Local Influence on Posterior Distributions under Multiplicative Modes of Perturbation

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.