The Annals of Statistics

A Unified Conditional Frequentist and Bayesian Test for Fixed and Sequential Simple Hypothesis Testing

James O. Berger, Lawrence D. Brown, and Robert L. Wolpert

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Preexperimental frequentist error probabilities are arguably inadequate, as summaries of evidence from data, in many hypothesis-testing settings. The conditional frequentist may respond to this by identifying certain subsets of the outcome space and reporting a conditional error probability, given the subset of the outcome space in which the observed data lie. Statistical methods consistent with the likelihood principle, including Bayesian methods, avoid the problem by a more extreme form of conditioning. In this paper we prove that the conditional frequentist's method can be made exactly equivalent to the Bayesian's in simple versus simple hypothesis testing: specifically, we find a conditioning strategy for which the conditional frequentist's reported conditional error probabilities are the same as the Bayesian's posterior probabilities of error. A conditional frequentist who uses such a strategy can exploit other features of the Bayesian approach--for example, the validity of sequential hypothesis tests (including versions of the sequential probability ratio test, or SPRT) even if the stopping rule is incompletely specified.

Article information

Ann. Statist., Volume 22, Number 4 (1994), 1787-1807.

First available in Project Euclid: 11 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62A20
Secondary: 62A15

Likelihood principle conditional frequentist Bayes factor likelihood ratio significance Type I error Bayesian statistics stopping rule principle


Berger, James O.; Brown, Lawrence D.; Wolpert, Robert L. A Unified Conditional Frequentist and Bayesian Test for Fixed and Sequential Simple Hypothesis Testing. Ann. Statist. 22 (1994), no. 4, 1787--1807. doi:10.1214/aos/1176325757.

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