Title: Consistency and testability

Abstract

Bayesian consistency theorems come in (at least) three distinct types, e.g. Doob's prior-almost-sure consistency on Polish spaces, Schwartz's Hellinger consistency with KL-priors and the `tailfree' weak consistency of Dirichlet posteriors. We ask the question how these notions are related and argue that one characterizes them most conveniently using tests. We show that the existence of Bayesian tests is equivalent with Doob-like consistency of the posterior and show that Bayesian tests exist in much greater abundance than uniform tests. As examples we consider hypothesis testing problems like Cover's rational mean problem, tests for smoothness in Sobolev classes and tests for connectedness or cyclicality in networks. To achieve frequentist posterior consistency, we combine Bayesian tests with a prior condition that generalises Schwartz's KL-condition and accommodates weak consistency, e.g. involving the `tailfree' property of the Dirichlet distribution and others.

Location, KdV Meeting Room (Science Park 105-107, F3.20)