Gaussian Fields for Approximate Inference in Layered Sigmoid Belief Networks

Abstract

Local "belief propagation" rules of the sort proposed by Pearl [15] are guaranteed to converge to the correct posterior probabilities in singly connected graphical models. Recently, a number of researchers have em(cid:173) pirically demonstrated good performance of "loopy belief propagation"(cid:173) using these same rules on graphs with loops. Perhaps the most dramatic instance is the near Shannon-limit performance of "Turbo codes", whose decoding algorithm is equivalent to loopy belief propagation. Except for the case of graphs with a single loop, there has been little theo(cid:173) retical understanding of the performance 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. The related "max-product" belief propagation algorithm finds the max(cid:173) imum posterior probability estimate for singly connected networks. We show that, even for non-Gaussian probability distributions, the conver(cid:173) gence points of the max-product algorithm in loopy networks are max(cid:173) ima over a particular large local neighborhood of the posterior proba(cid:173) bility. These results help clarify the empirical performance results and motivate using the powerful belief propagation algorithm in a broader class of networks.