# $H(X)$ vs. $H(f(X))$

## TL;DR

This paper derives tight bounds on the entropy of a function of a random variable when the function is not one-to-one, improving existing bounds and exploring scenarios where this is relevant.

## Contribution

It provides new tight bounds on $H(f(X))$ for non-injective functions and introduces an improved lower bound on distribution entropy based on probability ratio constraints.

## Key findings

- Tight bounds on $H(f(X))$ for non-one-to-one functions.
- An improved lower bound on distribution entropy based on max-min probability ratio.
- Illustrations of scenarios where entropy bounds are significant.

## Abstract

It is well known that the entropy $H(X)$ of a finite random variable is always greater or equal to the entropy $H(f(X))$ of a function $f$ of $X$, with equality if and only if $f$ is one-to-one. In this paper, we give tights bounds on $H(f(X))$ when the function $f$ is not one-to-one, and we illustrate a few scenarios where this matters. As an intermediate step towards our main result, we prove a lower bound on the entropy of a probability distribution, when only a bound on the ratio between the maximum and the minimum probability is known. Our lower bound improves previous results in the literature, and it could find applications outside the present scenario.