Correctness of Belief Propagation in Gaussian Graphical Models of Arbitrary Topology

TLDR

Graphical models encode variable dependencies via graphs, and while Pearl’s belief propagation converges in singly connected graphs, loopy belief propagation has shown strong empirical performance—e.g., Turbo codes—yet theoretical guarantees are limited to single-loop cases. This study investigates belief propagation in arbitrary-topology Gaussian graphical models. The authors derive an analytical expression linking true posteriors to loopy propagation results and establish sufficient convergence conditions. They prove that when loopy belief propagation converges, it yields correct posterior means for all topologies, thereby extending its applicability beyond single-loop networks.

Abstract

Graphical models, such as Bayesian networks and Markov random fields, represent statistical dependencies of variables by a graph. Local "belief propagation" rules of the sort proposed by Pearl (1988) 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 we refer to 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 sufficient 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. These results motivate using the powerful belief propagation algorithm in a broader class of networks and help clarify the empirical performance results.