Local sensitivity diagnostics for Bayesian inference

We investigate diagnostics for quantifying the effect of small changes to the prior distribution over a k-dimensional parameter space. We show that several previously suggested diagnostics, such as the norm of the Fréchet derivative, diverge at rate nk/2 if the base prior is an interior point in the class of priors, under the density ratio topology. Diagnostics based on φ-divergences exhibit similar asymptotic behavior. We show that better asymptotic behavior can be obtained by suitably restricting the classes of priors. We also extend the diagnostics to see how various marginals of the prior affect various marginals of the posterior.