Predictive discrepancy statistic ($T(X)$)

A measurable summary of a possible dataset is chosen to detect disagreement with a specified predictive model. Compare its observed value with its distribution under replicated datasets. Large-discrepancy statistics can be checked with $\mathbb{P}(T(X^{\mathrm{rep}})\ge T(x))$. A posterior predictive check integrates over posterior parameter uncertainty; a fully specified null instead supplies a fixed reference law.