Learning model structures based on marginal model structures of ...

<strong>Learning</strong> <strong>model</strong> <strong>structures</strong> <strong>based</strong> on marginal<strong>model</strong> <strong>structures</strong> of undirected graphsbySung-Ho KimBK21 Research Report09 - 04March 25, 2009DEPARTMENT OF MATHEMATICAL SCIENCES

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1G 1 2 G2133 4Figure 1: Two <strong>model</strong> <strong>structures</strong> of graphical <strong>model</strong>s.& Kim (1999)]. The relationship becomes more explicit when the distribution is multivariate normal.Let X be a normal random vector. The precision matrix of the conditional distribution of asubvector X 1 given the remaining part of X is the same as the X 1 part of the precision matrix of X[Section 5.7, Whittaker (1990)]. Marginals of a joint probability distribution are not in general representedas parts of the joint distribution. However, there is a way that we can express explicitly therelationship between joint and marginal distributions under the assumption that the joint (as againstmarginal) probability <strong>model</strong> is graphical and decomposable (Kim, 2006b).In addressing the issue of information reuse in the form of combining graphical <strong>model</strong> <strong>structures</strong>,we can not help using independence graphs and related theories to derive desired results withmore clarity and refinement. The conditional independence embedded in a distribution can be expressedto some level of satisfaction by a graph in the form of graph-separateness [see, for example,the separation theorem in p. 67, Whittaker (1990)]. We instrument the notion of conditional independencewith some particular sets of random variables in a <strong>model</strong>, where the sets form a basis ofthe <strong>model</strong> structure so that the Markov property among the variables of the <strong>model</strong> may be preservedbetween the <strong>model</strong> and its marginals. The sets are called prime separators for decomposable graphsand self-connected (SC) separators for non-decomposable graphs and defined in sections 2 and 4respectively.It is shown that if we are given a graphical <strong>model</strong> with its independence graph, G, and some ofits marginal <strong>model</strong>s, then we can find, under the assumption that the graphical <strong>model</strong> is decomposable,a graph, say H, which is not smaller than G and in which the graph-separateness in the givenmarginal <strong>model</strong>s is preserved. This graph-separateness is substantiated by the prime separators andSC-separators which are found in the graphs of the marginal <strong>model</strong>s. In combining marginal <strong>model</strong>sinto H, we see to it that these prime separators appear as the only prime separators in H when thegraphs are decomposable (Kim & Lee, 2008). In this paper, we will extend the graphical combinationof marginal <strong>model</strong>s to the case where the marginal <strong>model</strong>s are not necessarily decomposable.There have been a number of papers applying marginal <strong>model</strong>s for data analysis during the last15 years or so. Some of the applications are for parameter estimation of a <strong>model</strong> <strong>based</strong> on medicaldata from crossover experiments (Balagtas, Becker & Lang, 1995), for estimating joint probabilitiesby applying the iterative proportional fitting technique (Molenberghs & Lesaffre (1999), for analyzingsociological data (Becker, 1994; Becker, Minick & Yang, 1998), and for analyzing contingencytable data with ordinal response variables (Colombi & Forcina, 2001). In most of these applicationsof marginal <strong>model</strong>s to multivariate statistical problems, we impose structural restrictions on certainsubsets of variables that are involved in a given data set (Liang, Zeger & Qaqish, 1992; Glonek &McCullagh, 1995; Bergsma, 1997; Bartolucci & Forcina, 2002; Bergsma & Rudas, 2002; Rudas& Bergsma, 2004). There also have been remarkable improvements in learning graphical <strong>model</strong>sin the form of a Bayesian network (Pearl 1986, 1988; Meek, 1995; Spirtes, Glymour & Scheines,2000; Neapolitan, 2004) from data. This learning however is mainly instrumented by heuristicsearching algorithms since the <strong>model</strong> searching is usually NP-hard (Chickering, 1996). In our pro-2

