An intuitive rearranging of the Yates covariance decomposition for probabilistic verification of forecasts with the Brier score
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866914372817780736 |
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| author | Vieira, Bruno Hebling |
| author_facet | Vieira, Bruno Hebling |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_05544 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An intuitive rearranging of the Yates covariance decomposition for probabilistic verification of forecasts with the Brier score Vieira, Bruno Hebling Methodology Machine Learning Applications 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. |
| title | An intuitive rearranging of the Yates covariance decomposition for probabilistic verification of forecasts with the Brier score |
| topic | Methodology Machine Learning Applications |
| url | https://arxiv.org/abs/2603.05544 |