From $p$-Values to Posterior Probabilities of Hypothesis

TL;DR

This paper presents a method to convert p-values into posterior probabilities of hypotheses using an adjusted -eplog(p) transformation, applicable to p-values and pseudo p-values, with extensions to linear models.

Contribution

It introduces an adjustment to the -eplog(p) transformation to approximate exact Bayes factors from p-values and pseudo p-values, including for linear models.

Findings

Provides a refined transformation for better posterior probability estimation.

Extends the method to pseudo p-values and linear models.

Improves the interpretability of p-values in Bayesian context.

Abstract

Minimum Bayes factors are commonly used to transform two-sided p-values to lower bounds on the posterior probability of the null hypothesis, as in Pericchi et al. (2017). In this article, we show posterior probabilities of hypothesis by transforming the commonly used -eplog(p), proposed by Vovk (1993) and Sellke et al. (2001). This is achieved after adjusting this minimum Bayes factor with the information to approximate it to an exact Bayes factor, not only when p is a p-value but also when p is a pseudo p-value in the sense of Casella and Berger (2001). Additionally, we show the fit to a refined version to linear models.