By Winter P. A.

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1t then triangulates the moral graphGM to obtain a decomposable cover G' (see Fig. 7c). The algorithm operates in a simple, greedy manner based on the fact that a graph is triangulated if and only if all of its nodes can be eliminated, where a node can be elimi nated whenever all of its neighbors are pairwise-linked. Whenever a node is eliminated, it and its neighbors define a clique in the junction tree that is eventually constructed. Thus, we can triangulate a graph and generate the cliques for the junction tree by eliminating nodes in some order, adding links if necessary.

Just as with the inference problem, the Viterbi algorithm provides an ef ficient, locally recursive method for Solving this problem with complexity Nm2, and again, as with the inference problem, the graphical modeling ap proach provides an automatic technique for determining efficient solutions to the MAP problem for arbitrary models, if an efficient solution is possible given the structure of the model. 5 Inference and MAP Algorithms for DPiNs ________ MAP algorithms for DPINs and UPINS are quite similar: the UPIN case involves some subtleties not encountered in DPINs, and so discussion of UPIN inference and MAP algorithms is deferred until Sec tion 7.

In Appendix B it is shown that Dawid's algorithm, when applied to HMM( l,l), is exactly equivalent to the standard Viterbi algorithm. Once again the equivalence is not surprising: Dawid's method and the Viterbi algorithm are both direct applications of dynamic programming to the MAP problem. However, once again, the important point is that Dawid's algorithm is specified for the general case of arbitrary PIN structures and can thus be directly applied to more complex HMMs than HMM( l,l) (such as those discussed later in Section 8).

### 2-3 graphs which have Vizings adjacency property by Winter P. A.

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