Loop series expansion with propagation diagrams
  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. 
Yusuke Watanabe and Kenji Fukumizu