Bayesian Analysis

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

Isabelle Smith and André Ferrari

Full-text: Open access

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.

Article information

Source
Bayesian Anal., Volume 9, Number 4 (2014), 939-962.

Dates
First available in Project Euclid: 21 November 2014

Permanent link to this document
https://projecteuclid.org/euclid.ba/1416579186

Digital Object Identifier
doi:10.1214/14-BA877

Mathematical Reviews number (MathSciNet)
MR3293963

Zentralblatt MATH identifier
1327.62165

Keywords
hypothesis testing PLR p-value likelihood ratio frequentist and Bayesian reconciliation Lindley’s paradox invariance

Citation

Smith, Isabelle; Ferrari, André. Equivalence between the Posterior Distribution of the Likelihood Ratio and a p-value in an Invariant Frame. Bayesian Anal. 9 (2014), no. 4, 939--962. doi:10.1214/14-BA877. https://projecteuclid.org/euclid.ba/1416579186


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