posed method of structural learning, we will assume that only the information which is embeddedin a given set of marginal <strong>model</strong> <strong>structures</strong> is available.This paper is organized in 7 sections. After introducing graphical terminologies and notation insection 2, we derive a result which shows how two sets of graphs, where the graphs in one set aresome type of subgraphs of the graphs in the other set, are related in stochastic context and introducea type of graph, called a combined <strong>model</strong> structure (CMS), with regard to the relationship of thetwo sets of graphs. In section 4, we consider some types of separators of undirected graphs, called aself-connected separator and a prime separator, and use them to further investigate the relationshipbetween the two sets of graphs. We then consider the notion of graphical compatibility (Dawid &Studeny, 1999) in section 5 as a necessary relationship between graphs and show existence of aCMS of set of graphs when the compatibility condition is satisfied among the graphs. In section6, we propose a combination method of graphs as a way of information reuse from a given set ofmarginal <strong>model</strong>s. Finally, in section 7, we close the paper with some discussion and concludingremarks.2 NOTATION AND PRELIMINARIESWe will consider only undirected graphs in the paper. We denote a graph by G = (V, E), where Vis the set of the indexes of the variables involved in G and E is a collection of ordered pairs, eachpair representing that the nodes of the pair are connected by an edge. Since G is undirected, that(u, v) is in E is the same as that (v, u) is in E. If (u, v) ∈ E, we say that u is a neighbor node of oradjacent to v or vice versa. We say that a set of nodes of G forms a complete subgraph of G if everypair of nodes in the set is adjacent to each other. If every node in A is adjacent to all the nodes in B,we will say that A is adjacent to B. The set of all the neighbor nodes of a node v in G is denoted bybd G (v); if v becomes a set, A say, we define bd G (A) = ∪ v∈A bd G (v) \ A. We define the closure ofa set A as cl G (A) = bd G (A) ∪ A. We denote by C(G) the set of cliques of G.A path of length n is a sequence of nodes u = v 0 , · · · , v n = v such that (v i , v i+1 ) ∈ E,i = 0, 1, · · · , n − 1 and u ≠ v. If u = v, the path is called an n-cycle. If u ≠ v and u and v areconnected by a path, we write u ⇋ v. We define the connectivity component of u as[u] = {v ∈ V ; v ⇋ u} ∪ {u}.We say that a path, v 1 , · · · , v n , v 1 ≠ v n , is intersected by A if A∩{v 1 , · · · , v n } ̸= ∅ and neitherof the end nodes of the path is in A. We say that nodes u and v are separated by A if all the pathsfrom u and v are intersected by A. In the same context, we say that, for three disjoint sets A, B,and C, A is separated from B by C if all the paths from A to B are intersected by C and write〈A|C|B〉 G . The complement of a set A is denoted by A c and the cardinality of a set A by |A|. Fortwo collection of sets, A and B, we write A ≼ B if, for every set a in A, there exists a set b in Bsuch that a ⊆ b.For A ⊂ V , we define an induced subgraph of G confined to A as GAind = (A, E ∩ (A × A)).We also define a graph, called a Markovian subgraph of G confined to A, which is formed fromby completing the boundaries in G of the connectivity components of the complement of Aand denote it by G A . If G ′ is a Markovian subgraph of G, we write G ′ ⊆ M G.If G = (V, E), G ′ = (V, E ′ ), and E ′ ⊆ E, then we say that G ′ is an edge-subgraph of G andwrite G ′ ⊆ e G. If G ′ is a subgraph of G, we call G a supergraph of G ′ . For a graph G, we will denotethe set of nodes of G by V (G).The cliques are elementary graphical components and so we will call the intersection of neighboringcliques a prime separator of the decomposable graph G. The prime separators in a decomposablegraph may be extended to separators of prime graphs in some graphs, where the prime graphsGAind3

are defined as the maximal subgraphs without a complete separator in Cox & Wermuth (1999). If Gis not decomposable, separators are not obtained as intersections of neighboring cliques.3 MARGINAL MODELS AND MARKOVIAN SUBGRAPHSSuppose that we are given a probability <strong>model</strong>, P , with its interaction graph, G, and some of itsmarginal <strong>model</strong>s are also given whose interaction graphs are Markovian subgraphs of G. In this section,we will look into the relationship between P and the marginal <strong>model</strong>s through the relationshipbetween G and the Markovian subgraphs.A distribution P is said to be globally Markov with respect to a graph G if, for a triple (A, B, C)of disjoint subsets A, B, C of V , random vectors X A and X B are conditionally independent givenan outcome of random vector X C whenever A is separated from B by C in G.In addition to the global Markov property, we will consider another property for a probabilitydistribution. A distribution P with probability function f is said to be factorized (FA) accordingto G [Section 3.2, Lauritzen (1996)] if for all c ∈ C(G) there exist non-negative functions ψ c thatdepend on x through x c only such thatf(x) =∏ψ c (x).c∈C(G)We will denote the collection of the distributions that are globally Markov with respect to G byM G (G).For a probability distribution P of X V , let the logarithm of the density of P be expanded intointeraction terms and let the set of the maximal domain sets of these interaction terms be denoted byΓ(P ), where maximality is in the sense of set-inclusion. We will call the set, Γ(P ), the generatingclass of P and denote by G(Γ(P )) = (V, E) the interaction graph of P which satisfies, under thehierarchy assumption for probability <strong>model</strong>s,(u, v) ∈ E ⇐⇒ {u, v} ⊆ a for some a ∈ Γ(P ).It is well known in literature (Pearl & Paz, 1987) that if a probability distribution on X V ispositive, then the three types of Markov property, pairwise Markov (PM), locally Markov (LM),and globally Markov (GM) properties relative to an undirected graph, are equivalent. Furthermore,for any probability distribution, it holds that(F A) =⇒ (GM) =⇒ (LM) =⇒ (P M) (1)[see Proposition 3.8 in Lauritzen (1996)]. Under the positivity condition of the probability distribution,we have (F A) ⇐⇒ (P M) by Hammersley and Clifford (1971). From this and expression (1),it follows, under the positivity condition, that(F A) ⇐⇒ (GM).For notation convenience, we will write M(G) instead of M G (G) and we will simply say that adistribution P is Markov with respect to G when P ∈ M G (G). For A ⊂ V , we denote by J A thecollection of the connectivity components in GA ind c and letβ(J A ) = {bd(B); B ∈ J A }.4

