Bayesian Model Assessment Using Pivotal Quantities

Suppose that S(Y, theta) is a function of data Y and a model parameter theta, and suppose that the sampling distribution of S(Y, theta) is invariant when evaluated at theta(0), the true (i.e., data-generating) value of theta. Then S(Y, theta) is a pivotal quantity, and it follows from simple probability calculus that the distribution of S(Y, theta(0)) is identical to the distribution of S(Y, theta(Y)), where theta(Y) is a value of theta 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.