Comparative investigation of three Bayesian p values

Bayesian p values are a popular and important class of approaches for Bayesian model checking. They are used to quantify the degree of surprise from the observed data given the specified data model and prior distribution. A systematic investigation is conducted to compare three Bayesian p values - the posterior predictive p value, the sampled posterior p value and the calibrated posterior predictive p value. Their general computation costs are compared, and several examples that incorporate both simple and complex Bayesian models are used to compare their frequency properties. It is recommended to use the sampled posterior p value because it is computationally least expensive and safest.