Let P A be the marginal of P on X A . We then define ¯Γ(P A ) asFrom this, it follows that¯Γ(P A ) = (Γ(P ) ∩ A) ∪ β(J A ). (2)β(J A ) ≼ ¯Γ(P A ) ≼ C(G(¯Γ(P A ))).The second ≼ holds since it is possible that, for some B ∈ J A , bd(B) is a strict subset of a cliquein G(¯Γ(P A )).The following result is immediate from (2).THEOREM 1. For a distribution P of X V and A ⊆ V ,G(¯Γ(P A )) = G(P ) A .Proof. By definition, the interaction graph corresponding to the right hand side of (2) is G(P ) A .Thus the result follows.From this theorem and the fact that Γ(P A ) ≼ ¯Γ(P A ), we haveCOROLLARY 1. For a distribution P of X V and A ⊆ V ,P A ∈ M(G(P ) A ).From Theorem 1, we can also derive a result concerning both the relationship between a distributionP and a graph G and the relationship between P A and G A .COROLLARY 2. For a distribution P of X V and A ⊆ V , suppose that P ∈ M(G) for a graph G.ThenP A ∈ M(G A ).Proof. Since P ∈ M(G), we have G(P ) ⊆ e G. This implies that G(P ) A ⊆ e G A . So, by Corollary1, we have the desired result.We call G A a Markovian subgraph of G in the context of Corollary 2.For A ⊆ V , we define M(G) A and L(G A ) asandM(G) A = {P A ; P ∈ M(G)}L(G A ) = {P ; P A ∈ M(G A )}.M(G) A is the set of the marginal distributions on X A of a distribution P which is Markov withrespect to G; L(G A ) is the set of the distributions of X V whose marginal P A on X A is Markov withrespect to G A .By definition and Corollary 2, we have the following:L(G) = M(G),M(G) ⊆ L(G A ), (by Corollary 2) (3)P ∈ L(G A ) ⇐⇒ P A ∈ M(G A )5

andM(G) A ⊆ M(G A ).The last expression holds since, if a distribution Q is in M(G) A , it means that Q = P A for somedistribution P in M(G), and so, by Corollary 2, it follows that Q ∈ M(G A ).It follows from (3) that, for A, B ⊆ V ,M(G) ⊆ L(G A ) ∩ L(G B ).We will derive a generalized version of this result below.Let V be a set of subsets of V . We will define another set of distributions,˜L(G A , A ∈ V) = {P ; P A ∈ M(G A ), A ∈ V}.˜L(G A , A ∈ V) is the set of the distributions each of whose marginals is Markov with respect to itscorresponding Markovian subgraph of G.THEOREM 2. For a collection V of subsets of V with a graph G,M(G) ⊆ ˜L(G A , A ∈ V).Proof. Let P ∈ M(G). Then, by (3), P ∈ L(G A ) for A ∈ V. By definition, P A ∈ M(G A ). Sincethis holds for all A ∈ V, it follows that P ∈ ˜L(G A , A ∈ V). This completes the proof.Theorem 2 shows the relationship between a graphical <strong>model</strong> with its graph G and a set ofMarkovian subgraphs of G. The set M(G) of the probability distributions each of which is Markovwith respect to G is contained in the set ˜L(G A , A ∈ V) of the distributions each of which has itsmarginals Markov with respect to their corresponding Markovian subgraphs G A , A ∈ V. This resultsheds light on our efforts in searching for M(G) since it can be found as a subset of ˜L(G A , A ∈ V).Let G = (V, E) be the graph of a graphical <strong>model</strong> and let V 1 , V 2 , · · · , V m be subsets of V . Them Markovian subgraphs, G V1 , G V2 , · · · , G Vm , may be regarded as the <strong>structures</strong> of m sub<strong>model</strong>s ofthe graphical <strong>model</strong>. In this context, we may refer to a Markovian subgraph as a marginal <strong>model</strong>structure. For simplicity, we write G i = G Vi .DEFINITION 1. Suppose there are m Markovian subgraphs, G 1 , · · · , G m . Then we say that graphH of a set of variables V is a combined <strong>model</strong> structure (CMS) of G 1 , · · · , G m , if the followingconditions hold:(i) ∪ m i=1 V i = V.(ii) H Vi = G i , for i = 1, · · · , m. That is, G i are Markovian subgraphs of H.We will call H a maximal CMS of G 1 , · · · , G m if adding any edge to H invalidates condition(ii) for at least one i = 1, · · · , m.Let M be the collection of G(P ) A , A ∈ V. We can construct a maximal CMS, H ∗ say, byadding edges, if any, to G in such a way that condition (ii) of Definition 1 is satisfied. SinceH ∗ A = G(P ) A ,if we put G = G(P ) in Theorem 2, we end up with a summarizing expression,M(G(P )) ⊆ M(H ∗ ) ⊆ ˜L (G(P ) A , A ∈ V) , (4)6

