Improving variational methods via pairwise linear response identities
This work addresses a specific technical problem in statistical inference for researchers in variational methods, offering incremental improvements.
The paper tackled the inconsistency of linear response estimates in variational approximations by introducing covariance constraints, which improved inference of marginal probability distributions, particularly for the Bethe approximation and its generalizations.
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.