Improving variational methods via pairwise linear response identities

TL;DR

This paper introduces covariance constraints in variational inference methods to improve the consistency and accuracy of marginal probability estimates, especially for Bethe approximations, through a variational framework and message passing algorithms.

Contribution

It proposes a novel approach to enforce covariance consistency in variational methods, enhancing inference accuracy for complex probabilistic models.

Findings

Covariance constraints improve marginal distribution estimates.

The method ensures linear response consistency with variational parameters.

Simple constraints suffice for Bethe approximation improvements.

Abstract

Inference methods are often formulated as variational approximations: these approximations allow easy evaluation of statistics by marginalization or linear response, but these estimates can be inconsistent. We show that by introducing constraints on covariance, one can ensure consistency of linear response with the variational parameters, and in so doing inference of marginal probability distributions is improved. For the Bethe approximation and its generalizations, improvements are achieved with simple choices of the constraints. The approximations are presented as variational frameworks; iterative procedures related to message passing are provided for finding the minima.