Fusion and Propagation in Graphical Belief Models

Graphical models give a clear and concise way of describing dependencies among many variables. Only relationships among variables which all share a common hyperedge must be modeled, considerably simplifying both the modeling and the computational task. Graphical models have been studied by Pearl [1986a,1986b], Moussouris[1974], and Lauritzen and Spiegelhalter[1988] in the Bayesian case, and Kong[1986a], Shafer, Shenoy, and Mellouli [1986] and Shenoy and Shafer[1986] in the Belief Function case. Belief functions are a generalization of probability that allow ways to express total ignorance, Bayesian prior probability distributions, conditional probability distributions (likelihoods), logical relationships (production rules) and observations. All these diverse types of knowledge can be combined with a uniform fusion rule, the direct sum operator. Belief functions can be simply restricted to a smaller frame and easily extended to a larger frame without adding additional information. The theory of belief functions is developed in Shafer[1976,1982] and Kong[1986a]. By a simple procedure given here and in Kong[1986b],we can transform the model hypergraph into a tree of closures. This is a tree of “chunks” of the original problem, each “chunk” can be computed independently of all other chunks except its neighbors. Each node in the tree of closures passes messages (expressed as belief functions) to its neighbors consisting of the local information fused with all the information that has propagated through the other branches of the tree. Using this propagation algorithm along with the fusion algorithm given by the direct sum operator, we can easily compute marginal beliefs, and can save considerable computational cost over the brute force approach. Key Concepts: Graphical Models, Belief Functions, Bayesian Models, Fusion and Propagation, Probability in Expert Systems, Triangulated Graphs.