2003, Volume 65, Pt. 3, 501--512

**Coherence Of The Posterior
Predictive P-Value Based On The Posterior Odds**

By

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.