Markov equivalence for ancestral graphs

30 Pages·2009·0.28 MB·English

TheAnnalsofStatistics
2009,Vol.37,No.5B,2808–2837
DOI:10.1214/08-AOS626
©InstituteofMathematicalStatistics,2009
MARKOV EQUIVALENCE FOR ANCESTRAL GRAPHS
BY R. AYESHA ALI,1,2 THOMAS S. RICHARDSON3 AND PETER SPIRTES
UniversityofGuelph,UniversityofWashingtonandCarnegie-MellonUniversity
Ancestral graphs can encode conditional independence relations that
ariseindirectedacyclicgraph(DAG)modelswithlatentandselectionvari-
ables. However,for any ancestral graph, there may be severalother graphs
towhichitisMarkovequivalent.Westateandproveconditionsunderwhich
twomaximalancestralgraphsareMarkovequivalenttoeachother,thereby
extendinganalogousresultsforDAGsgivenbyotherauthors.Thesecondi-
tionsleadtoanalgorithmfordeterminingMarkovequivalencethatrunsin
timethatispolynomialinthenumberofverticesinthegraph.
1. Introduction. A graphical Markov model is a set of distributions with in-
dependence structure described by a graph consisting of vertices and edges. The
independencemodelassociatedwithagraphisthesetofconditionalindependence
relationsencodedbythegraphthroughaglobalMarkovproperty.Ingeneral,dif-
ferent graphs may encode the same independence model. In this paper, we con-
sider a particular class of graphs, called ancestral graphs, and characterize when
twographsencodethesamesetsofconditionalindependencerelations.
The class of ancestral graphs is motivated in the following way. We suppose
our observed data were generated by a process represented by a directed acyclic
graph(DAG)withafixedsetofvariables.ThecausalinterpretationofsuchaDAG
is described by [18] and [14]. However, in general, we may only have observed
a subset of these variables in a specific sub-population. Hence, some variables in
theunderlyingDAGarenotobserved(“latent”),whileothervariables,specifying
the specific sub-population from which our data were sampled, are conditioned
upon(“selectionvariables”).
Even though the underlying model is a DAG, the conditional independence
structureholdingamongtheobservedvariables,conditionalontheselectionvari-
ables,cannotalwaysberepresentedbyaDAGcontaining onlytheobservedvari-
ables.Forthispurpose,themoregeneralclassofancestralgraphsisrequired[see
Figure 2(ii) and Definition 2.1]. The statistical models associated with ancestral
ReceivedApril2008.
1SupportedbytheNationalUniversityofSingapore2004–2005.
2SupportedinpartbyNSERCGrantRG326951-06.
3Supported in part by the William and Flora Hewlett Foundation, NSF Grants DMS-99-72008,
DMS-05-05865,EPAGrantCR827970-01-0andNIHGrantR01AI032475.
AMS2000subjectclassifications.Primary68T30,05C75;secondary68T37.
Keywordsandphrases.Directed acyclic graphs, discriminating path, inducing path, Markov
equivalence,polynomial-timealgorithm.
2808
MARKOVEQUIVALENCEFORANCESTRALGRAPHS 2809
FIG.1. (i)Aseeminglyunrelatedregressionmodeland(ii)aMarkovequivalentDAGmodel.
graphsretainmanyofthedesirablepropertiesthatareassociatedwithDAGmod-
els.
LikeDAGs,twodifferentancestralgraphscanrepresentthesamesetofcondi-
tional independence relations, and hence distributions. Such graphs are said to be
Markovequivalent.Agraphicalcharacterizationofthecircumstancesunderwhich
graphsareMarkovequivalentisofimportanceforseveralreasons:
• Markov equivalent graphs lead to identical likelihoods because the sets of dis-
tributionsobeyingtheMarkovpropertyassociatedwiththegraphsarethesame.
Thus, for the purposes of interpreting a model, it is often important to charac-
terizethosefeaturesthatarecommontoallthegraphsinagivenclass(see[18]
and[13]).
