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

James Berger, Larry Brown, and Robert Wolpert

Pre-experimental Frequentist error probabilities are arguably
inadequate, as summaries of evidence from data,
in many hypothesis-testing settings. The Conditional
Frequentist responds 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.
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