The Annals of Statistics

Uniformly most powerful Bayesian tests

Valen E. Johnson

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Uniformly most powerful tests are statistical hypothesis tests that provide the greatest power against a fixed null hypothesis among all tests of a given size. In this article, the notion of uniformly most powerful tests is extended to the Bayesian setting by defining uniformly most powerful Bayesian tests to be tests that maximize the probability that the Bayes factor, in favor of the alternative hypothesis, exceeds a specified threshold. Like their classical counterpart, uniformly most powerful Bayesian tests are most easily defined in one-parameter exponential family models, although extensions outside of this class are possible. The connection between uniformly most powerful tests and uniformly most powerful Bayesian tests can be used to provide an approximate calibration between $p$-values and Bayes factors. Finally, issues regarding the strong dependence of resulting Bayes factors and $p$-values on sample size are discussed.

Article information

Ann. Statist., Volume 41, Number 4 (2013), 1716-1741.

First available in Project Euclid: 5 September 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62A01: Foundations and philosophical topics 62F03: Hypothesis testing 62F05: Asymptotic properties of tests 62F15: Bayesian inference

Bayes factor Jeffreys–Lindley paradox objective Bayes one-parameter exponential family model Neyman–Pearson lemma nonlocal prior density uniformly most powerful test Higgs boson


Johnson, Valen E. Uniformly most powerful Bayesian tests. Ann. Statist. 41 (2013), no. 4, 1716--1741. doi:10.1214/13-AOS1123.

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