Comment: Bayesian Checking of the Second Levels of Hierarchical Models

This article extends Bayarri and Berger’s (1999) proposal for model evaluation using “partial posterior” p values to the evaluation of second-stage model assumptions in hierarchical models. Applications focus on normal-normal hierarchical models, although the final example involves an application to a beta-binomial model in which the distribution of the test statistic is assumed to be approximately normal. The notion of using partial posterior p values is potentially appealing because it avoids what the authors refer to as “double use” of the data, that is, use of the data for both fitting model parameters and evaluating model fit. In classical terms, this phenomenon is synonymous to masking and is widely known to reduce the power of test statistics for diagnosing model inadequacy. In the present context, masking is avoided by defining the reference distribution of a test statistic t by the partial posterior distribution, defined as