Credal sets, i.e., closed convex sets of probability measures, provide a natural framework to represent aleatoric and epistemic uncertainty in machine learning. Yet how to quantify these two types of uncertainty for a given credal set, particularly in multiclass classification, remains underexplored. In this paper, we propose a distance-based approach to quantify total, aleatoric, and epistemic uncertainty for credal sets. Concretely, we introduce a family of such measures within the framework of Integral Probability Metrics (IPMs). The resulting quantities admit clear semantic interpretations, satisfy natural theoretical desiderata, and remain computationally tractable for common choices of IPMs. We instantiate the framework with the total variation distance and obtain simple, efficient uncertainty measures for multiclass classification. In the binary case, this choice recovers established uncertainty measures, for which a principled multiclass generalization has so far been missing. Empirical results confirm practical usefulness, with favorable performance at low computational cost.

翻译：置信集（即概率测度的闭凸集）为机器学习中随机不确定性与认知不确定性的表示提供了自然框架。然而，对于给定的置信集（特别是在多分类场景中），如何量化这两类不确定性仍未被充分研究。本文提出一种基于距离的方法来量化置信集的总不确定性、随机不确定性与认知不确定性。具体而言，我们在积分概率度量（IPMs）框架下引入了一族此类度量。所得量具备清晰的语义解释，满足自然的理论期望性质，并且对于常见的IPM选择仍保持计算可行性。我们以全变差距离实例化该框架，获得了适用于多分类任务的简洁高效不确定性度量。在二分类情形下，该选择复现了已有的不确定性度量——此前一直缺乏这类度量的原则性多分类推广。实验结果证实了其实用价值，在低计算成本下展现出优越性能。