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The Posterior distribution of the Likelihood Ratio (PLR) is proposed byDempster in 1974 for significance testing in the simple vs composite hypothesescase. In this hypotheses test case, classical frequentist and Bayesianhypotheses tests are irreconcilable, as emphasized by Lindley's paradox, Berger& Selke in 1987 and many others. However, Dempster shows that the PLR (withinner threshold 1) is equal to the frequentist p-value in the simple Gaussiancase. In 1997, Aitkin extends this result by adding a nuisance parameter andshowing its asymptotic validity under more general distributions. Here weextend the reconciliation between the PLR and a frequentist p-value for afinite sample, through a framework analogous to the Stein's theorem frame inwhich a credible (Bayesian) domain is equal to a confidence (frequentist)domain.  This general reconciliation result only concerns simple vs compositehypotheses testing. The measures proposed by Aitkin in 2010 and Evans in 1997have interesting properties and extend Dempster's PLR but only by adding anuisance parameter. Here we propose two extensions of the PLR concept to thegeneral composite vs composite hypotheses test. The first extension can bedefined for improper priors as soon as the posterior is proper. The secondextension appears from a new Bayesian-type Neyman-Pearson lemma and emphasizes,from a Bayesian perspective, the role of the LR as a discrepancy variable forhypothesis testing.

Generalizations related to hypothesis testing with the Posterior distribution of the Likelihood Ratio