe
G
ancestral relationships—and hence causal relationships under the standard interpretation—among
O. So, ifG isthecausalDAGforhO;Li, itisfairtocallM thecausalMAGforO. Henceforth
G
when we speak of a MAG over O representing a DAG over hO;Li, we mean that the MAG is the
outputoftheaboveconstructionprocedureappliedtotheDAG.
Different causal DAGs may correspond to the same causal MAG. So essentially a MAG rep-
resents a set of DAGs that have the exact same d-separation structures and ancestral relationships
among the observed variables. A causal MAG thus carries uncertainty about what the true causal
DAGis,butalsorevealsfeaturesthatmustbesatisfiedbytheunderlyingcausalDAG.
There is then a natural causal interpretation of the edges in MAGs, derivative from the causal
interpretationofDAGs. AdirectededgefromAtoBinaMAGmeansthatAisacauseofB(which
is a shorthand way of saying that there is a causal pathway from A to B in the underlying DAG);
a bi-directed edge between A and B means that A is not a cause of B and B is not a cause of A,
whichimpliesthatthereisalatentcommoncauseofAandB(i.e.,thereisalatentvariableLinthe
underlyingDAGsuchthatthereisadirectedpathfromLtoAandadirectedpathfromLtoB6).
WeborrowasimpleexamplefromSpirtesetal.(1993)toillustratevariousconceptsandresults
inthispaper. Supposeweareabletoobservethefollowingvariables: Income(I),Parents’smoking
habits (PSH), Smoking (S), Genotype (G) and Lung cancer (L). The data, for all we know, are
generatedaccordingtoanunderlyingmechanismwhichmightinvolveunobservedcommoncauses.
Suppose, unknown to us, the structure of the causal mechanism is the one in Figure 2, where Pro-
fessionisanunmeasuredcommoncauseofIncomeandSmoking.7
Genotype
Parents’ smoking habits
Income Smoking Lung Cancer
Profession
Figure2: AcausalDAGwithalatentvariable.
6.Notethatalatentcommoncauseisnotnecessarilyacommondirectcauseasdefinedonpage4. ThepathfromLto
A,forexample,mayincludeotherobservedvariables.
7.This example is used purely for illustrative purposes, so we will not worry why Profession is not observed but
Genotypeis.Theexactdomainsofthevariablesdonotmattereither.
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ThecausalMAGthatcorrespondstothecausalDAGisdepictedinFigure3(a)—whichsyntac-
tically happens to be a DAG in this case. This MAG can represent some other DAGs as well. For
example,itcanalsorepresenttheDAGwithanextralatentcommoncauseofPSH andS.
G G
PSH PSH
I S L I S L
(a) (b)
Figure3: TwoMarkovEquivalentMAGs.
IngeneralaMAGisstillnotfullytestablewithobservationaldata. JustasdifferentDAGscan
sharetheexactsamed-separationfeaturesandhenceentailtheexactsameconditionalindependence
constraints,differentMAGscanentailtheexactsameconstraintsbythem-separationcriterion. This
is known as Markov equivalence. Several characterizations of the Markov equivalence between
MAGsareavailable(SpirtesandRichardson,1996;Alietal.,2004;ZhangandSpirtes,2005;Zhao
et al., 2005). For the purpose of the present paper, it suffices to note that, as is the case with
DAGs, all Markov equivalent MAGs have the same adjacencies and usually some common edge
orientationsaswell. Forexample,thetwoMAGsinFigure3areMarkovequivalent.
ThismotivatesthefollowingrepresentationofequivalenceclassesofMAGs. Letpartialmixed
graphs denote the class of graphs that can contain four kinds of edges: !, $, (cid:14)(cid:0)(cid:0)(cid:14) and (cid:14)!, and
hencethreekindsofendmarksforedges: arrowhead(>),tail((cid:0))andcircle((cid:14)).
Definition3(PAG) Let[M]betheMarkovequivalenceclassofanarbitraryMAGM. Thepartial
ancestralgraph(PAG)for[M],P ,isapartialmixedgraphsuchthat
[M]
i. P hasthesameadjacenciesasM (andanymemberof[M])does;
[M]
ii. AmarkofarrowheadisinP ifandonlyifitissharedbyallMAGsin[M];and
[M]
iii. AmarkoftailisinP ifandonlyifitissharedbyallMAGsin[M].8
[M]
Basically a PAG represents an equivalence class of MAGs by displaying all common edge marks
shared by all members in the class and displaying circles for those marks that are not common,
much in the same way that a so-called Pattern (a.k.a. a PDAG or an essential graph) represents an
equivalenceclassofDAGs(see,e.g.,Spirtesetal.,1993,chap. 5; Chickering,1995; Anderssonet
al., 1997). Forinstance, thePAGforourrunningexampleisdrawninFigure4, whichdisplaysall
thecommonalitiesamongMAGsthatareMarkovequivalenttotheMAGsinFigure3.
