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 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.

• Type: Conference
• Year: 2007
• Where: NIPS
• Authors: Erik B. Sudderth, Martin J. Wainwright, Alan S. Willsky
• Keywords: Information Technology, Lower Bounds, NIPS 2007, Upper Bounds, fixed Points