Variance estimation for Brier Score decomposition

TL;DR

This paper develops methods to estimate the variance of Brier Score decomposition components, improving understanding of uncertainty in probabilistic forecast evaluation.

Contribution

It derives variance expressions for both traditional and bias-corrected estimators of Brier Score components, enhancing the assessment of forecast quality.

Findings

Variance estimates are validated with artificial data.

Bias correction increases the variance of estimators.

Application to meteorological data demonstrates practical utility.

Abstract

The Brier Score is a widely-used criterion to assess the quality of probabilistic predictions of binary events. The expectation value of the Brier Score can be decomposed into the sum of three components called reliability, resolution, and uncertainty which characterize different forecast attributes. Given a dataset of forecast probabilities and corresponding binary verifications, these three components can be estimated empirically. Here, propagation of uncertainty is used to derive expressions that approximate the sampling variances of the estimated components. Variance estimates are provided for both the traditional estimators, as well as for refined estimators that include a bias correction. Applications of the derived variance estimates to artificial data illustrate their validity, and application to a meteorological prediction problem illustrates a possible use case. The observed increase of variance of the bias-corrected estimators is discussed.