  Loopy belief propagation (LBP), which is equivalent to the Bethe
approximation in statistical mechanics, is a message-passing-type inference
method that is widely used to analyze systems based on Markov random fields
(MRFs). In this paper, we propose a message-passing-type method to analytically
evaluate the quenched average of LBP in random fields by using the replica
cluster variation method. The proposed analytical method is applicable to
general pair-wise MRFs with random fields whose distributions differ from each
other and can give the quenched averages of the Bethe free energies over random
fields, which are consistent with numerical results. The order of its
computational cost is equivalent to that of standard LBP. 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 the numerical results for natural images.
