On the optimality of solutions of the max-product belief-propagation algorithm in arbitrary graphs

TLDR

Graphical models such as Bayesian networks and Markov random fields use local message‑passing algorithms like max‑product belief propagation, which is known to converge to the optimal assignment on trees and has shown good empirical performance on loopy graphs, yet theoretical understanding beyond simple cycle or single‑loop graphs remains limited. This work proves a general result about the fixed points of max‑product belief propagation on arbitrary graphs with arbitrary probability distributions. The authors show that the assignment derived from a fixed point is a neighborhood maximum of the posterior probability, where the neighborhood consists of all assignments differing only on node subsets that form at most a single loop, and they illustrate this analysis with examples. Consequently, the max‑product assignment is guaranteed to have higher posterior probability than all assignments in this potentially exponentially large neighborhood.

Abstract

Graphical models, such as Bayesian networks and Markov random fields (MRFs), represent statistical dependencies of variables by a graph. The max-product "belief propagation" algorithm is a local-message-passing algorithm on this graph that is known to converge to a unique fixed point when the graph is a tree. Furthermore, when the graph is a tree, the assignment based on the fixed point yields the most probable values of the unobserved variables given the observed ones. Good empirical performance has been obtained by running the max-product algorithm (or the equivalent min-sum algorithm) on graphs with loops, for applications including the decoding of "turbo" codes. Except for two simple graphs (cycle codes and single-loop graphs) there has been little theoretical understanding of the max-product algorithm on graphs with loops. Here we prove a result on the fixed points of max-product on a graph with arbitrary topology and with arbitrary probability distributions (discrete- or continuous-valued nodes). We show that the assignment based on a fixed point is a "neighborhood maximum" of the posterior probability: the posterior probability of the max-product assignment is guaranteed to be greater than all other assignments in a particular large region around that assignment. The region includes all assignments that differ from the max-product assignment in any subset of nodes that form no more than a single loop in the graph. In some graphs, this neighborhood is exponentially large. We illustrate the analysis with examples.