Linear Response Methods for Accurate Covariance Estimates from Mean Field Variational Bayes

TL;DR

This paper introduces Linear Response Variational Bayes (LRVB), a method that enhances mean field variational Bayes by providing accurate covariance and uncertainty estimates without restrictive assumptions.

Contribution

The paper generalizes linear response techniques to variational Bayes, enabling accurate covariance estimation in a broad class of models, including non-conjugate cases.

Findings

LRVB provides significantly improved covariance estimates over standard MFVB.

The method is simple, analytic, and applicable to large-scale models.

LRVB demonstrates high accuracy on both simulated and real datasets.

Abstract

Mean field variational Bayes (MFVB) is a popular posterior approximation method due to its fast runtime on large-scale data sets. However, it is well known that a major failing of MFVB is that it underestimates the uncertainty of model variables (sometimes severely) and provides no information about model variable covariance. We generalize linear response methods from statistical physics to deliver accurate uncertainty estimates for model variables---both for individual variables and coherently across variables. We call our method linear response variational Bayes (LRVB). When the MFVB posterior approximation is in the exponential family, LRVB has a simple, analytic form, even for non-conjugate models. Indeed, we make no assumptions about the form of the true posterior. We demonstrate the accuracy and scalability of our method on a range of models for both simulated and real data.