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Mean field variational Bayes (MFVB) is a popular posterior approximationmethod due to its fast runtime on large-scale data sets. However, it is wellknown that a major failing of MFVB is that it underestimates the uncertainty ofmodel variables (sometimes severely) and provides no information about modelvariable covariance. We develop a fast, general methodology for exponentialfamilies that augments MFVB to deliver accurate uncertainty estimates for modelvariables -- both for individual variables and coherently across variables.MFVB for exponential families defines a fixed-point equation in the means ofthe approximating posterior, and our approach yields a covariance estimate byperturbing this fixed point. Inspired by linear response theory, we call ourmethod linear response variational Bayes (LRVB). We also show how LRVB can beused to quickly calculate a measure of the influence of individual data pointson parameter point estimates. We demonstrate the accuracy and scalability ofour method by learning Gaussian mixture models for both simulated and realdata.

Covariance Matrices and Influence Scores for Mean Field Variational Bayes