4 research outputs found
Truncating the loop series expansion for Belief Propagation
Recently, M. Chertkov and V.Y. Chernyak derived an exact expression for the
partition sum (normalization constant) corresponding to a graphical model,
which is an expansion around the Belief Propagation solution. By adding
correction terms to the BP free energy, one for each "generalized loop" in the
factor graph, the exact partition sum is obtained. However, the usually
enormous number of generalized loops generally prohibits summation over all
correction terms. In this article we introduce Truncated Loop Series BP
(TLSBP), a particular way of truncating the loop series of M. Chertkov and V.Y.
Chernyak by considering generalized loops as compositions of simple loops. We
analyze the performance of TLSBP in different scenarios, including the Ising
model, regular random graphs and on Promedas, a large probabilistic medical
diagnostic system. We show that TLSBP often improves upon the accuracy of the
BP solution, at the expense of increased computation time. We also show that
the performance of TLSBP strongly depends on the degree of interaction between
the variables. For weak interactions, truncating the series leads to
significant improvements, whereas for strong interactions it can be
ineffective, even if a high number of terms is considered.Comment: 31 pages, 12 figures, submitted to Journal of Machine Learning
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Approximate inference on planar graphs using loop calculus and belief progagation
We introduce novel results for approximate inference on planar graphical models using the loop calculus framework. The loop calculus (Chertkov and Chernyak, 2006b) allows to express the exact partition function Z 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 in detail both the loop series and the Pfaffian series for models with binary variables and pairwise interactions, and show that the first term of the Pfaffian series 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