Abstract: Hierarchical models are increasingly used in many applications. Along with this increase use comes a desire to investigate whether the model is compatible with the observed data. Bayesian methods are well suited to eliminate the many (nuisance) parameters in these complicated models; in this paper we investigate Bayesian methods for model checking. Since we contemplate model checking as a preliminary, exploratory analysis, we concentrate in objective Bayesian methods in which careful specification of an informative prior distribution is avoided. Numerous examples are given and different proposals are investigated. Key words and phrases: Model checking; model criticism; objective Bayesian methods; p-values. 1.

We would like to thank the discussants for the valuable insights and for commenting on important aspects of model checking that we did not touch in our paper. Our goal was modest (but crucial): to select an appropriate distribution with which to judge the compatibility of the data with a hypothesized (hierarchical) model, when the test statistic is not ancillary and an improper prior is used for the hyperparameters. Since it is important to emphasize that this is by no means the only aspect of model checking, the discussants ’ complementary contributions and comments are all most welcome. The specific technical contributions of Evans and Johnson are also appreciated, since their developments in this area were not mentioned in our review. Several discussants have highlighted the importance of graphical displays in model checking. We will not comment on this because we entirely agree. We similarly agree with most of the discussants’ other comments, although in this rejoinder we mainly concentrate on disagreements. Our comments are organized around the main topics that arise in the discussions. We keep the same notation and terminology used in the paper (although it does conflict with the notation used by some of the discussants).