  The Bethe approximation is a successful method for approximating partition
functions of probabilistic models associated with a graph. Recently, Chertkov
and Chernyak derived an interesting formula called Loop Series Expansion, which
is an expansion of the partition function. The main term of the series is the
Bethe approximation while other terms are labelled by subgraphs called
generalized loops. In this paper, we derive a loop series expansion of binary
pairwise Markov random fields with propagation diagrams, which describe rules
how first messages and secondary messages propagate. Our approach allows to
express the Loop Series in the form of a polynomial with coefficients positive
integers. Using the propagation diagrams, we establish a new formula that shows
a relation between the exact marginal probabilities and their Bethe
approximations.
