Bayesian model assessment using pivotal quantities

Suppose that S( Y,Θ) is a function of data Y and a model parameter Θ, and suppose that the sampling distribution of S( Y,Θ) is invariant when evaluated at Θ₀, the "true" (i.e., data-generating) value of Θ. Then S( Y,Θ) is a pivotal quantity, and it follows from simple probability calculus that the distribution of S( Y,Θ₀) is identical to the distribution of S( Y,ΘY), where ΘY is a value of Θ drawn from the posterior distribution given Y. This fact makes it possible to define a large number of Bayesian model diagnostics having a known sampling distribution. It also facilitates the calibration of the joint sampling of model diagnostics based on pivotal quantities.