• When viewed as a Gaussian path diagram (see [15], Section 8.1), different
(maximal)ancestralgraphscorrespondtodifferentparametrizationsofthesame
GaussianMarkovmodel.However,someparametrizationsmaybesimplertofit
than others. For example, the model corresponding to the graph in Figure 1(i),
inthe Gaussian case, is an example of aseemingly unrelated regression (SUR)
model(see[23]).Ingeneral,therearenoclosedformexpressionsfortheMLEs
for SUR models, iterative fitting methods are required and there may be multi-
ple solutions to the likelihood equations (see [8]). However, the graph in Fig-
ure 1(i) is Markov equivalent to Figure 1(ii), which is a DAG. Gaussian DAG
modelshaveclosedformMLEs,andthelikelihoodisunimodal(see[12]).Con-
sequently, none of the problems which may arise for general Gaussian SUR
modelsapplytothespecificmodelcorrespondingtoFigure1(i)(seealso[7]).
In this paper, we provide necessary and sufficient graphical conditions under
which two ancestral graphs are Markov equivalent. Though other characteriza-
tions have been given previously in [24] and [19], the criterion given here is the
first which leads to an algorithm that runs in time polynomial in the size of the
graph. Reference [22] solved the Markov equivalence problem for DAGs. Refer-
ences[2,3]and[9]solvedtheproblemofrepresentingMarkovequivalenceclasses
forDAGs,whichweleaveforfuturework.
Section 2 defines the class of ancestral graphs and outlines the motivation for
the class. Section 3 contains the main result of the paper. Discussion and relation
topriorworkareinSection4.TheAppendixcontainsalgorithmicdetails.
2810 R.A.ALI,T.S.RICHARDSONANDP.SPIRTES
FIG.2. (i)ADAGwithalatentvariableH.(ii)Theancestralgraphresultingfrommarginalizing
overH includesabi-directededgebetweenPcpandCD4.
2. Ancestral graphs. The basic motivation for ancestral graphs is to enable
one to model the independence structure over the observed variables that results
from a DAG containing latent or selection variables without explicitly including
such variables in the model. To illustrate this, consider the DAG shown in Fig-
ure 2(i) in which Azt, Pcp, Ap and CD4 are observed variables, while H is unob-
served. Azt and Ap represent treatments given to AIDS patients (see Robins [17],
Section2).Pcpisanopportunistic infection thatoftenafflictsAIDSpatients, and
CD4 can be viewed as a measure of disease progression. Supposing development
of Pcp was a side-effect of taking Azt, then the DAG given in Figure 2(i) incor-
porates the assumption that Azt and Ap are both randomized, Pcp and CD4 are
responsescorrelatedbyunderlyinghealthstatusH,and,further,thatAzt doesnot
affect CD4. The DAG implies the following conditional independence relations
overtheobservedvariables:
Azt⊥⊥Ap, CD4, Ap⊥⊥Azt, Pcp.
These relations can be derived from the DAG in Figure 2(i) via d-separation
(see [12] or [22]). Also, note that other valid independence statements, such as
Azt⊥⊥CD4,canbederivedfromthetwostatementsgivenabove.Thecorrespond-
ing ancestral graph that represents these same conditional independence relations
is shown in Figure 2(ii). (See Section 2.2 for the definition of an ancestral graph
and Section 2.3 for the Markov property.) However, there is no DAG on the four
observed variables which represents all and only these conditional independence
relations.
Asthis example suggests, bi-directed edges (≺−(cid:4))may arise fromunobserved
parents.Likewise,undirectededges(−−−)mayarisefromchildrenthathavebeen
conditioned on in the selected sub-population from which the sample is taken
(see[4]and[5]).However,bi-directedandundirectededgesmayalsoariseinother
contexts,wherebothmarginalizationandconditioningarepresent.Reference[16]
providesadetaileddiscussionontheinterpretationofedgesinanancestralgraph.
2.1. Basic graphical notation and terminology. We use the following termi-
nologytodescriberelationsbetweenverticesinamixedgraphG,whichmaycon-
MARKOVEQUIVALENCEFORANCESTRALGRAPHS 2811
tainthreetypesofedge.
⎧ ⎫ ⎧ ⎫ ⎧ ⎫
⎪⎪⎨a−−−b⎪⎪⎬ ⎪⎪⎨neighbor⎪⎪⎬ ⎪⎪⎨a∈neG(b)⎪⎪⎬
a≺−(cid:4)b spouse a∈sp (b)
If inG, thena isa ofb and G .
