用合理评分规则统一量化回归模型不确定性。
Uncertainty Quantification for Regression using Proper Scoring Rules
- 基于合理评分规则构建回归不确定性量化框架
- 推导出在参数假设下的闭式解，可由模型集成估计
- 自然分解出认知与随机不确定性，适合安全关键场景
机器学习预测的不确定性量化对可靠决策至关重要，尤其在安全关键应用中。尽管不确定性量化理论近年来取得显著进展，主要基于合理评分规则，但这些成果多集中于分类任务，回归任务仍具挑战。本文提出一种基于合理评分规则（如CRPS、对数评分、平方误差、二次评分）的统一回归不确定性量化框架。在常见的参数假设下，推导出相应不确定性度量的闭式表达式，并说明如何通过模型集成进行估计。所提出的度量能自然分解为认知不确定性和随机不确定性。该框架恢复了基于预测方差和微分熵的流行回归不确定性度量。在合成数据和真实世界回归数据集上的广泛评估，为选择可靠的不确定性度量提供了指导。
原文摘要 · Abstract (English)
Quantifying uncertainty of machine learning model predictions is essential for reliable decision-making, especially in safety-critical applications. Recently, uncertainty quantification (UQ) theory has advanced significantly, building on a firm basis of learning with proper scoring rules. However, these advances were focused on classification, while extending these ideas to regression remains challenging. In this work, we introduce a unified UQ framework for regression based on proper scoring rules, such as CRPS, logarithmic, squared error, and quadratic scores. We derive closed-form expressions for the resulting uncertainty measures under practical parametric assumptions and show how to estimate them using ensembles of models. In particular, the derived uncertainty measures naturally decompose into aleatoric and epistemic components. The framework recovers popular regression UQ measures based on predictive variance and differential entropy. Our broad evaluation on synthetic and real-world regression datasets provides guidance for selecting reliable UQ measures.
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