where the first inequality follows since G(P ) ⊆ e H ∗ . Since P ∈ M(G(P )), expression (4) impliesthat P is also Markov relative to H ∗ .If two nodes u and v are separated in a Markovian subgraph of a graph G, then so are theyin G by the property of a graph. We can extend this result to disjoint sets. Let C G (A) denote thecollection of the cliques which include nodes of A in G.THEOREM 3. (Theorem 4.2, Kim & Lee (2008)) Let G ′ = (V ′ , E ′ ) be a Markovian subgraph of Gand suppose that, for three disjoint subsets A, B, C of V ′ , 〈A|B|C〉 G ′. Then(i) 〈A|B|C〉 G ;(ii) For W ∈ C G (A) and W ′ ∈ C G (C), 〈W |B|W ′ 〉 G .4 MARKOVIAN SUBGRAPHS OF UNDIRECTED GRAPHSConsider two Markovian subgraphs of G on A and B, G A and G B . Then by the transitivity propertyof the Markovian marginalization (Kim, 2006b), (G A ) B is also a Markovian subgraph of G, andsimilarly for (G B ) A . Furthermore, we can see, by definition, that(G A ) B = (G B ) A = G A∩B . (5)DEFINITION 2. For three disjoint and exhaustive subsets, A, B, and C, of V = V (G), we will callC a self-connected (SC) separator in G, if the following conditions hold:(i) 〈A|C|B〉 G .(ii) G indCis connected.(iii) GCind does not contain any n-cycle (n > 3) nor a clique of G which consists of more than twonodes.(iv) GA∪C ind and GindB∪Ceach consists of n-cycles (n > 3) or cliques of G only.According to the definition, we can see that, a SC-separator is given as a union of some intersectionsof n-cycles (n > 3) or cliques. For example, in Figure 2, the SC-separator, {1, 2, 4, 5, 6},is the union of the intersections of the following two pairs of cycles:12347856Figure 2: An undirected graph of 8 nodes.7

(a) (b) (c) (d)3559575 228 26112434 13 4 13555934722586114 13 4 124Figure 3: Undirected graphs and their Markovian subgraphs. The slant “=” on node v means that vis removed from the graph.pair 1 : {1, 2, 4, 5, 6, 8} and {1, 2, 3, 4}pair 2 : {1, 2, 4, 5, 6, 8} and {3, 4, 5, 6}In this respect, it follows that, if G is a decomposable graph, then all of its SC-separators are primeseparators.Note that condition (iv) in the definition does not imply condition (ii). For example, if, in Figure2, we let A = {3, 4}, B = {7, 8}, and C = {1, 2, 5, 6}, then conditions (i), (iii), (iv) are satisfiedfor the three sets, but (ii) is not.If G is Markovian-marginalized over a node v which is included in a SC-separator of the graphG, then new SC-separators are created in G V \{v} . For example, in column (b) of Figure 3, node3 is removed from the graph at the top and the resultant Markovian subgraph is given at the bottom,where {1, 2, 5} and {2, 4, 5} are new SC-separators. In column (c), node 2 is removed andthe removal yields a new SC-separator, {3, 5}, in the Markovian subgraph. In column (d), theset {3, 5} is a SC-separator and removal of node 3 creates a new SC-separator {1, 5}. Note that{1, 3, 5, 8, 9} forms a 5-cycle and {2, 3, 4, 5, 6, 7} forms a clique and that the removal creates a newclique, {1, 2, 4, 5, 6, 7}, and a new SC-separator {1, 5}. On the other hand, removal of a node whichis not a member of a SC-separator does not create any new SC-separator as we see in column (a) ofFigure 3.Let M and S be, respectively, a set of nodes to be removed and a set of nodes which formSC-separators. Since Markovian subgraphs are not dependent upon the order of node-removal, wecan begin node-removal with the nodes in M ∩ S or with the nodes in M \ S. The only difference isthat the removal of a node in M \ S simply reduces the size of a cycle or a clique while the removalof a node in M ∩ S creates new SC-separators.THEOREM 4. Let G ′ be a Markovian subgraph of an undirected graph G. If A is a SC-separator inG ′ , then there exists a SC-separator, S, in G such that A ∩ S ≠ ∅.Proof. Since A is a SC-separator in G ′ , we can find disjoint sets, B and C, in V (G ′ ) \ A suchthat A ∪ B ∪ C = V (G ′ ) and 〈B|A|C〉 G ′. Then, by Theorem 3, it follows that 〈B|A|C〉 G . LetD = V (G) \ V (G ′ ). Then, by the property of an undirected graph, we have 〈B|A ∪ D|C〉 G . Now,we have only to show (i) that the set A ∪ D is itself a SC-separator or (ii) that A ∪ D contains a8

