On the Existence of Uniformly Most Powerful Bayesian Tests With Application to Non-Central Chi-Squared Tests

Abstract

Uniformly most powerful Bayesian tests (UMPBT’s) are an objective class of Bayesian hypothesis tests that can be considered the Bayesian counterpart of classical uniformly most powerful tests. Because the rejection regions of UMPBT’s can be matched to the rejection regions of classical uniformly most powerful tests (UMPTs), UMPBT’s provide a mechanism for calibrating Bayesian evidence thresholds, Bayes factors, classical significance levels and p-values. The purpose of this article is to expand the application of UMPBT’s outside the class of exponential family models. Specifically, we introduce sufficient conditions for the existence of UMPBT’s and propose a unified approach for their derivation. An important application of our methodology is the extension of UMPBT’s to testing whether the non-centrality parameter of a chi-squared distribution is zero. The resulting tests have broad applicability, providing default alternative hypotheses to compute Bayes factors in, for example, Pearson’s chi-squared test for goodness-of-fit, tests of independence in contingency tables, and likelihood ratio, score and Wald tests.