8.ThisdefineswhatZhang(2006,pp.71)callscompleteormaximallyorientedPAGs.Inthispaper,wedonotconsider
PAGsthatfailtodisplayallcommonedgemarksinanequivalenceclassofMAGs(as,e.g.,allowedinSpirtesetal.,
1999),sowewillsimplyuse‘PAG’tomean‘maximallyorientedPAG’.
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G
PSH
I S L
Figure4: ThePAGinourfive-variableexample.
Different PAGs, representing different equivalence classes of MAGs, entail different sets of
conditionalindependenceconstraints. HenceaPAGisinprinciplefullytestablebytheconditional
independencerelationsamongtheobservedvariables. AssumingthecausalMarkovconditionand
itsconverse,thecausalFaithfulnesscondition,9thereisaprovablycorrectindependence-constraint-
based algorithm to learn a PAG from an oracle of conditional independence relations (Spirtes et
al., 1999; Zhang, 2006, chap. 3).10 Score-based algorithms for learning PAGs are also under
investigation.
Directedpathsandancestors/descendantsinaPAGaredefinedinthesamewayasinaMAG.In
addition,apathbetweenX andY,hX =V ;:::;V =Yi,iscalledapossiblydirectedpathfromX to
0 n
Y11 ifforevery0<i(cid:20)n,theedgebetweenV andV isnotintoV . CallX apossibleancestor
i(cid:0)1 i i(cid:0)1
ofY (andY apossibledescendantofX)ifX =Y orthereisapossiblydirectedpathfromX toY in
thePAG.12 Forexample,inFigure4,thepathhI;S;Liisapossiblydirectedpath,andI isapossible
ancestorofL. WeusePossibleAn (Y)todenotethesetofpossibleancestorsofY inP.
P
Inpartialmixedgraphstwoanaloguesofm-connectingpathswillplayarolelater. Let pbeany
path in a partial mixed graph, andW be any (non-endpoint) vertex on p. LetU andV be the two
verticesadjacenttoW on p.W isacollideron pif,asbefore,boththeedgebetweenU andW and
theedgebetweenV andW areintoW (i.e.,haveanarrowheadatW,U(cid:3)!W (cid:3)V). W iscalleda
definite non-collider on p if the edge betweenU andW or the edge betweenV andW is out ofW
9.We have introduced the causal Markov condition in its factorization form. In terms of d-separation, the causal
Markovconditionsaysthatd-separationinacausalDAGimpliesconditionalindependenceinthe(pre-intervention)
populationdistribution.ThecausalFaithfulnessconditionsaysthatd-connectioninacausalDAGimpliesconditional
dependenceinthe(pre-intervention)populationdistribution. Giventheexactcorrespondencebetweend-separation
relationsamongtheobservedvariablesinthecausalDAGandm-separationrelationsinthecausalMAG,thetwo
conditions imply that conditional independence relations among the observed variables correspond exactly to m-
separationinthecausalMAG,whichformsthebasisofconstraint-basedlearningalgorithms.
10.ItisessentiallytheFCIalgorithm(Spirtesetal.,1999),butwithslightmodifications(Zhang,2006,chap.3). The
implementedFCIalgorithmintheTetradIVpackage(http://www.phil.cmu.edu/projects/tetrad/tetrad4.html)isthe
modifiedversion.Bytheway,ifwealsotakeintoaccountthepossibilityofselectionbias,thenweneedtoconsider
abroaderclassofMAGswhichcancontainundirectededges,andtheFCIalgorithmneedstobeaugmentedwith
additionaledgeinferencerules(Zhang,2006,chap.4;forthcoming).
11.ItisnamedapotentiallydirectedpathinZhang(2006,pp.99).Thepresentterminologyismoreconsistentwiththe
namesforotherrelatednotions,suchaspossibleancestor,possiblym-connectingpath,etc.
