Testing Normal Means: The Reconcilability of the P Value and the Bayesian Evidence

The problem of reconciling the frequentist and Bayesian evidence in testing statistical hypotheses has been extensively studied in the literature. Most of the existing work considers cases without the nuisance parameters which is not the frequently encountered situation since the presence of the nuisance parameters is very common in practice. In this paper, we consider the reconcilability of the Bayesian evidence against the null hypothesis H0 in terms of the posterior probability of H0 being true and the frequentist evidence against H0 in terms of the P value in testing normal means where the nuisance parameters are present. The reconcilability of evidence can be obtained both for testing a normal mean and for the Behrens-Fisher problem.