Two simple examples for understanding posterior p-values whose distributions are far from uniform

Posterior predictive p-values do not in general have uniform dis- tributions under the null hypothesis (except in the special case of ancillary test variables) but instead tend to have distributions more concentrated near 0.5. From di! erent perspectives, such nonuniform distributions have been portrayed as desirable (as reflecting an ability of vagu ep rior distri- butions to nonetheless yield accurate posterior predictions) or undesirable (as making it more di" cult to reject a false model). We explor et his ten- sion through two simple normal-distribution examples. In one example, we argue that the low power of the posterior predictive check is desirable from as tatistical perspective; in the other, the posterior predictive check seems inappropriate. Our conclusion is that the relevance of the p-value depends on the applied context, a point which (ironically) can be see ne ven in these two toy examples.