Isoseparation and robustness in finite parameter Bayesian inference

Under a new family of separations the distance between two posterior densities is the same as the distance between their prior densities whatever the observed likelihood when that likelihood is strictly positive. Local versions of such separations form the basis of a weak topology having close links to the Euclidean metric on the natural parameters of two exponential family densities. Using these local separation measures it is shown that when the tails of the approximating density have appropriate properties, the variation distance between an approximating posterior density to a genuine density can be bounded explicitly. These bounds apply irrespective of whether the prior densities are grossly misspecified with respect to variation distance and irrespective of the form or the validity of the observed likelihood