  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, since this choice is made by the modeler and
is often somewhat subjective. A different, equally subjectively plausible
choice of prior may result in a substantially different posterior, and so
different conclusions drawn from the data. Were this to be the case, our
conclusions would not be robust to the choice of prior. To determine whether
our model is robust, we must quantify how sensitive our posterior is to
perturbations of our prior. Despite the importance of the problem and a
considerable body of literature, generic, easy-to-use methods to quantify
Bayesian robustness are still lacking.
  Abstract In this paper, we demonstrate that powerful measures of robustness
can be easily calculated from Variational Bayes (VB) approximate posteriors. We
begin with local robustness, which measures the effect of infinitesimal changes
to the prior on a posterior mean of interest. In particular, we show that the
influence function of Gustafson (2012) has a simple, easy-to-calculate closed
form expression for VB approximations. We then demonstrate how local robustness
measures can be inadequate for non-local prior changes, such as replacing one
prior entirely with another. We propose a simple approximate non-local
robustness measure and demonstrate its effectiveness on a simulated data set.