⎪⎪⎩a−−(cid:4)b⎪⎪⎭ ⎪⎪⎩ parent ⎪⎪⎭ ⎪⎪⎩a∈paG(b)⎪⎪⎭
a≺−−b child a∈ch (b)
G
(For a formal set-theoretic definition of mixed graphs see [15], Appendix.) Two
vertices that are connected by some edge are said to be adjacent. Note that the
three edge types should be considered as distinct symbols, and that all the mixed
graphs we consider in this paper are simple in that they have at most one edge
between each pair of vertices. If there is an edge a−−(cid:4)b or a≺−(cid:4)b, then there is
said to be an arrowhead at b on this edge. Conversely, if there is an edge a−−(cid:4)b
or a−−−b, then there is said to be a tail at a. We also do not allow a vertex to be
adjacenttoitself.Werestrictattentiontographswithfinitevertexsets.
Apathπ betweentwoverticesx andy inasimplemixedgraphGisasequence
of distinct vertices π =(cid:6)x,v ,...,v ,y(cid:7) such that each vertex in the sequence is
1 k
adjacent to its predecessor and its successor; x and y are the endpoints of π; all
other vertices on the path are nonendpoints of π. If a and b are distinct vertices
on π, then the portion of π between a and b is called a section of π, denoted
π(a,b). Note that we use both π(a,b) and π(b,a) to represent the same section
ofπ.Apathoftheformx−−(cid:4)···−−(cid:4)y,onwhicheveryedgeisoftheform −−(cid:4),
with the arrowheads pointing toward y, is a directed path from x to y. A directed
pathfromx toy,togetherwithanedgey−−(cid:4)x∈G,iscalledadirectedcycle.
2.2. Definition of ancestral graphs. DAGs are directed graphs in which di-
rected cycles are not permitted. Similarly, certain configurations of edges are not
permittedinancestralgraphs:
DEFINITION 2.1. A graph, which may contain undirected (−−−), directed
(−−(cid:4))orbi-directededges(≺−(cid:4))isancestralif:
(a) therearenodirectedcycles;
(b) wheneverthereisanedge x≺−(cid:4)y,thenthereisnodirectedpathfrom x to y,
orfromy tox;
(c) ifthereisanundirectededgex−−−y thenx andy havenospousesorparents.
Conditions(a)and(b)maybesummarizedbysayingthat,ifx andy arejoined
by an edge and there is an arrowhead at x, then x is not an ancestor of y; this is
themotivationfortheterm“ancestral.”
A vertex a is said to be an ancestor of a vertex b if either there is a directed
path a−−(cid:4)···−−(cid:4)b from a to b or a=b. Further, if a is an ancestor of b, then b
issaidtobeadescendantofa.
A vertex a is said to be anterior to a vertex b if a =b or there is a path μ
betweena andb,onwhicheveryedgeiseitheroftheformc−−−d orc−−(cid:4)d,with
2812 R.A.ALI,T.S.RICHARDSONANDP.SPIRTES
d between c and b on μ;suchapath μ issaidtobeananteriorpathfrom a to b.
By (c) in Definition 2.1, the configuration −−(cid:4)c−−− never occurs in an ancestral
graph;hence,everyanteriorpathtakestheform
a−−−···−−−c−−(cid:4)···−−(cid:4)b,
where a =c and c=b are possible. We use an(x), de(x) and ant(x) to denote,
respectively,theancestorsofx,thedescendantsofx andtheverticesanteriortox.
Weapplythesedefinitionsdisjunctivelytosets.Forexample,
an(X)={a|a isanancestorofbforsomeb∈X},
ant(X)={a|a isanteriortobforsomeb∈X}.
By definition, X ⊆an(X)⊆ant(X). Note that every DAG is an ancestral graph,
sinceclauses(b)and(c)aretriviallysatisfied.
In the next lemma and elsewhere, we will make use of the shorthand notation
x?−(cid:4)y to indicate that either x−−(cid:4)y or x≺−(cid:4)y. Similarly, x?−−y indicates that
eitherx≺−−y orx−−−y,whilex?−?y indicatesanyedge.
LEMMA2.2. Leta,b,cbeverticesinanancestralgraphGwithaandcadja-
cent.Ifa?−(cid:4)b−−(cid:4)c,then a?−(cid:4)c.Inparticular,iftheedgeendsat a onthe (cid:6)a,b(cid:7)
and (cid:6)a,c(cid:7) edgesdiffer,thenwehave c≺−−a≺−(cid:4)b;otherwise,either c≺−−a−−(cid:4)b,
orc≺−(cid:4)a≺−(cid:4)b.
