The Bethe Partition Function of Log-supermodular Graphical Models

TL;DR

This paper proves that for binary graphical models with log-supermodular potentials, the Bethe partition function always provides a lower bound on the true partition function, confirming a longstanding conjecture.

Contribution

The paper establishes that the Bethe approximation lower bounds the true partition function for a broad class of models with log-supermodular potentials, extending previous results.

Findings

Bethe partition function lower bounds the true partition function for these models

The proof introduces a new variant of the 'four functions' theorem

Confirms the conjecture by Sudderth, Wainwright, and Willsky

Abstract

Sudderth, Wainwright, and Willsky have 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 log-supermodular, 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.