Approximate Posterior Distributions

Abstract

This paper proposes the use of approximate posterior distributions resulting from operational prior distributions chosen with regard to the realized likelihood function. L.J. Savage's “precise measurement” is generalized for approximation in terms of an arbitrary operational prior density, including mixed-type prior distributions with positive probabilities on singular subsets. A new approximation is also given relating such distributions to absolutely continuous distributions with high local concentrations of density. Mixed-type distributions constructed from the natural conjugate prior distributions are proposed and illustrated in the normal-sampling case for unified Bayesian inference in testing and estimation contexts.