Bayesian Statistics: An Introduction, 4th Edition

4.2 One-sided hypothesis tests

4.2.1 Definition

A hypothesis testing situation of the type described in Section 4.1 is said to be one-sided if the set Θ of possible values of the parameter θ is the set of real numbers or a subset of it and either

or

From the Bayesian point of view, there is nothing particularly special about this situation. The interesting point is that this is one of the few situations in which classical results, and in particular the use of P-values, has a Bayesian justification.

4.2.2 P-values

This is one of the places where it helps to use the ‘tilde’ notation to emphasize which quantities are random. If where is known and the reference prior is used, then the posterior distribution of θ given is . Consider now the situation in which we wish to test versus . Then, ...