Resolving some paradoxes and problems with Bayesian precise hypothesis testing.

Bayesian hypothesis testing of a precise null hypothesis suffers from a paradox discovered by Jeffreys (1939), Lindley (1957) and Bartlett (1957). This paradox appears to indicate that the usual priors, both proper and improper, are inappropriate for testing precise null hypotheses, and lead to difficulties in specifying prior distributions that could be widely accepted as appropriate in this situation. This paper considers an alternative hypothesis testing procedure and derives the Bayes factor for this procedure, which turns out to be B = p(?0 | x)/sup?[p(?i | x)], the ratio of the posterior density function evaluated at the value in the null hypothesis, ?0, and evaluated at its supremum. This leads to a Bayesian hypothesis testing procedure in which the Jeffreys-Lindley-Bartlett paradox does not occur. Further, under the proposed procedure, the prior does not depend on the hypotheses to be tested, there is no need to place non-zero mass on a particular point in a continuous distribution, and the same hypothesis testing procedure applies for all continuous and discrete distributions. Further, the resulting test procedure is robust to reasonable variations in the prior, uniformly most powerful and easy to interpret correctly in practice. Several examples are given to illustrate the use and performance of the test. A justification for the proposed procedure is given based on the argument that scientific inference always at least implicitly involves and requires precise alternative working hypotheses.