Classical p-values and the Bayesian posterior probability that the hypothesis is approximately true

Abstract

This paper relates p-values for the hypothesis that θ=c to the Bayesian posterior probability that the hypothesis is approximately true, in the sense that θ∈[c−ϵ,c+ϵ] for a selected ϵ>0. In a setup with a continuous prior for θ, the results show that a larger (respectively, smaller) p-value does not necessarily correspond to a higher (respectively, lower) probability that θ is close to c. Therefore, the results suggest caution about common ways of using p-values, specifically the use of small p-values as a key standard in empirical research.