WemakeuseofthispropertyinSections3.7and3.9.
PROOF OF LEMMA 2.2. Suppose, for a contradiction, that there is a tail at c
on the (cid:6)a,c(cid:7) edge. Since, by hypothesis, there is an arrowhead at b on the (cid:6)b,c(cid:7)
edge, a−−−c is ruled out by Definition 2.1(c), so a≺−−c. But then G violates
Definition 2.1(b), since a?−(cid:4)b−−(cid:4)c−−(cid:4)a. Hence, a?−(cid:4)c. The conclusion then
followsfromnotingthattheconfigurationa−−(cid:4)b−−(cid:4)c≺−(cid:4)a isnotancestral. (cid:2)
2.3. Them-separationcriterion. Inanancestralgraph,anonendpointvertexv
on a path is said to be a collider if two arrowheads meet at v (i.e., −−(cid:4)v≺−−,
≺−(cid:4)v≺−(cid:4), ≺−(cid:4)v≺−− or −−(cid:4)v≺−(cid:4)). All other nonendpoint vertices on a path
arenoncolliders(i.e.,−−−v−−−,−−−v−−(cid:4),−−(cid:4)v−−(cid:4),≺−−v−−(cid:4),or≺−(cid:4)v−−(cid:4)).
These definitions of collider and noncollider are direct extensions of the corre-
sponding definitions for DAGs. A path along which every nonendpoint is a col-
lider is called a collider path. A path comprised of 3 vertices is called a triple. In
an ancestral graph, a triple is either a collider or a noncollider; we refer to this as
the type of the triple. Hence, if (cid:6)a,b,c(cid:7) forms a triple, then (cid:6)c,b,a(cid:7) and (cid:6)a,b,c(cid:7)
areofthesametype.
Reference[22]introducedd-separation,asetofgraphicalconditionsbywhich
conditional independence relations could be read from a DAG. Reference [15]
applied a natural extension of Pearl’s d-separation criterion, called m-separation,
toancestralgraphs.
MARKOVEQUIVALENCEFORANCESTRALGRAPHS 2813
DEFINITION 2.3. Leta andbbedistinctverticesinanancestralgraphG,and
letZ beasubsetofverticeswitha,b∈/Z.Apathπ betweena andb issaidtobe
m-connectinggivenZ ifthefollowinghold:
(i) nononcollideronπ isinZ;and,
(ii) everycollideronπ isanancestorofavertexinZ.
Two vertices a and b are said to be m-separated given Z in G if there is no
pathm-connectinga andb givenZ inG.Likewise,setsAandB arem-separated
givenZ inGif,foreverypaira∈Aandb∈B,a andb arem-separatedgivenZ.
Forexample,intheancestralgraphinFigure2(ii),Azt andApare m-separated
givenCD4.Definition2.3isanextensionoftheoriginaldefinitionofd-separation
for DAGs in that the notions of “collider” and “noncollider” now allow for bi-
directed and undirected edges; the definition of ancestor is unchanged. Further-
more, d-separation is equivalent to m-separation for DAGs. The following result
isuseful.
LEMMA 2.4. In an ancestral graph G, if π is a path m-connecting a and b
given Z, c is on π (a (cid:9)= c (cid:9)= b) and there is an arrowhead at c on the section
π(a,c),theneitherc∈an(Z)orπ(c,b)isadirectedpathfromc tob.
PROOF. Supposetheresultisfalse.Let c bethevertexclosestto b satisfying
the premise of the lemma but not the conclusion. If c is a collider on π, then, by
definitionofm-connection,c∈an(Z)whichisacontradiction.Letd bethevertex
after c on π(c,b). If c is anoncollider on π then, byDefinition 2.1(c), c−−(cid:4)d. If
d ∈an(Z) or π(d,b) forms a directed path from d to b, then, clearly, c satisfies
the conclusion of the lemma. But, if d ∈/ an(Z) and π(d,b) is not a directed path
tob,thend satisfiesthepremiseofthelemma(andhence,cisnottheclosestsuch
vertextob),againacontradiction. (cid:2)
2.4. Formalindependencemodels. AnindependencemodeloverafinitesetV
isasetIofternaryrelations(cid:6)X,Y |Z(cid:7)whereX,Y andZaredisjointsubsetsofV,
whileXandY arenotempty;thefirsttwoargumentsaretreatedsymmetrically,so
that(cid:6)X,Y |Z(cid:7)∈Iiff(cid:6)Y,X|Z(cid:7)∈I.Theinterpretationof(cid:6)X,Y |Z(cid:7)∈IisthatX
and Y are independent given Z [see [20], Chapter 2]. The independence model
associatedwithanancestralgraph,I (G),isdefinedviam-separationasfollows
m
I (G)≡{(cid:6)X,Y |Z(cid:7)|X ism-separatedfromY givenZ inG}.
