A conservation law for posterior predictive variance

TL;DR

This paper introduces a conservation law for posterior predictive variance in Bayesian hierarchical models, providing multiple expressions to analyze and interpret the variance for model assessment.

Contribution

It derives a set of expressions for posterior predictive variance using the law of total variance, aiding in understanding and assessing hierarchical Bayesian models.

Findings

Multiple expressions for posterior predictive variance are derived.

The approach helps identify key contributors to prediction interval length.

Examples demonstrate practical applications in model assessment.

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.