SC-separator as a subset in G.In case (i), we have the desired result. In case (ii), there are two possibilities. One possibilityis that there is a SC-separator A ′ in A ∪ D such that A ⊆ A ′ , and the other that A is itself a SCseparatorin G. In the former situation, at least one node is removed from A ′ in the marginalizationof G, and in the latter situation the removal takes place outside the neighborhood of A in G. Inthe latter situation, A itself is a SC-separator in G; and in the former situation, if node v ∈ A ′ isremoved from G, all of its neighbor nodes become adjacent to each other, which means that newSC-separators are created in G v c as in panels (b), (c), and (d) in Figure 3, where v c = V (G) \ {v}.If multiple nodes, v 1 , · · · , v r , are removed from A ′ \ A, we can see by the same argument that wehave new SC-separators in G R where R = V (G) \ {v 1 , · · · , v r }. This completes the proof.From this theorem, we can see that a SC-separator, S say, in a Markovian subgraph of G meansthat there is a SC-separator in G which shares at least one node with S. An analogous but moretangible result holds when G is decomposable. In the theorem below, χ(G) is the set of all the primeseparators of a decomposable graph G.THEOREM 5. (Theorem 4 of Kim (2006b)) Let there be Markovian subgraphs G i , i = 1, 2, · · · , m,of a decomposable graph G. Then(i)∪ m i=1χ(G i ) ⊆ χ(G);(ii) for any maximal CMS H,∪ m i=1χ(G i ) = χ(H).The above two theorems say that,(a) when G is decomposable, every prime separator that is found in a Markovian subgraph of G isalso found in G; but(b) when G is not decomposable, every SC-separator, A say, that is found in a Markovian subgraphof G has at least one SC-separator in G which shares at least one node with A.There is another noteworthy difference between the two types of graphs. In a decomposablegraph, if a node which is included in a prime separator is removed, then a new clique is formedby the nodes of the cliques that share the prime separator. This means that the prime separatordisappears with no trace left. On the other hand, if a node which is included in a SC-separator isremoved from a non-decomposable graph, then new SC-separators are created as shown in Figure3 unless the SC-separator is shared by neighboring cliques only. From this, we can see that primeseparators in a decomposable graph may easily be lost in its Markovian subgraphs when at least oneof the nodes in a prime separator is removed. On the other hand, node-removal from a SC-separator,S say, in a non-decomposable graph create new SC-separators in the Markovian subgraph whichshare nodes with S if the number of the removed nodes from S is less than |S|.5 GRAPHICAL COMPATIBILITY AND EXISTENCE OF CMS’SFor C ⊆ V (G) ∩ V (H), G and H are said to be C-compatible (Dawid & Studeny, 1999) ifG C = H C .For graphs, G 1 , · · · , G k , and sets of nodes, A 1 , · · · , A k−1 , if G i and G i+1 are A i -compatible fori = 1, 2, · · · , k − 1, then we say that G 1 and G k are compatible with regard to G i , i = 2, · · · , k − 1.9

