An Asymptotic Viewpoint on High-Dimensional Bayesian Testing*

Abstract

Abstract The Bayesian point-null testing problem is studied asymptotically under a high-dimensional normal means model. Associated procedures are intended for application in finite dimensional settings, but their behavior is studied as the dimensionality grows arbitrarily. Thus, ‘high-dimensionality’ is formulated as an asymptotic concept, and interest is in the behavior of the procedures ‘in the limit’. The approach is to allow the prior null probability to vary with dimension and with prior dispersion parameters, and to guide its parameterization so that the posterior null probability behaves in accordance with Bayesian consistency concepts. Among issues studied are the objectivity of setting the prior null probability to one-half, the Jeffreys–Lindley paradox, and the influence of smoothness constraints. Relevance to applications in functional data analysis and goodness-of-fit testing is also discussed.