Alternatives to post-processing posterior predictive p-values

Abstract

The posterior predictive p value (ppp) was invented as a Bayesian counterpart to classical p values. The methodology can be applied to discrepancy measures involving both data and parameters and can, hence, be targeted to check for various modeling assumptions. The interpretation can, however, be difficult since the distribution of the ppp value under modeling assumptions varies substantially between cases. A calibration procedure has been suggested, treating the ppp value as a test statistic in a prior predictive test. In this paper, we suggest that a prior predictive test may instead be based on the expected posterior discrepancy, which is somewhat simpler, both conceptually and computationally. Since both these methods require the simulation of a large posterior parameter sample for each of an equally large prior predictive data sample, we furthermore suggest to look for ways to match the given discrepancy by a computation‐saving conflict measure. This approach is also based on simulations but only requires sampling from two different distributions representing two contrasting information sources about a model parameter. The conflict measure methodology is also more flexible in that it handles non‐informative priors without difficulty. We compare the different approaches theoretically in some simple models and in a more complex applied example.