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 are guaranteed to converge to the correct posterior probabilities in singly connected graphical models. Recently, a number of researchers have empirically demonstrated good performance of "loopy belief propagation" -- 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 theoretical 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 maximum posterior probability estimate for singly connected networks. We show that, even for non-Gaussian probability distributions, the convergence 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.