Covariances, Robustness, and Variational Bayes

Variational Bayes (VB) is an approximate Bayesian posterior inference technique that is increasingly popular due to its fast runtimes on large-scale datasets. However, even when VB provides accurate posterior means for certain parameters, it often mis-estimates variances and covariances. Furthermore, prior robustness measures have remained undeveloped for VB. By deriving a simple formula for the effect of infinitesimal model perturbations on VB posterior means, we provide both improved covariance estimates and local robustness measures for VB, thus greatly expanding the practical usefulness of VB posterior approximations. The estimates for VB posterior covariances rely on a result from the classical Bayesian robustness literature relating derivatives of posterior expectations to posterior covariances. Our key assumption is that the VB approximation provides good estimates of a select subset of posterior means -- an assumption that has been shown to hold in many practical settings. In our experiments, we demonstrate that our methods are simple, general, and fast, providing accurate posterior uncertainty estimates and robustness measures with runtimes that can be an order of magnitude smaller than MCMC.