Uncertainty functionals revisited: Concavity and Jensen's inequality

TL;DR

This paper investigates the mathematical properties of uncertainty functionals, focusing on the relationship between concavity and Jensen's inequality in infinite measurable spaces, and clarifies their foundational role in uncertainty quantification.

Contribution

It establishes that concavity alone is insufficient for Jensen's inequality in infinite spaces and provides conditions under which the inequality holds, enhancing the theoretical foundation of uncertainty analysis.

Findings

Concavity is necessary but not sufficient for Jensen's inequality in infinite spaces.

Provides sufficient conditions for Jensen's inequality to hold.

Contributes to the mathematical foundation of uncertainty quantification.

Abstract

This article presents a theoretical study of uncertainty functionals on general measurable spaces. These functionals are fundamental in experimental design and global sensitivity analysis, where they are used to quantify variability and information content in probabilistic models. As first articulated in DeGroot's seminal 1962 article, a natural requirement is that uncertainty should decrease on average when additional information is obtained. This requirement is equivalent to the probabilistic form of Jensen's inequality on the space of probability measures. Our main results show that concavity is necessary but not sufficient for Jensen's inequality to hold whenever the underlying measurable space is infinite. We also provide practicable sufficient conditions under which the desired property holds. These results contribute to a clearer mathematical foundation for uncertainty quantification. Several open questions are formulated.