Uncertainty Quantification Via the Posterior Predictive Variance

TL;DR

This paper introduces a method to decompose and analyze the sources of uncertainty in Bayesian predictive models using the law of total variance, aiding in better understanding and assessment of predictive uncertainty.

Contribution

It presents multiple expansions for the posterior predictive variance, enabling detailed analysis of uncertainty sources and their contributions in Bayesian models.

Findings

The expansions help identify key contributors to predictive uncertainty.

The approach quantifies uncertainty across different terms and orders.

Illustrations demonstrate improved model assessment techniques.

Abstract

We use the law of total variance to generate multiple expansions for the posterior predictive variance. These expansions are sums of terms involving conditional expectations and conditional variances and provide a quantification of the sources of predictive uncertainty. Since the posterior predictive variance is fixed given the model, it represents a constant quantity that is conserved over these expansions. The terms in the expansions can be assessed in absolute or relative sense to understand the main contributors to the length of prediction intervals. We quantify the term-wise uncertainty across expansions varying in the number of terms and the order of conditionates. In particular, given that a specific term in one expansion is small or zero, we identify the other terms in other expansions that must also be small or zero. We illustrate this approach to predictive model assessment in several well-known models.