Local influence on posterior distributions under multiplicative modes of perturbation

Abstract

Any unperturbed and perturbed posterior density can formally be linked by a mixture. Many divergences between the unperturbed and perturbed posterior density - global measures of influence of the perturbation - are then essentially determined by the Fisher information with respect to the mixing parameter evaluated at the unperturbed density. It is investigated which aspect of change this Fisher information - commonly interpreted as local measure of influence - captures in assessing influence of the perturbation. Under multiplicative modes of perturbation it is nicely interpretable as unperturbed posterior variance of the (log-)perturbation function