m
TheindependencerelationsinI (G)comprisetheglobalMarkovpropertyforG.
m
2814 R.A.ALI,T.S.RICHARDSONANDP.SPIRTES
2.5. Probability distributions obeying a formal independence model. We as-
sociate a set of probability distributions with a formal independence model I by
usingthefinitesetV toindexacollectionofrandomvariables(Xν)ν∈V takingval-
uesinprobabilityspaces((cid:3)ν)ν∈V.Inalltheexamplesweconsider,theprobability
spaces are either real finite-dimensional vector spaces or finite discrete sets. For
×
A⊆V, we let (cid:3)A ≡ v∈A((cid:3)ν), (cid:3)≡(cid:3)V and XA ≡(Xν)ν∈A. We will assume
theexistenceofregularconditionalprobabilitymeasuresthroughout.
A distribution P on (cid:3) is said to obey the independence model I over V if, for
alldisjointsetsA,B,Z (AandB arenotempty),
(cid:6)A,B |Z(cid:7)∈I (cid:11)⇒ A⊥⊥B |Z[P],
wherewehaveusedthe(⊥⊥)notationof[6],andtheusualshorthandthatAdenotes
bothavertexsetandtherandomvariableX .Thus,adistributionP obeysI (G)
A m
if,foralldisjointsubsetsofV,sayX,Y,Z (X andY notempty),
X ism-separatedfromY givenZ inG (cid:11)⇒ X⊥⊥Y |Z[P].
Notethat,ifP obeysI,therestillmaybeindependencerelationsthatarenotinI
thatalsoholdinP.
2.6. Marginalizingandconditioning. InSection4.1of[15]operationsofmar-
ginalizing and conditioning are introduced for formal independence models. If P
∗
obeysIandI istheindependencemodelobtainedbyformallymarginalizingover
variables in L and conditioning on variables in S, then P(XV\(L∪S) |XS) obeys
∗
theindependencemodelI [P(X )a.e.](seeTheorem7.1of[15],AppendicesA
S
andBof[10]).
InSection4.2of[15],agraphicaltransformationcorrespondingtomarginaliz-
ingandconditioningisgivensuchthattheindependencemodelassociatedwiththe
transformedgraphistheindependencemodelobtainedbymarginalizingandcon-
ditioningtheindependencemodel I (G)oftheoriginalgraph(seeTheorem4.18
m
in[15]).Thus,inparticular,if G isaDAGwithobservedvariables O,latentvari-
ablesLandselectionvariablesS,thentheancestralgraphformedbythegraphical
transformation applied to G represents those conditional independence relations
impliedtoholdamongtheobservedvariablesO,conditionalontheselectionvari-
ables[P(X )a.e.].
S
3. Markovequivalence. Weintroducethefollowing.
DEFINITION 3.1. Two ancestral graphs G1 and G2 with the same vertex set
aresaidtobeMarkovequivalent,denotedG ∼G ,ifforalldisjointsetsA,B,Z
1 2
(A,B notempty),AandB arem-separatedgiven Z inG ifandonlyifAandB
1
arem-separatedgivenZ inG ;thatis,I (G )=I (G ).
2 m 1 m 2
ThegraphsinFigure3areMarkovequivalent,asareG andG inFigure4.The
1 2
set of all ancestral graphs that encode the same set of conditional independence
statementsformsaMarkovequivalenceclass.
MARKOVEQUIVALENCEFORANCESTRALGRAPHS 2815
FIG.3. (a)Thepath(cid:6)a,c,d,b(cid:7)isanexampleofaninducingpathinanancestralgraph.(b)Amax-
imalancestralgraphMarkovequivalentto(a).
