On the xed points of the max - product algorithm

Graphical models, such as Bayesian networks and Markov random elds, 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 xed point when the graph is a tree. Furthermore, when the graph is a tree, the assignment based on the xedpoint yields the most probable a posteriori (MAP) values of the unobserved variables given the observed ones. Recently, 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 xed 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 the xed-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 di er 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. MERL-TR-99-39 January 2000 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 nonpro t 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 Information Technology Center America; 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 Information Technology Center America. All rights reserved. Copyright c Mitsubishi Electric Information Technology Center America, 2000 201 Broadway, Cambridge, Massachusetts 02139 W. T. Freeman is with MERL, Mitsubishi Electric Research Labs., 201 Broadway, Cambridge, MA 02139. E-mail: [email protected]. Y. Weiss is with Computer Science Division, 485 Soda Hall, UC Berkeley, Berkeley, CA 94720-1776. E-mail: [email protected] On the xed points of the max-product algorithm William T. Freeman, Yair Weiss