Bounds on the Jensen Gap, and Implications for Mean-Concentrated Distributions

TL;DR

This paper derives bounds on Jensen's inequality gap based on function growth and moments, with implications for mean-concentrated distributions like i.i.d. averages and statistical mechanics.

Contribution

It introduces new bounds on Jensen's gap that depend solely on function growth and distribution moments, applicable to mean-concentrated scenarios.

Findings

Bounds depend only on growth properties and moments

Applicable to mean-concentrated distributions

Useful for statistical mechanics and sample averages

Abstract

This paper gives upper and lower bounds on the gap in Jensen's inequality, i.e., the difference between the expected value of a function of a random variable and the value of the function at the expected value of the random variable. The bounds depend only on growth properties of the function and specific moments of the random variable. The bounds are particularly useful for distributions that are concentrated around the mean, a commonly occurring scenario such as the average of i.i.d. samples and in statistical mechanics.