The mathematical structures of loopy belief propagation are reviewed for graphical models in probabilistic information processing in the stand point of cluster variation method. An extension of adaptive TAP approaches is given by introducing a generalized scheme of the cluster variation method. Moreover the practical message update rules in loopy belief propagation are summarized also for quantum systems. It is suggested that the loopy belief propagation can be reformulated for quantum electron systems by using density matrices of ideal quantum lattice gas system.
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