Bayesian model assessment using pivotal quantities

Abstract

Suppose that $S({\bf Y},\Theta)$ is a function of data ${\bf Y}$ and a model parameter $\Theta$, and suppose that the sampling distribution of $S({\bf Y},\Theta)$ is invariant when evaluated at $\Theta_0$, the "true" (i.e., data-generating) value of $\Theta$. Then $S({\bf Y},\Theta)$ is a pivotal quantity, and it follows from simple probability calculus that the distribution of $S({\bf Y},\Theta_0)$ is identical to the distribution of $S({\bf Y},\Theta_{\bf Y})$, where $\Theta_{\bf Y}$ is a value of $\Theta$ drawn from the posterior distribution given ${\bf Y}$. 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.