Propagating belief functions in qualitative Markov trees

Abstract This article is concerned with the computational aspects of combining evidence within the theory of belief functions. It shows that by taking advantage of logical or categorical relations among the questions we consider, we can sometimes avoid the computational complexity associated with brute-force application of Dempster's rule. The mathematical setting for this article is the lattice of partitions of a fixed overall frame of discernment. Different questions are represented by different partitions of this frame, and the categorical relations among these questions are represented by relations of qualitative conditional independence or dependence among the partitions. Qualitative conditional independence is a categorical rather than a probabilistic concept, but it is analogous to conditional independence for random variables. We show that efficient implementation of Dempster's rule is possible if the questions or partitions for which we have evidence are arranged in a qualitative Markov tree—a tree in which separations indicate relations of qualitative conditional independence. In this case, Dempster's rule can be implemented by propagating belief functions through the tree.