Denote V (G i ) by V i and suppose that (G 1 ) A1 = (G 2 ) A1 and (G 2 ) A2 = (G 3 ) A2 . If there exist graphs,G ′ and G ′′ , such thatG ′ V 1= G 1 and G ′ V 2= G 2andthen it follows that G ′ A 2= G ′′ A 2sinceG ′′V 2= G 2 and G ′′V 3= G 3 ,G ′ A 2= (G ′ V 2) A2 = G 2 A 2= G 3 A 2= (G ′′V 3) A2 = G ′′ A 2,where the first and the last equality hold by (5) and the inequality, A 2 ⊆ V 2 ∩ V 3 .If we assume that G 1 , · · · , G k are Markovian subgraphs of an undirected graph G, then theremust exist such graphs as G ′ and G ′′ for every pair of Markovian subgraphs that share at least onenode.Let 〈i〉 = {1, 2, · · · , i}. Suppose that V i ∩ V i+1 ≠ ∅ for i = 1, 2, · · · , k − 1 and that we havea graph G 〈j〉 for 1 < j < k whose Markovian subgraphs are G i , i = 1, 2, · · · , j. Then there mustexist G 〈j+1〉 of which G 〈j〉 and G j+1 are Markovian subgraphs. Otherwise, the assumption for Gbecomes invalid. We state this in a formal manner below.THEOREM 6. If two graphs, G and H, are C-compatible for C = V (G) ∩ V (H), then there existsa CMS of G and H.Proof. When G = H, the result is trivial since a graph is a CMS of itself. Suppose that|V (G)\V (H)| = 1. Then we can construct a graph H 1 of which G and H are Markovian subgraphsas described below.Let {α} = V (G) \ V (H). Then we can think of the following three cases:(i) bd G (α) = C and there exists a connectivity component g in H for which C ⊆ cl H (g).(ii) bd G (α) = C and there does not exist any connectivity component as in (i) but a connectivitycomponent g ′ for which ∅ ⊂ cl H (g ′ ) ∩ C ⊂ C.(iii) bd G (α) ⊂ C, i.e., 〈α|bd G (α)|C \ cl G (α)〉 G .In case (i): In this case, C ⊆ cl H (g). So node α may be attached to any clique in H that isconnected to C in H, in such a way that h ∪ {α} may form a new clique in H 1 .In case (ii): In this case, α is attached to H to form H 1 such that bd H 1(α) = C.In case (iii): If there exists a connectivity component g in H such that bd G (α) ⊆ cl H (g) andC \ bd G (α) = C \ cl H (g), then α can be attached to any clique in H that is connected to bd G (α) toform a new clique in H 1 . If there is no such a connectivity component in H, then we attach α to Hsuch that bd H 1(α) = C.Now suppose that |V (G) \ V (H)| > 1. Let G 0 = G C and V (G) \ V (H) = {α 1 , · · · , α k }. LetC i = C ∪ {α 1 , · · · , α i } and G i = G Ci for i = 1, · · · , k, where G i ⊆ M G i+1 , i = 0, 1, · · · , k − 1.Then, by the transitivity property of Markovian subgraphs (Theorem 6 in Kim (2006b)), we haveG i ⊆ M G, i = 0, 1, · · · , k.By applying the above argument, we can obtain a graph H i of which H i−1 and G i are Markoviansubgraphs for i = 1, · · · , k, where H 0 = H. By the transitivity property of Markovian subgraphs,we have H ⊆ M H k . Therefore, H and G = G k are Markovian subgraphs of H k . This completesthe proof.10

This theorem can be extended to a set of graphs where each graph is compatible with at leastone of the other graphs of the set as shown in the following corollary.COROLLARY 3. For graphs G i , i = 1, 2, · · · , m, let G 〈i〉 be a graph of which G j is a Markovian subgraph,j ≤ i. If G 〈i〉 and G i+1 are C i -compatible with C i = V (G 〈i〉 )∩V (G i+1 ), i = 1, 2, · · · , m−1,then there exists a CMS of G i , i = 1, 2, · · · , m.Proof. Since G 〈m−1〉 and G m are C m−1 -compatible by the condition of the corollary, thereexists, by Theorem 6, a CMS, I, of G 〈m−1〉 and G m . By the transitivity property of Markoviansubgraphs, I is a CMS of G i , i = 1, 2, · · · , m.6 MARKOVIAN COMBINATION OF MARGINAL MODELSIn the proof of Theorem 6, we considered, to show existence of a CMS, how we can add an edge betweena node in V (G)\V (H) and another node in H with no confliction with the node-separatenessthat is found in at least one of the graphs. The two graphs in Figure 1 are {1, 3}-compatible andtheir CMS’s are as in Figure 4. As for the two graphs in Figure 1, consider adding edges betweennode 4 in V (G 2 ) \ V (G 1 ) and some nodes in G 1 . Because of the node-separateness in G 1 , node 4can only be adjacent to nodes 1 and 2 or to nodes 2 and 3 as in Figure 4.Since a CMS, H say, of a pair of compatible graphs, G ′ and G ′′ say, is obtained in the formof attaching the nodes in V (G ′ ) \ V (G ′′ ) (or V (G ′′ ) \ V (G ′ )) to G ′′ (or G ′ ), it may be regarded ascombining the two graphs together. We will call this combination a Markovian combination in thesense thatM(H) ⊆ ˜L(G ′ , G ′′ );in other words, a probability <strong>model</strong> P which is globally Markov with respect to H has its marginals,P V (G ′ ) and P V (G ′′ ), globally Markov with respect to G ′ and G ′′ respectively.Since a maximal CMS has a better property than CMS’s in the context of Theorem 5, we willpropose a combination method for maximal CMS’s <strong>based</strong> on a set of marginal <strong>model</strong> <strong>structures</strong>.In the combination, it is imperative that node-separateness is preserved between a graph and itsMarkovian subgraph. This is reflected in the combination process in such a way that the followingcondition is satisfied:[Separateness condition ] Let M be a set of Markovian subgraphs of G and H a maximal CMS ofM. If two nodes are in a graph in M and they are not adjacent in the graph, then neither arethey in H. Otherwise, adjacency of the nodes in H is determined by checking separateness ofthe nodes in M.Two main rules of Markovian combination are ‘union’ and ‘check of separateness.’ We willdescribe each of them below.11234234Figure 4: Two CMS’s of the graphs in Figure 1.11

