28 Posterior and Prior Predictive Checks
Posterior predictive checks are a way of measuring whether a model does a good job of capturing relevant aspects of the data, such as means, standard deviations, and quantiles (Donald B. Rubin 1984; Andrew Gelman, Meng, and Stern 1996). Posterior predictive checking works by simulating new replicated data sets based on the fitted model parameters and then comparing statistics applied to the replicated data set with the same statistic applied to the original data set.
Prior predictive checks evaluate the prior the same way. Specifically, they evaluate what data sets would be consistent with the prior. They will not be calibrated with actual data, but extreme values help diagnose priors that are either too strong, too weak, poorly shaped, or poorly located.
Prior and posterior predictive checks are two cases of the general concept of predictive checks, just conditioning on different things (no data and the observed data, respectively). For hierarchical models, there are intermediate versions, as discussed in the section on hierarchical models and mixed replication.