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

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