Equivalence between the Posterior Distribution of the Likelihood Ratio and a p-value in an Invariant Frame

Abstract

The Posterior distribution of the Likelihood Ratio (PLR) is proposed by Dempster in 1973 for significance testing in the simple vs. composite hypothesis case. In this hypothesis test case, classical frequentist and Bayesian hypothesis tests are irreconcilable, as emphasized by Lindley’s paradox, Berger & Selke in 1987 and many others. However, Dempster shows that the PLR (with inner threshold 1) is equal to the frequentist p-value in the simple Gaussian case. In 1997, Aitkin extends this result by adding a nuisance parameter and showing its asymptotic validity under more general distributions. Here we extend the reconciliation between the PLR and a frequentist p-value for a finite sample, through a framework analogous to the Stein’s theorem frame in which a credible (Bayesian) domain is equal to a confidence (frequentist) domain.