Local robustness of Bayesian parametric inference and observed likelihoods

Abstract

Here a new class of local separation measures over prior densities is studied and their usefulness for examining prior to posterior robustness under a sequence of observed likelihoods, possibly erroneous, illustrated. It is shown that provided an approximation to a prior distribution satisfies certain mild smoothness and tail conditions then prior to posterior inference for large samples is robust, irrespective of whether the priors are grossly misspecified with respect to variation distance and irrespective of the form or the validity of the observed likelihood. Furthermore it is usually possible to specify error bounds explicitly in terms of statistics associated with the posterior associated with the approximating prior and asumed prior error bounds. These results apply in a general multivariate setting and are especially easy to interpret when prior densities are approximated using standard families or multivariate prior densities factorise