#### Could Fisher, Jeffreys and Neyman have agreed on testing?

James Berger

Ronald Fisher advocated testing using p-values; Harold Jeffreys
proposed use of objective posterior probabilities of hypotheses;
and Jerzy Neyman recommended testing with fixed error
probabilities. Each was quite critical of the other approaches.
Most troubling for statistics and science is that the three
approaches can lead to quite different practical conclusions.
We focus on discussion of the conditional frequentist
approach to testing, which is argued to provide the basis for a
methodological unification of the approaches of Fisher, Jeffreys
and Neyman. The idea is to follow Fisher, in using p-values to
define the `strength of evidence' in data, and to follow his
approach of conditioning on strength of evidence; then follow
Neyman by computing Type I and Type II error probabilities, but do
so conditional on the strength of evidence in the data. The
resulting conditional frequentist error probabilities equal the
objective posterior probabilities of the hypotheses advocated by
Jeffreys.

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