An asymptotic viewpoint on high-dimensional Bayesian testing

Abstract

The Bayesian point-null testing problem is studied asymptotically under a high-dimensional normal-means model. A noninformative prior structure is proposed for general problems, and then refined for the specialized contexts of goodness-of-fit testing and functional data analysis. The associated tests are demonstrated on existing data sets and shown to provide a cornerstone for a toolbox of detailed analysis tools. The conceptual approach is to allow the prior null probability to vary with dimension and with prior dispersion parameters, then to guide its parametrization so that the posterior null probability behaves in accordance with Bayesian asymptotic-consistency concepts. Among the theoretical issues studied are the objectivity of setting the prior null probability to one-half, the Jeffreys-Lindley paradox, and the influence of smoothness constraints.