Abstract
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.
