TY - GEN

T1 - Correctness of belief propagation in Gaussian graphical models of arbitrary topology

AU - Weiss, Yair

AU - Freeman, William T.

PY - 2000

Y1 - 2000

N2 - 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 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 mat, even for non-Gaussian probability distributions, the convergence points of the max-product algorithm in loopy networks are maxima over a particular large local neighborhood of the posterior probability. These results help clarify the empirical performance results and motivate using the powerful belief propagation algorithm in a broader class of networks.

AB - 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 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 mat, even for non-Gaussian probability distributions, the convergence points of the max-product algorithm in loopy networks are maxima over a particular large local neighborhood of the posterior probability. These results help clarify the empirical performance results and motivate using the powerful belief propagation algorithm in a broader class of networks.

UR - http://www.scopus.com/inward/record.url?scp=84898973486&partnerID=8YFLogxK

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AN - SCOPUS:84898973486

SN - 0262194503

SN - 9780262194501

T3 - Advances in Neural Information Processing Systems

SP - 673

EP - 679

BT - Advances in Neural Information Processing Systems 12 - Proceedings of the 1999 Conference, NIPS 1999

PB - Neural information processing systems foundation

T2 - 13th Annual Neural Information Processing Systems Conference, NIPS 1999

Y2 - 29 November 1999 through 4 December 1999

ER -