12.Thequalifier’possible/possibly’isusedtoindicatethatthereissomeMAGrepresentedbythePAGinwhichthe
correspondingpathisdirected,andX isanancestorofY. Thisisnothardtoestablishgiventhevalidprocedurefor
constructingrepresentativeMAGsfromaPAGpresentedinLemma4.3.6ofZhang(2006)orTheorem2ofZhang
(forthcoming).
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(i.e.,hasatailatW,U W(cid:3)(cid:0)(cid:0)(cid:3)V orU(cid:3)(cid:0)(cid:0)(cid:3)W !V),orbothedgeshaveacirclemarkatW and
thereisnoedgebetweenU andV (i.e.,U(cid:3)(cid:0)(cid:0)(cid:14)W(cid:14)(cid:0)(cid:0)(cid:3)V,whereU andV arenotadjacent).13 The
firstanalogueofm-connectingpathisthefollowing:
Definition4(Definitem-connectingpath) In a partial mixed graph, a path p between two ver-
tices X and Y is a definite m-connecting path relative to a (possibly empty) set of vertices Z
(X;Y 2= Z)ifeverynon-endpointvertexon piseitheradefinitenon-collideroracolliderand
i. everydefinitenon-collideron pisnotamemberofZ;
ii. everycollideron pisanancestorofsomememberofZ.
It is not hard to see that if there is a definite m-connecting path between X and Y given Z in a
PAG,thenineveryMAGrepresentedbythePAG,thecorrespondingpathisanm-connectingpath
between X andY given Z. For example, in Figure 4 the path hI;S;Gi is definitely m-connecting
given L, and this path is m-connecting given L in every member of the equivalence class. A quite
surprisingresult is that if there is an m-connectingpath betweenX andY given Z in a MAG, then
theremustbeadefinitem-connectingpath(notnecessarilythesamepath)betweenX andY given
ZinitsPAG,whichwewilluseinSection5.
Anotheranalogueofm-connectingpathisthefollowing:
Definition5(Possiblym-connectingpath) Inapartialmixedgraph,apath pbetweenverticesX
andY ispossiblym-connectingrelativetoa(possiblyempty)setofverticesZ(X;Y 2= Z)if
i. everydefinitenon-collideron pisnotamemberofZ;
ii. everycollideron pisapossibleancestorofsomememberofZ.
Obviously a definite m-connecting path is also a possibly m-connecting path, but not necessarily
viceversa. Inparticular,onapossiblym-connectingpathitisnotrequiredthatevery(non-endpoint)
vertexbeofa“definite”status. Figure5providesanillustration. ThegraphontherightisthePAG
fortheequivalenceclassthatcontainstheMAGontheleft(inthiscase,unfortunately,noinforma-
tiveedgemarkisrevealedinthePAG).InthePAG,thepathhX;Y;Z;Wiisapossiblym-connecting
path but not a definite m-connecting path relative to fY;Zg, because Y and Z are neither collid-
ers nor definite non-colliders on the path. Note that in the MAG, hX;Y;Z;Wi is not m-connecting
relative to fY;Zg. In fact, X and W are m-separated by fY;Zg in the MAG. So unlike a definite
m-connecting path, a mere possibly m-connecting path in a PAG does not necessarily correspond
toam-connectingpath(orimplytheexistenceofam-connectingpath)inarepresentativeMAGin
theequivalenceclass.14
As we will see, the main result in Section 4 is formulated in terms of absence of possibly m-
connecting paths (what we will call, for want of a better term, definite m-separation), whereas the
13.’*’isusedaswildcardthatdenotesanyofthethreepossiblemarks:circle,arrowhead,andtail. Whenthegraphisa
PAGforsomeequivalenceclassofMAGs,thequalifier’definite’isusedtoindicatethatthevertexisanon-collider
onthepathineachandeveryMAGrepresentedbythePAG,eventhoughthecirclesmaycorrespondtodifferent
marksindifferentMAGs.ThereasonwhyU(cid:3)(cid:0)(cid:0)(cid:14)W(cid:14)(cid:0)(cid:0)(cid:3)V isadefinitenon-colliderwhenUandV arenotadjacent
isbecauseifitwereacollider,itwouldbesharedbyallMarkovequivalentMAGs,andhencewouldbemanifestin
thePAG.
14.ThiscaseisevenmoreextremeinthatineveryMAGthatbelongstotheequivalenceclass,XandW arem-separated
byY andZ. Sothisexamplecanbeusedtoshowthatthedo-calculusdevelopedinSection4isnotyetcomplete,
thoughitisnotclearhowserioustheincompletenessis.
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