135713 517214682G G 1 G 23 57edge appearing in both of G 1 and G2edge between the nodes that appearin only one of G 1 and G2628 4 63 517828284(a) Union64 6(b) Check of separatenessFigure 5: Markovian combination of graphs. The Markovian subgaphs G 1 and G 2 of G are combinedin two steps, union and check of separateness. Different colors are used for G 1 (in blue) and G 2 (inred). When an edge appears in both of the graphs, it is in black; an edge is colored green when itstwo nodes are not in the same graph of G 1 or G 2 .Union. Suppose we have two Markovian subgraphs, G 1 and G 2 , of a graph G. If nodes u and v arenot separated in any of G 1 and G 2 , we put an edge between the two nodes. We do the samefor all the pairs of nodes that are not separated in any of the subgraphs.If two nodes are not in the same subgraph, then we put an edge between them. If two nodesare shared by G 1 and G 2 and they are connected by an edge in one subgraph but not in theother, we leave them separated. We denote the graph resulting from this operation by G ∗ .Check of separateness. We check if the separateness that is found in G 1 and G 2 holds in G ∗ also. Ifan edge in G ∗ is in conflict with the separateness of some pair of nodes, we remove the edgefrom G ∗ . We denote the graph resulting from this operation by G ∗∗ .This combining process is illustrated in Figure 5. Note that in panel (a), edges, (3,4), (3,7), (4,5),(5,7), are created since the nodes in each of the pairs are not in the same graph of G 1 or G 2 . Two ofthe edges are removed in panel (b) since their existence are in conflict with the node-separatenessthat is embedded in G 1 and G 2 . The combined result contains two edges more than the true graphG in Figure 5. It is interesting to note in this figure that G 1 and G 2 are decomposable while none ofG and the combined graph is. This is an example that the Markovian combination of decomposablegraphs does not necessarily produce a decomposable combined graph.Another illustration is given in Figure 6 where the graph G is not a chain of cycles as in Figure5 but a more general form of undirected graphs. A 4-cycle {3, 4, 7, 8} is surrounded by a 7-cycle{1, 2, 5, 6, 9, 10, 11} in the graph. The combined graph which appears in panel (b) contains allthe edges in the graph G in addition to the edges (1,4), (1,7), (5,7), and (7,10). These four edgesappeared in G 1 or G 2 and are not in conflict with any node-separateness that is found in G 1 and G 2 .Note, in Figures 5 and 6, that the black edges in panel (a) which appear in both of the graphs, G 1 andG 2 in each of the figures, are preserved in the combined graph in panel (b). This is consequentialon the fact that the adjacency of a pair of nodes in both of G 1 and G 2 is in no conflict with the12

1212G5346785349111078134G 1 6951778 102 4G11210edge appearing in both of G 1 and G2edge between the nodes that appearin only one of G 1 and G25917382 4911106(a) Union11611(b) Check of separatenessFigure 6: Markovian combination of graphs. In panel (a), there are 12 green edges for the pairs ofnodes that do not appear in the same graph of G 1 or G 2 . Three of the green edges remain in panel(b).node-separateness in both of the graphs.The combined graphs which are obtained through the two operations are maximal CMS’s of agiven set of Markovian subgraphs as shown in the theorem below.THEOREM 7. The combination process by the two operations of Union and Check of separatenessproduces a maximal CMS.Proof. Let M be a set of Markovian subgraphs of a graph. The “Union” operation puts an edgebetween a pair of nodes, u and v say, unless u and v are both in a graph in M and separated therein.Denote the graph from this operation by G ∗ . It is obvious that G ′ ⊆ e G ∗ V (G ′ ) for every G′ ∈ M.The “Check of separateness” operation removes edges from G ∗ in such a way that the followingcondition is satisfied for every G ′ in M:For any pair of non-adjacent nodes u and v in G ′ and a set C in G ′ which is disjoint with {u, v},〈u|C|v〉 G ′ if and only if 〈u|C|v〉 G ∗. (6)Denote a graph obtained from this check-of-separateness by G ∗∗ . Then any pair of non-adjacentnodes, u and v say, in G ∗∗ mean either (i) that they are non-adjacent in at least one of the graphsin M or (ii) that they belong to different graphs each and putting an edge between the nodes incursconflicts with the node-separateness in some of the graphs in M.Therefore, adding any edge to G ∗∗ into another graph G ′′ disqualifies G ′′ as a CMS of the graphsin M. This means that G ′′ is a maximal CMS of the graphs in M.13

