2003, Volume 65, Pt. 3, 501--512
Coherence Of The Posterior Predictive P-Value Based On The Posterior Odds
JULI\'AN DE LA HORRA, Universidad Aut\'onoma de Madrid, Spain and
MAR\'IA TERESA RODR\'IGUEZ-BERNAL, Universidad Carlos III de Madrid, Spain
SUMMARY. It is well-known that classical $p$-values sometimes behave incoherently for testing hypotheses in the sense that, when $\Theta_0\subset\Theta'_0,$ the support given to $\Theta_0$ is greater than or equal to the support given to $\Theta'_0.$. This problem is also found for posterior predictive $p$-values (a Bayesian-motivated alternative to classical $p$-values). In this paper, it is proved that, under some conditions, the posterior predictive $p$-value based on the posterior odds is coherent, showing that the choice of a suitable discrepancy variable is crucial.
AMS (1991) subject classification. 62F15, 62F03.
Key words and phrases. Coherence, posterior odds, $p$-value, posterior predictive $p$-value.