On the minima of bethe free energy in gaussian distributions

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. © 2008 Springer-Verlag Berlin Heidelberg.