Bayesian point null hypothesis testing via the posterior likelihood ratio

Neyman-Pearson or frequentist inference and Bayes inference are most clearly differentiated by their approaches to point null hypothesis testing. With very large samples, the frequentist and Bayesian conclusions from a classical test of significance for a point null hypothesis can be contradictory, with a small frequentist P -value casting serious doubt on the null hypothesis, but a large Bayes factor or Bayesian Information Criterion (BIC) in favour of the null hypothesis. We follow the Bayesian approach by Dempster (1974, 1997) and Aitkin (1997) and use the likelihood ratio between the null and alternative hypotheses to provide a different evaluation of the point null hypothesis, one in which frequentist and Bayesian conclusions are much closer. In this paper, we outline the key ideas behind this approach to hypothesis testing and contrast this approach with frequentist P -values and Bayesian analyses based on usual Bayes factors.