# Two-Moment Inequalities for R\'enyi Entropy and Mutual Information

## TL;DR

This paper introduces new two-moment inequalities for Re9nyi entropy and mutual information, providing tighter bounds and improved methods for information measure estimation using moment-based techniques.

## Contribution

It presents novel two-moment inequalities for Re9nyi entropy and a new approach to upper bounding mutual information via variance-based integrals.

## Key findings

- Bounds on Re9nyi entropy improve with two moments.
- New upper bounds on mutual information relate to variance of conditional density.
- Method enhances estimation accuracy with modest complexity increase.

## Abstract

This paper explores some applications of a two-moment inequality for the integral of the $r$-th power of a function, where $0 < r< 1$. The first contribution is an upper bound on the R\'{e}nyi entropy of a random vector in terms of the two different moments. When one of the moments is the zeroth moment, these bounds recover previous results based on maximum entropy distributions under a single moment constraint. More generally, evaluation of the bound with two carefully chosen nonzero moments can lead to significant improvements with a modest increase in complexity. The second contribution is a method for upper bounding mutual information in terms of certain integrals with respect to the variance of the conditional density. The bounds have a number of useful properties arising from the connection with variance decompositions.