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Approximate Inference on Planar Graphs using Loop Calculus and Belief Propagation
, 2010
"... We introduce novel results for approximate inference on planar graphical models using the loop calculus framework. The loop calculus (Chertkov and Chernyak, 2006a) allows to express the exact partition function of a graphical model as a finite sum of terms that can be evaluated once the belief propa ..."
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We introduce novel results for approximate inference on planar graphical models using the loop calculus framework. The loop calculus (Chertkov and Chernyak, 2006a) allows to express the exact partition function of a graphical model as a finite sum of terms that can be evaluated once the belief propagation (BP) solution is known. In general, full summation over all correction terms is intractable. We develop an algorithm for the approach presented in Chertkov et al. (2008) which represents an efficient truncation scheme on planar graphs and a new representation of the series in terms of Pfaffians of matrices. We analyze the performance of the algorithm for models with binary variables and pairwise interactions on grids and other planar graphs. We study in detail both the loop series and the equivalent Pfaffian series and show that the first term of the Pfaffian series for the general, intractable planar model, can provide very accurate approximations. The algorithm outperforms previous truncation schemes of the loop series and is competitive with other state of the art methods for approximate inference.
Graph polynomials and approximation of partition functions with Loopy Belief Propagation
, 2009
"... 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 fu ..."
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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.