An intuitive rearranging of the Yates covariance decomposition for probabilistic verification of forecasts with the Brier score

TL;DR

This paper presents a simplified algebraic rearrangement of the Yates covariance decomposition for the Brier score, clarifying the conditions for optimal probabilistic forecasts and making evaluation more transparent.

Contribution

It introduces a new algebraic rearrangement of the Yates covariance decomposition that clearly separates the components affecting forecast quality.

Findings

The rearranged decomposition consists of three non-negative terms.

Optimal forecasts match outcome variance, correlation, and mean.

Deviations from these conditions increase the Brier score.

Abstract

Proper scoring rules are essential for evaluating probabilistic forecasts. We propose a simple algebraic rearrangement of the Yates covariance decomposition of the Brier score into three independently non-negative terms: a variance mismatch term, a correlation deficit term, and a calibration-in-the-large term. This rearrangement makes the optimality conditions for perfect forecasting transparent: the optimal forecast must simultaneously match the variance of outcomes, achieve perfect positive correlation with outcomes, and match the mean of outcomes. Any deviation from these conditions results in a positive contribution to the Brier score.