A framework for ordering composite beliefs in belief networks

Given a belief network with evidence, the task of finding the l most probable explanations (MPE) in the belief network is that of identifying and ordering the l most probable instantiations of the nonevidence nodes. Although many approaches have been proposed for solving this problem, most work only for restricted topologies (e.g., singly connected belief networks). In this paper we will present a framework, optimal factoring, for finding the l MPEs in arbitrary belief networks. Under this framework, efficiently finding the MPE in a belief network can be considered as the problem of finding an ordering of the distributions of the belief network and efficiently combining them. We will discuss the essence of the problem of finding the MPE, and present an optimal algorithm for singly connected belief networks and an efficient algorithm for multiply connected belief networks. We will also discuss the problem of finding the MPE for a subset of variables of a belief network under this framework