From p -Values to Posterior Probabilities of Null Hypotheses.

Minimum Bayes factors are commonly used to transform two-sided p -values to lower bounds on the posterior probability of the null hypothesis, in particular the bound -e·p·log(p). This bound is easy to compute and explain; however, it does not behave as a Bayes factor. For example, it does not change with the sample size. This is a very serious defect, particularly for moderate to large sample sizes, which is precisely the situation in which p -values are the most problematic. In this article, we propose adjusting this minimum Bayes factor with the information to approximate an exact Bayes factor, not only when p is a p -value but also when p is a pseudo- p -value. Additionally, we develop a version of the adjustment for linear models using the recent refinement of the Prior-Based BIC.