A conservation law for posterior predictive variance

Abstract

We use the law of total variance to generate multiple expressions for the posterior predictive variance in Bayesian hierarchical models. These expressions are sums of terms involving conditional expectations and conditional variances. Since the posterior predictive variance is fixed given the hierarchical model, it represents a constant quantity that is conserved over the various expressions for it. The terms in the expressions can be assessed in absolute or relative terms to understand the main contributors to the length of prediction intervals. Also, sometimes these terms can be intepreted in the context of the hierarchical model. We show several examples, closed form and computational, to illustrate the uses of this approach in model assessment.