Is the Volume of a Credal Set a Good Measure for Epistemic Uncertainty?

Yusuf Sale, Michele Caprio, Eyke Hüllermeier

Keywords: Aleatoric and epistemic uncertainty, credal sets, geometry, uncertainty measure, high-dimensional probability, uncertainty quantification, uncertainty representation

TL;DR: We show that the volume of a credal set is a good measure for epistemic uncertainty in a binary classification setting, while it ceases to be so in multi-class setting.

Abstract: 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 in the case of binary classification, the volume of the geometric representation of a credal set is a good measure of epistemic uncertainty, while for multi-class classification it ceases to be appealing. Our theoretical findings highlight the crucial role of specifying and employing appropriate measures of uncertainty in machine learning tasks and generally call for awareness of possible pitfalls.