A convex analysis approach to tight expectation inequalities

TL;DR

This paper introduces a convex analysis framework to derive tight bounds on expectations of functions of random vectors, connecting expectation inequalities with convex hull properties and optimization, and providing analytical and numerical methods for specific cases.

Contribution

It establishes a novel convex analysis approach linking expectation inequalities to convex hulls, enabling sharp bounds and discrete solutions, extending the moment problem.

Findings

Derived sharp bounds for expectation inequalities.

Connected expectation bounds to convex hull properties.

Developed analytical and numerical methods for specific expectation problems.

Abstract

In this work, we investigate the question of how knowledge about expectations $\mathbb{E}(f_i(X))$ of a random vector $X$ translate into inequalities for $\mathbb{E}(g(X))$ for given functions $f_i$, $g$ and a random vector $X$ whose support is contained in some set $S\subseteq \mathbb{R}^n$. We show that there is a connection between the problem of obtaining tight expectation inequalities in this context and properties of convex hulls, allowing us to rewrite it as an optimization problem. The results of these optimization problems not only arrive at sharp bounds for $\mathbb{E}(g(X))$ but in some cases also yield discrete probability measures where equality holds. We develop an analytical approach that is particularly suited for studying the Jensen gap problem when the known information are the average and variance, as well as a numerical approach for the general case, that reduces the problem to a convex optimization; which in a sense extends known results about the moment problem.