The Bayes / Non-Bayes Compromise : A Brief

Some forms of compromise between Bayesian and nonBayesian statistics date back perhaps to Laplace, but the concept of such a compromise seems to have been not fully explicit until much more recently. There is an analogy with the explicit way in which Egon Pearson introduced the nonnull hypotheses and the earlier and less explicit use by Fisher. Pearson ( 1939, p. 242) acknowledged that a letter from "Student" had stimulated him to be more explicit about non-null hypotheses than was customary among P value devotees, and that this suggestion "formed the basis of all the later joint researches of Neyman and myself." Student was very familiar with Bayes's theorem, which uses explicit nonnull hypotheses. So the influence of Bayesianism on the Neyman-Pearson technique seems to have been fairly direct. An explicit mention of the compromise was published 35 years ago, hidden in a paper on saddle-point methods (Good 1957, pp. 862-863). The basic idea was that a Bayesian model not necessarily a good one, could be used to compute a Bayes factor F against a (sharp or point) null hypothesis, and that F then could be used as a significance criterion; that is, its distribution under the null hypothesis could be used for computing P values. A subsidiary suggestion was that F should lie in the range