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

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