Correctness of Belief Propagation in Gaussian Graphical Models of Arbitrary Topology Correctness of Belief Propagation in Gaussian Graphical Models of Arbitrary Topology

Graphical models, such as Bayesian networks and Markov random elds represent statistical dependencies of variables by a graph. Local \belief propagation" rules of the sort proposed by Pearl [20] are guaranteed to converge to the correct posterior probabilities in singly connected graphs. Recently good performance has been obtained by using these same rules on graphs with loops, a method known as \loopy belief propagation". Perhaps the most dramatic instance is the near Shannon-limit performance of \Turbo codes", whose decoding algorithm is equivalent to loopy propagation. Except for the case of graphs with a single loop, there has been little theoretical understanding of loopy propagation. Here we analyze belief propagation in networks with arbitrary topologies when the nodes in the graph describe jointly Gaussian random variables. We give an analytical formula relating the true posterior probabilities with those calculated using loopy propagation. We give su cient conditions for convergence and show that when belief propagation converges it gives the correct posterior means for all graph topologies, not just networks with a single loop. The related \max-product" algorithm nds the maximumposterior probability estimate for singly connected networks. We show that, even for non-Gaussian probability distributions, the xed points of the max-product algorithm in loopy networks are at least local maxima of the posterior probability. These results motivate using the powerful belief propagation algorithm in a broader class of networks, and help clarify the empirical performance results. Sumbitted to Neural Computation. Preliminary version appeared in Proc. NIPS 99