3.1. Markov equivalence for DAGs. References [9] and [22] gave simple
graphical conditions for determining whether two DAGs are Markov equivalent.
Atripleofvertices(cid:6)a,b,c(cid:7)issaidtobeunshieldedifa andcarenotadjacentand
shielded otherwise.(AtripleisdefinedinSection2.3.)
THEOREM 3.2. TwoDAGsareMarkovequivalentifandonlyiftheyhavethe
sameadjacenciesandthesameunshieldedcolliders.
ThattwoMarkovequivalentDAGshavethesameadjacenciesisadirectconse-
quenceofthefactthatDAGssatisfyapairwiseMarkovproperty.
PROPOSITION 3.3 ([12], page 50). In a DAG D, if a and b are not adjacent
andb∈/an(a),thena isd-separatedfromb byV \(de(b)∪{a}).
ThisisaconsequenceofthelocalMarkovpropertyforDAGs[12],appliedtob,
whichimpliesthat b is d-separated from V \(pa(b)∪de(b)) by pa(b); b∈/an(a)
implies a∈/de(b) and pa(b)⊆V \(de(b)∪{a}).Notethat,byacyclicity, forany
pair a, b, either b ∈/ an(a) or a ∈/ an(b). Consequently, in a DAG, every miss-
ingedgeimpliesaconditionalindependencebetweenthenonadjacentvertices.In
general, no such pairwise property holds for ancestral graphs. For example, there
is no set that m-separates a and b in the graph in Figure 3(a). This motivates the
followingsection.
3.2. Maximalancestralgraphs.
DEFINITION3.4. AnancestralgraphGissaidtobemaximalif,foreverypair
ofnonadjacentvertices(a,b),thereexistsasetZ (a,b∈/Z)suchthata andb are
m-separatedconditionalonZ.
FIG. 4. G1,G2,G3 havethesameadjacenciesandthesameunshieldedcolliders,butG1 andG3
arenotMarkovequivalent.π=(cid:6)x,q,b,y(cid:7)formsadiscriminatingpathforbineverygraph.
2816 R.A.ALI,T.S.RICHARDSONANDP.SPIRTES
Thesegraphsaremaximalinthesensethatnoadditionaledgemaybeaddedto
thegraphwithoutchangingtheassociatedindependencemodel.Inanonmaximal
ancestralgraphtwononadjacentverticesa andb,forwhichnom-separatingsetZ
exists,willbejoinedbyaninducingpath.
DEFINITION3.5. Aninducingpathπ betweenverticesaandbinanancestral
graph G is a path on which every nonendpoint vertex is both a collider on π and
anancestorofatleastoneoftheendpoints,a,b.
For a proof, see [15], Corollary 4.3, where the definition given here is termed
a“primitive”inducingpath;theconceptwasintroducedbyVermaandPearl[21].
Note that, strictly speaking, an inducing “path” π =(cid:6)a,v ,...,v ,b(cid:7) is a collec-
1 k
tionofpaths:thecolliderpathπ,togetherwithdirectedpathsfromeachvertexv ,
i
1≤i ≤k, to one of the endpoints. The name “inducing” path refers to the fact
that given any set Z (a,b∈/ Z) π is m-connecting given Z. If there is some ver-
tex v ∈/ an(Z), then there is an m-connecting path involving one or more of the
i
directedpaths,otherwisethepathπ itselfism-connecting.
Figure 3(a) shows an example of a nonmaximal ancestral graph. The path
(cid:6)a,c,d,b(cid:7)formsaninducingpathbetweenaandb.Byaddingthebi-directededge
a≺−(cid:4)b,thegraphismademaximalwithoutchangingtheassociatedindependence
model(whichisempty),asshowninFigure3(b).Asisthecaseinthisexample,in
general,if π =(cid:6)a,v ,...,v ,b(cid:7) isaninducing path,thenonlyabi-directed edge
1 k
a≺−(cid:4)b may be added while obeying Definition 2.1. Since there are arrowheads
presentata andb,addinganundirectededgeisruledoutby(c);addingadirected
edge would violate (b) since we would either have a≺−(cid:4)v −−(cid:4)···−−(cid:4)b−−(cid:4)a or
1
b≺−(cid:4)v −−(cid:4)···−−(cid:4)a−−(cid:4)b.
k
By[15],Theorem5.1,foreverynonmaximalancestralgraphGthereisaunique
maximalancestralgraph G¯ ofwhichitisasubgraph;infact, G¯ =G[∅∅ andthus G¯
may be constructed in polynomial time. Consequently, the problem of character-
izing Markov equivalence for ancestral graphs naturally reduces to that of char-
acterizing equivalence in the case where both graphs are maximal. Except where
noted,intheremainderofthispaper,wewillrestrictattentiontomaximalancestral
graphs(MAGs).
