Decomposing Probabilistic Scores: Reliability, Information Loss and Uncertainty
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| Format: | Preprint |
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2026
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| _version_ | 1866908904838922240 |
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| author | Charpentier, Arthur Machado, Agathe Fernandes |
| author_facet | Charpentier, Arthur Machado, Agathe Fernandes |
| contents | Calibration is a conditional property that depends on the information retained by a predictor. We develop decomposition identities for arbitrary proper losses that make this dependence explicit. At any information level $\mathcal A$, the expected loss of an $\mathcal A$-measurable predictor splits into a proper-regret (reliability) term and a conditional entropy (residual uncertainty) term. For nested levels $\mathcal A\subseteq\mathcal B$, a chain decomposition quantifies the information gain from $\mathcal A$ to $\mathcal B$. Applied to classification with features $\boldsymbol{X}$ and score $S=s(\boldsymbol{X})$, this yields a three-term identity: miscalibration, a {\em grouping} term measuring information loss from $\boldsymbol{X}$ to $S$, and irreducible uncertainty at the feature level. We leverage the framework to analyze post-hoc recalibration, aggregation of calibrated models, and stagewise/boosting constructions, with explicit forms for Brier and log-loss. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_15232 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Decomposing Probabilistic Scores: Reliability, Information Loss and Uncertainty Charpentier, Arthur Machado, Agathe Fernandes Machine Learning Statistics Theory Calibration is a conditional property that depends on the information retained by a predictor. We develop decomposition identities for arbitrary proper losses that make this dependence explicit. At any information level $\mathcal A$, the expected loss of an $\mathcal A$-measurable predictor splits into a proper-regret (reliability) term and a conditional entropy (residual uncertainty) term. For nested levels $\mathcal A\subseteq\mathcal B$, a chain decomposition quantifies the information gain from $\mathcal A$ to $\mathcal B$. Applied to classification with features $\boldsymbol{X}$ and score $S=s(\boldsymbol{X})$, this yields a three-term identity: miscalibration, a {\em grouping} term measuring information loss from $\boldsymbol{X}$ to $S$, and irreducible uncertainty at the feature level. We leverage the framework to analyze post-hoc recalibration, aggregation of calibrated models, and stagewise/boosting constructions, with explicit forms for Brier and log-loss. |
| title | Decomposing Probabilistic Scores: Reliability, Information Loss and Uncertainty |
| topic | Machine Learning Statistics Theory |
| url | https://arxiv.org/abs/2603.15232 |