Loop series expansion with propagation diagrams

Details Loop series expansion with propagation diagrams Loop series expansion with propagation diagrams

2008-08-08

arxiv.org/abs/0808.1155v2

J.Phys.A42:045001,2009

Physics

Mathematical Physics

17 pages, 14 figures. Accepted for publication in the Journal of Physics A: Mathematical and Theoretical

Scientific paper

10.1088/1751-8113/42/4/045001

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.

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