3.3. NecessaryconditionsforMarkovequivalence.
PROPOSITION 3.6. If G1, G2 are MAGs and G1 ∼G2, then G1 and G2 have
thesameadjacenciesandunshieldedcolliders.
PROOF. Since G1 is maximal, for each pair of nonadjacent vertices (x,y)
in G , there is some set Z such that x and y are m-separated given Z in G .
1 1
If x and y are adjacent in G , then they are not m-separated by Z, contradict-
2
ing G ∼G . So, adjacencies in G are a subset of those in G . By a symmetric
1 2 1 2
argument,theadjacenciesinG areasubsetofthoseinG .
2 1
MARKOVEQUIVALENCEFORANCESTRALGRAPHS 2817
Suppose, for a contradiction, that (cid:6)a,b,c(cid:7) is an unshielded collider in G but
1
notinG .SinceG ismaximal,forsomesetZ,a andcarem-separatedbyZ,and
2 1
b∈/ Z. If (cid:6)a,b,c(cid:7) is a noncollider in G then a and c are m-connected given Z,
2
which is a contradiction. Hence, every unshielded collider in G is present in G .
1 2
Theconclusionfollowsbysymmetry. (cid:2)
Animportantconsequenceofthispropositionisthatif G and G aremaximal
1 2
andMarkovequivalent,thenasequenceofverticesformingapathinG alsoforms
1
apathinG andvice-versa,thoughtheedge-typesonthesepathsmaydiffer.Con-
2
sequently, when G ∼G , we will often refer to the path π∗ in G corresponding
1 2 2
toagivenpathπ inG .
1
AkeydifferencebetweenDAGsandMAGsisthathavingthesameadjacencies
and the same unshielded colliders, though necessary, are no longer sufficient for
Markov equivalence. Consider the graphs shown in Figure 4. G and G contain
1 3
the same adjacencies and the same unshielded colliders, but these two graphs are
not Markov equivalent to each other. In G , x is m-separated from y given q; but
1
accordingtoG ,xism-connectedtoygivenq.Infact,inanygraphMarkovequiv-
3
alentto G , (cid:6)q,b,y(cid:7) formsashieldedcollider.(Thereisonlyonesuchgraph, G ,
1 2
so {G ,G } forms a Markov equivalence class.) However, in general, it is clearly
1 2
not necessary that two graphs have all of the same shielded colliders in order for
them to be Markov equivalent. Much of the remainder of this paper will focus
onidentifyingthe“relevant”setofcollidersforjudgingMarkovequivalence.The
mainresultofthispaperfollows.
THEOREM 3.7. If G1, G2 are MAGs, then G1 ∼G2 if and only if G1 and G2
havethesameadjacenciesandthesamecolliderswithorder.
Thesetof“colliderswithorder”withinagraphisdefinedrecursivelyinDefin-
ition3.11inthenextsection.TheproofconcludesinSection3.10.
3.4. Discriminating paths in maximal ancestral graphs. A discriminating
path, if present in two Markov equivalent MAGs, implies that a certain shielded
triplewillbeofthesametypeinbothgraphs.
DEFINITION 3.8 [18]. Apathπ =(cid:6)x,q1,...,qp,b,y(cid:7) (p≥1)isadiscrimi-
natingpathfor(cid:6)q ,b,y(cid:7)inaMAGGif:
p
(i) x isnotadjacenttoy,and,
(ii) everyvertexq (1≤i≤p)isacollideronπ,andaparentofy.
i
Wewilloftenrefertoasectionπ(x,y)ofsomepathπ asadiscriminatingpath
forb,therebyimplicitlyspecifyingthetriple(cid:6)q ,b,y(cid:7)=π(q ,y).Byconvention,
p p
we order the endpoints of the discriminating path so it is the second endpoint (in