On the fixed points of the max-product algorithm

Graphical models, such as Bayesian networks and Markov random fields represent statistical dependencies of variables by a graph. The max-product b̈elief propagationälgorithm is a localmessage 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 is guaranteed to yields the most probable a posteriori (MAP) values of the unobserved variables given the observed ones. Here we prove a result on the fixed points of max-product on a graph with arbitrary toplogy and with arbitrary probability distributions (discrete or continuous valued nodes). We show that the assignment based on the fixed-point is a n̈eighborhood maximumöf 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. This work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to copy in whole or in part without payment of fee is granted for nonprofit educational and research purposes provided that all such whole or partial copies include the following: a notice that such copying is by permission of Mitsubishi Electric Research Laboratories, Inc.; an acknowledgment of the authors and individual contributions to the work; and all applicable portions of the copyright notice. Copying, reproduction, or republishing for any other purpose shall require a license with payment of fee to Mitsubishi Electric Research Laboratories, Inc. All rights reserved. Copyright c ©Mitsubishi Electric Research Laboratories, Inc., 1999 201 Broadway, Cambridge, Massachusetts 02139