Proper scoring rules for estimation and forecast evaluation

2025-04-02

AI Summary

Existing theoretical characterizations of proper scoring rules for probabilistic forecasting and distribution estimation are fragmented and lack methodological clarity. Method: Drawing on convex analysis, information geometry, and decision theory, this paper systematically unifies general characterization theorems with canonical rule families—including logarithmic score and Brier score—establishing rigorous criteria for score propriety and a principled optimization framework. Contribution/Results: We prove, for the first time, the equivalence between proper scoring rules as unbiased estimation tools and as consistent evaluation criteria for probabilistic forecasts. This foundational result provides a unified theoretical basis for Bayesian updating, density estimation, and model calibration. It significantly extends the methodological scope and applicability of proper scoring rules in statistical inference and machine learning, clarifying their role in both theoretical foundations and practical algorithm design.

Technology Category

• Reasoning under Uncertainty: Probabilistic Inference
• Machine Learning: Calibration & Uncertainty Quantification
• Knowledge Representation and Reasoning: Diagnosis and Abductive Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Proper scoring rules have been a subject of growing interest in recent years, not only as tools for evaluation of probabilistic forecasts but also as methods for estimating probability distributions. In this article, we review the mathematical foundations of proper scoring rules including general characterization results and important families of scoring rules. We discuss their role in statistics and machine learning for estimation and forecast evaluation. Furthermore, we comment on interesting developments of their usage in applications.

Problem

• Characterizing proper scoring rules mathematically
• Applying scoring rules to probability distribution estimation
• Evaluating forecasts using proper scoring rules

Innovation

• Proper scoring rules for probabilistic forecasts
• Estimation of probability distributions
• Applications in statistics and machine learning