Statistical Analysis of Loopy Belief Propagation based on Replica Cluster Variation Method
The estimation of the statistical performances of signal processing systems that use Bayesian frameworks and Markov random fields (MRFs), such as Bayesian image restoration, is often reduced to the statistical mechanical analysis of spin models in random fields. Since many such systems have been implemented using the loopy belief propagation (LBP), which is equivalent to the Bethe approximation in statistical mechanics, in order to estimate their practical performances, we have to evaluate the statistical behavior of LBP in random fields. In this paper, we propose a message-passing type of method that allows quenched averages over random fields of Bethe free energies in general pair-wise MRFs to be analytically evaluated by using the replica cluster variation method. In the latter part of this paper, we describe the application of the proposed method to Bayesian image restoration, in which we observed that our theoretical results are in good agreement with numerical results.