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Loop series and Bethe variational bounds in attractive graphical models
, 2008
"... Variational methods are frequently used to approximate or bound the partition or likelihood function of a Markov random field. Methods based on mean field theory are guaranteed to provide lower bounds, whereas certain types of convex relaxations provide upper bounds. In general, loopy belief propaga ..."
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Cited by 33 (0 self)
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Variational methods are frequently used to approximate or bound the partition or likelihood function of a Markov random field. Methods based on mean field theory are guaranteed to provide lower bounds, whereas certain types of convex relaxations provide upper bounds. In general, loopy belief propagation (BP) provides often accurate approximations, but not bounds. We prove that for a class of attractive binary models, the so–called Bethe approximation associated with any fixed point of loopy BP always lower bounds the true likelihood. Empirically, this bound is much tighter than the naive mean field bound, and requires no further work than running BP. We establish these lower bounds using a loop series expansion due to Chertkov and Chernyak, which we show can be derived as a consequence of the tree reparameterization characterization of BP fixed points.
The Bethe partition function of logsupermodular graphical models
 In Neural Information Processing Systems
, 2012
"... Sudderth, Wainwright, and Willsky conjectured that the Bethe approximation corresponding to any fixed point of the belief propagation algorithm over an attractive, pairwise binary graphical model provides a lower bound on the true partition function. In this work, we resolve this conjecture in the a ..."
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Cited by 13 (1 self)
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Sudderth, Wainwright, and Willsky conjectured that the Bethe approximation corresponding to any fixed point of the belief propagation algorithm over an attractive, pairwise binary graphical model provides a lower bound on the true partition function. In this work, we resolve this conjecture in the affirmative by demonstrating that, for any graphical model with binary variables whose potential functions (not necessarily pairwise) are all logsupermodular, the Bethe partition function always lower bounds the true partition function. The proof of this result follows from a new variant of the “four functions ” theorem that may be of independent interest. 1
PRELIMINARY VERSION SUBJECT TO REVISION Loop Series and Bethe Variational Bounds in Attractive Graphical Models
"... Variational methods are frequently used to approximate or bound the partition or likelihood function of a Markov random field. Methods based on mean field theory are guaranteed to provide lower bounds, whereas certain types of convex relaxations provide upper bounds. In general, loopy belief propaga ..."
Abstract
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Variational methods are frequently used to approximate or bound the partition or likelihood function of a Markov random field. Methods based on mean field theory are guaranteed to provide lower bounds, whereas certain types of convex relaxations provide upper bounds. In general, loopy belief propagation (BP) provides often accurate approximations, but not bounds. We prove that for a class of attractive binary models, the so–called Bethe approximation associated with any fixed point of loopy BP always lower bounds the true likelihood. Empirically, this bound is much tighter than the naive mean field bound, and requires no further work than running BP. We establish these lower bounds using a loop series expansion due to Chertkov and Chernyak, which we show can be derived as a consequence of the tree reparameterization characterization of BP fixed points. 1