Correctness of Belief Propagation in Gaussian Graphical Models of Arbitrary Topology
Authors: William T. Freeman Yair Weiss
Published in: Advances in Neural Information Processing Systems 12  (NIPS 1999) (Main Conference Track), 1999

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. 
Problems involving probabilistic belief propagation arise in a wide variety of applications,  including error correcting codes,  speech recognition and medical diagnosis.  If the  graph  is  singly connected, there exist local message-passing schemes to  calculate the  posterior  probability of an unobserved variable given the observed variables.  Pearl [15] derived such  a scheme for singly connected Bayesian networks and showed that this "belief propagation"  algorithm is guaranteed to converge to the correct posterior probabilities (or "beliefs"). 
Several groups have recently reported excellent experimental results by running algorithms 
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Y.  Weiss  and W T.  Freeman 
equivalent to Pearl's algorithm on networks with loops [8, 13, 6].  Perhaps the most dramatic  instance of this performance is for "Turbo code"  [2]  error correcting codes.  These codes  have been described as "the most exciting and potentially important development in coding  theory  in  many years" [12]  and have recently been shown  [10,  11] to utilize an algorithm  equivalent to belief propagation in a network with loops. 
Progress in the analysis of loopy belief propagation has been made for the case of networks  with  a  single  loop  [17,  18,  4,  1].  For  these  networks,  it  can  be  shown  that  (1)  unless  all  the  compatabilities are  deterministic,  loopy belief propagation will converge.  (2) The  difference between the loopy beliefs and the true beliefs is related to the convergence rate  the faster the convergence the more exact the approximation and (3) If  of the messages - the hidden nodes are binary, then the loopy beliefs and the true beliefs are both maximized  by the same assignments, although the confidence in that assignment is wrong for the loopy  beliefs. 
In this paper we analyze belief propagation in  graphs of arbitrary topology, for nodes de(cid:173) scribing jointly Gaussian random variables.  We  give an exact formula relating the correct  marginal  posterior probabilities with  the  ones calculated using  loopy  belief propagation.  We  show that if belief propagation converges, then it will give the correct posterior means  for all graph  topologies, not just networks with  a single loop.  We  show that the covari(cid:173) ance estimates will  generally be  incorrect but present a relationship between the error in  the covariance estimates and the convergence speed.  For Gaussian or non-Gaussian vari(cid:173) ables,  we  show  that the  "max-product" algorithm,  which  calculates the MAP estimate in  singly connected networks, only converges to points that are maxima over a particular large  neighborhood of the posterior probability of loopy networks.