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

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