On the conservativeness of posterior predictive pp-values

Bayesian posterior predictive pp-values (ppp) tend to be conservative, in the sense of low rejection probability under the prior distribution. It is well known that a ppp has the distribution of a conditional expectancy of a uniform (0,1) variable. We show that this does not imply a tail probability inequality P(ppp⩽α)⩽αP(ppp⩽α)⩽α, even when αα tends to zero. In some special cases involving localization parameters, however, we show that ppp-values are in fact strictly conservative.