Coherence of the posterior predictive p-value based on the posterior odds

^aIt 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.