In any case, I have mixed feelings about the relevance of posterior predictive p-values for these people. I would definitely like them to do some model checks, and I continue to feel that some posterior predictive distribution is the best way to get a reference set to use to compare observed data in a model check. But I think more and more that p-values are a dead end. I guess what I’d really like of non-Bayesian statisticians is for them to make their assumptions more explicit—to express their assumptions in the form of generative models, so that then these models can be checked and improved. Right now things are so indirect: the method is implicitly based on assumps (or, to put it another way, the method will be most effective when averaging over data generating processes that are close to some ideal) but these assumps are not stated clearly or always well understood, which I think makes it difficult to choose among methods or to improve them in light of data.