On the Minima of Bethe Free Energy in Gaussian Distributions

Authors:
Yu Nishiyama;Sumio Watanabe
Affiliations:
Department of Computational Intelligence and Systems Science, Tokyo Institute of Technology, Yokohama, Japan 226-8503;Precision and Intelligence Laboratory, Tokyo Institute of Technology, Yokohama, Japan 226-8503
Venue:
ICAISC '08 Proceedings of the 9th international conference on Artificial Intelligence and Soft Computing
Year:
2006

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Abstract

Belief propagation (BP) is effective for computing marginal probabilities of a high dimensional probability distribution. Loopy belief propagation (LBP) is known not to compute precise marginal probabilities and not to guarantee its convergence. The fixed points of LBP are known to accord with the extrema of Bethe free energy. Hence, the fixed points are analyzed by minimizing the Bethe free energy.In this paper, we consider the Bethe free energy in Gaussian distributions and analytically clarify the extrema, equivalently, the fixed points of LBP for some particular cases. The analytical results tell us a necessary condition for LBP convergence and the quantities which determine the accuracy of LBP in Gaussian distributions. Based on the analytical results, we perform numerical experiments of LBP and compare the results with analytical solutions.