  The Bethe approximation, or loopy belief propagation algorithm is a
successful method for approximating partition functions of probabilistic models
associated with a graph. 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
labeled by subgraphs called generalized loops. In our recent paper, we derive
the loop series expansion in form of a polynomial with coefficients positive
integers, and extend the result to the expansion of marginals. In this paper,
we give more clear derivation for the results and discuss the properties of the
polynomial which is introduced in the paper.
