Brier-UDAM for Paired Prediction-Rule Comparisons: Latent-Risk Interpretation, Observable Contrasts, and Inference

Abstract

The Brier risk summarizes probability-prediction error but does not explain why two prediction rules differ. We develop Brier-UDAM, a framework for paired binary prediction-rule comparisons that separates the Brier-risk difference into mean-bias, dispersion, and Brier-scale alignment contributions. Although the observable contrast identity follows algebraically from Yates-type moment decompositions, the framework gives these terms a latent-risk interpretation and shows that paired contrasts can be identified from observed outcomes and paired predictions without estimating the conditional event-risk function. For two rules evaluated in the same target population, the exact identity $\Delta R = \Delta M + \Delta D + \Delta L$ holds, and the corresponding plug-in estimators reproduce the empirical paired Brier-risk difference exactly in finite samples. We derive influence-function-based inference for first-order regular contrasts and a projection-based confidence procedure for the mean-bias contrast when first-order inference degenerates. Simulations confirmed interpretable component behavior, recovery of the target contrasts, and generally satisfactory inference in regular settings; projection intervals retained coverage near degeneracy but were conservative. In a large public-data analysis of 30-day hospital readmission, models with similar held-out Brier risks showed substantially larger and opposing dispersion and alignment contributions, revealing cancellation that was hidden in the aggregate score. Brier-UDAM therefore provides a diagnostic account of paired Brier-risk differences, while calibration, discrimination, and information-based assessments continue to address their distinct performance questions.