Doing What Comes Naturally: Interpreting a Tail Area as a Posterior Probability or as a Likelihood Ratio

Consider a problem in which a certain statistic X has a specified distribution function F(x) if a given hypothesis H is true, and suppose that the hypothesis H is evaluated by calculating the tail area 1 – F(x) corresponding to the observed value x of the statistic X. Examples are given in which this tail area is equal to the posterior probability that H is true and in which it is equal to the likelihood ratio comparing H to a certain class K of alternatives. The purpose of these examples is to render the traditional statistical practice of calculating tail areas consonant with the principles of Bayesian statistics.