Adequate uncertainty representation and quantification have become imperative in various scientific disciplines, especially in machine learning and artificial intelligence. As an alternative to representing uncertainty via one single probability measure, we consider credal sets (convex sets of probability measures). The geometric representation of credal sets as $d$-dimensional polytopes implies a geometric intuition about (epistemic) uncertainty. In this paper, we show that the volume of the geometric representation of a credal set is a meaningful measure of epistemic uncertainty in the case of binary classification, but less so for multi-class classification. Our theoretical findings highlight the crucial role of specifying and employing uncertainty measures in machine learning in an appropriate way, and for being aware of possible pitfalls.

翻译：不确定性表示与量化已成为众多科学领域（尤其是机器学习和人工智能）的迫切需求。作为用单一概率测度表示不确定性的替代方案，我们考虑采用信任函数集合（概率测度的凸集）。信任函数集合作为$d$维多面体的几何表示，蕴含着关于（认知）不确定性的几何直觉。本文证明：在二分类问题中，信任函数集合几何表示的体积是认知不确定性有意义的度量，但在多分类问题中效果欠佳。我们的理论发现凸显了在机器学习中恰当指定和运用不确定性测度的关键作用，以及关注潜在陷阱的重要性。