2213 1⇐⇒3 1 3(1a)(1b)Interaction graphs (G 1 ) of X 1 , X 2 , X 3 G ′222134134 134⇐⇒(2a) (2a’) (2b)Interaction graphs (G 2 ) of X 1 , · · · , X 4 G ′′Figure 7: Some simple examples where each of the graphs in the right column are not a Markoviansubgraph of any of the graphs on the left-hand side of ⇐⇒.1347 FURTHER DISCUSSIONIn Theorem 2, we are given a set of Markovian subgraphs of G. But in reality, we are oftengiven a set of marginal <strong>model</strong> <strong>structures</strong> that are assumed to be interaction graphs of the marginal<strong>model</strong>s. The interaction graphs may not be Markovian subgraphs of the unknown G. In this case,maximal CMS’s may not contain G as an edge-subgraph. Simple examples of this situation aredisplayed in Figure 7. In the first row of the figure are two interaction graphs (G 1 ) for X 1 , X 2 , X 3and a subgraph G ′ which is not Markovian with respect to G 1 , and similarly in the second row forX 1 , · · · , X 4 . Under the hierarchy assumption for contingency tables, none of the graphical loglinear<strong>model</strong>s (1a), (2a), and (2a’) is compatible with the graphical sub<strong>model</strong>s at the right ends ofthe corresponding rows by Theorem 2.3 of Asmussen and Edwards (1983). The <strong>model</strong> G ′ in Figure7 is possible with the graphical log-linear <strong>model</strong> (1b) in the figure whenE[(P (X {1,3} = x {1,3} |X 2 )] = P (X 1 = x 1 )P (X 3 = x 3 ) for all x {1,3} ∈ X {1,3} , (7)where X i is the support of X i and X a = ∏ i∈a X i. The graphical log-linear <strong>model</strong> G ′′ in Figure 7 isalso possible from the graphical <strong>model</strong> (2b) in the figure. Instances of this phenomenon follow.Example 1. Probability distributions corresponding to some of the graphs in Figure 7. We willpresent contingency tables for which the pair of <strong>model</strong>s, (1b) and G ′ in Figure 7, are possible and soare the pair of <strong>model</strong>s, (2b) and G ′′ . When X i and X j are conditionally independent given X k , wewill simply write i ⊥j|k.(a) Concerning <strong>model</strong>s (1b) and G ′ :x 2 x 1 x 3 P (X = x) x 2 x 1 x 3 P (X = x)0 0 0 1/24 1 0 0 2/241 3/24 1 6/241 0 2/24 1 0 1/241 6/24 1 3/24This distribution satisfies that 1 ⊥3|2 and 1 ⊥3.(b) Concerning <strong>model</strong>s (2b) and G ′′ :14

x 2 x 3 x 1 x 4 P (X = x) x 2 x 3 x 1 x 4 P (X = x)0 0 0 0 1/42 1 0 0 0 3/421 2/42 1 1/421 0 2/42 1 0 6/421 4/42 1 2/421 0 0 2/42 1 0 0 6/421 4/42 1 2/421 0 1/42 1 0 3/421 2/42 1 1/42This distribution satisfies the conditional independencies displayed in graph (2b) in Figure 7.The marginal for X {1,3,4} satisfies the conditional independence 1 ⊥4|3.Although we have seen examples where subgraphs of graphical log-linear <strong>model</strong>s are not Markovian,Markovian subgraphs are usual situations under the hierarchy assumption for <strong>model</strong>s. As indicatedin (7), in order for a subgraph to be non-Markovian, a certain set of equations must be satisfiedbetween the set of parameters of a joint <strong>model</strong> and that of its interested non-Markovian subgraph.This implies that non-Markovian subgraphs are a rare situation under the hierarchy assumption aslong as interaction graphs are concerned. Furthermore, when the distribution is Normal, we can seeby its density function that the subgraphs are Markovian. Based on this point of view on Markoviansubgraphs, we have assumed in this paper that all the interaction graphs of subsets V i of randomvariables are Markovian.The combination of <strong>model</strong> <strong>structures</strong> is in two steps, Union and Check of separateness. Supposewe combine the graphs in M. At the ‘Union’ step, we put an edge between every pair of nodesunless there exists at least one graph in M where both of the nodes appear and are not adjacent; atthe ‘Check of separateness’ step, we then remove an edge when its existence is in conflict with thenode-separateness in the graphs in M. In this process, we don’t need data but the <strong>model</strong> <strong>structures</strong>.In this sense, the proposed method reuses the information that is embedded in the marginal <strong>model</strong><strong>structures</strong> for learning <strong>structures</strong> of a larger set of random variables which are involved in at leastone of the graphs in M.REFERENCESBalagtas, C.C., Becker, M.P. & Lang, J.B. (1995). Marginal <strong>model</strong>ling of categorical data fromcrossover experiments, Appl. Statist. 44, 63-77.Bartolucci, F. & Forcina, A. (2002). Extended RC association <strong>model</strong>s allowing for order restrictionsand marginal <strong>model</strong>ing, J. Am. Statist. Assoc. 97, 1192-9.Becker, M.P. (1994). Analysis of repeated categorical measurements using <strong>model</strong>s for marginaldistributions: an application to trends in attitudes on legalized abortion. In SociologicalMethodology, Ed. P.V. Marsden, pp. 229-65. Oxford: Blackwell.Becker, M.P., Minick, S. & Yang, I. (1998). Specifications of <strong>model</strong>s for cross-classified counts:comparisons of the log-linear <strong>model</strong> and marginal <strong>model</strong> perspectives, Sociological Methodsand Research 26, 511-29.15

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