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

arXiv: https://arxiv.org/abs/2603.05544 · HTML: https://mlsanity.com/paper/2603.05544/

## Key facts

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- First posted on arXiv 4 Mar 2026.

## Abstract (arXiv)

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.