  In Bayesian analysis, the posterior follows from the data and a choice of a
prior and a likelihood. One hopes that the posterior is robust to reasonable
variation in the choice of prior and likelihood, since this choice is made by
the modeler and is necessarily somewhat subjective. Despite the fundamental
importance of the problem and a considerable body of literature, the tools of
robust Bayes are not commonly used in practice. This is in large part due to
the difficulty of calculating robustness measures from MCMC draws. Although
methods for computing robustness measures from MCMC draws exist, they lack
generality and often require additional coding or computation.
  In contrast to MCMC, variational Bayes (VB) techniques are readily amenable
to robustness analysis. The derivative of a posterior expectation with respect
to a prior or data perturbation is a measure of local robustness to the prior
or likelihood. Because VB casts posterior inference as an optimization problem,
its methodology is built on the ability to calculate derivatives of posterior
quantities with respect to model parameters, even in very complex models. In
the present work, we develop local prior robustness measures for mean-field
variational Bayes(MFVB), a VB technique which imposes a particular
factorization assumption on the variational posterior approximation. We start
by outlining existing local prior measures of robustness. Next, we use these
results to derive closed-form measures of the sensitivity of mean-field
variational posterior approximation to prior specification. We demonstrate our
method on a meta-analysis of randomized controlled interventions in access to
microcredit in developing countries.
