# Decompositions of the mean continuous ranked probability score

arXiv: https://arxiv.org/abs/2311.14122 · HTML: https://mlsanity.com/paper/2311.14122/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Nov 2023.

## Built on by (most cited first, 9 of 9)

- [Proper scoring rules for estimation and forecast evaluation](https://mlsanity.com/paper/2504.01781/) (2025)
- [Distributional Regression with Tabular Foundation Models: Evaluating Probabilistic Predictions via Proper Scoring Rules](https://mlsanity.com/paper/2603.08206/) (2026)
- [Unveiling Stochasticity: Universal Multi-modal Probabilistic Modeling for Traffic Forecasting](https://mlsanity.com/paper/2604.16084/) (2026)
- [Hierarchies of Calibration: Classification meets Regression](https://mlsanity.com/paper/2606.03245/) (2026)
- [Trajectory learning for ensemble forecasts via the continuous ranked probability score: a Lorenz '96 case study](https://mlsanity.com/paper/2508.21664/) (2025)
- [Forecasting Commencing Enrolments Under Data Sparsity: A Zero-Shot Time Series Foundation Models Framework for Higher Education Planning](https://mlsanity.com/paper/2602.12120/) (2026)
- [Wasserstein-type Gaussian Process Regressions for Input Measurement Uncertainty](https://mlsanity.com/paper/2603.17271/) (2026)
- [Multi-Rater Calibrated Segmentation Models](https://mlsanity.com/paper/2605.02437/) (2026)
- [Suitable Measures for the Potential Operational Utility of AI NWP Rainfall Forecasts Over Africa](https://mlsanity.com/paper/2609.31775/) (2026)

## Abstract (arXiv)

The continuous ranked probability score (crps) is the most commonly used scoring rule in the evaluation of probabilistic forecasts for real-valued outcomes. To assess and rank forecasting methods, researchers compute the mean crps over given sets of forecast situations, based on the respective predictive distributions and outcomes. We propose a new, isotonicity-based decomposition of the mean crps into interpretable components that quantify miscalibration (MSC), discrimination ability (DSC), and uncertainty (UNC), respectively. In a detailed theoretical analysis, we compare the new approach to empirical decompositions proposed earlier, generalize to population versions, analyse their properties and relationships, and relate to a hierarchy of notions of calibration. The isotonicity-based decomposition guarantees the nonnegativity of the components and quantifies calibration in a sense that is stronger than for other types of decompositions, subject to the nondegeneracy of empirical decompositions. We illustrate the usage of the isotonicity-based decomposition in case studies from weather prediction and machine learning.