What Cannot be Learned with Bethe Approximations

TL;DR

This paper investigates the limitations of Bethe approximations in learning graphical model parameters, showing that certain empirical marginals cannot be recovered, thus revealing when Bethe-based learning methods are expected to fail.

Contribution

The paper provides conditions and bounds on empirical marginals that determine their learnability via Bethe approximations, highlighting the failure regimes.

Findings

Bethe approximations fail to recover certain empirical marginals.

Conditions for outer and inner bounds on Bethe learnable marginals.

Large class of marginals cannot be stable fixed points of belief propagation.

Abstract

We address the problem of learning the parameters in graphical models when inference is intractable. A common strategy in this case is to replace the partition function with its Bethe approximation. We show that there exists a regime of empirical marginals where such Bethe learning will fail. By failure we mean that the empirical marginals cannot be recovered from the approximated maximum likelihood parameters (i.e., moment matching is not achieved). We provide several conditions on empirical marginals that yield outer and inner bounds on the set of Bethe learnable marginals. An interesting implication of our results is that there exists a large class of marginals that cannot be obtained as stable fixed points of belief propagation. Taken together our results provide a novel approach to analyzing learning with Bethe approximations and highlight when it can be expected to work or fail.