# Estimating Mixture Entropy with Pairwise Distances

## TL;DR

This paper introduces a family of efficient, differentiable estimators for mixture entropy based on pairwise component distances, providing tight bounds and applications in mutual information estimation and optimization problems.

## Contribution

The authors propose a novel estimator family for mixture entropy using pairwise distances, including bounds with closed-form expressions for Gaussian mixtures, improving accuracy and computational efficiency.

## Key findings

- Estimator family includes lower and upper bounds on mixture entropy.
- Chernoff and Bhattacharyya distances provide tight bounds for specific cases.
- Numerical simulations show bounds are significantly tighter than existing methods.

## Abstract

Mixture distributions arise in many parametric and non-parametric settings -- for example, in Gaussian mixture models and in non-parametric estimation. It is often necessary to compute the entropy of a mixture, but, in most cases, this quantity has no closed-form expression, making some form of approximation necessary. We propose a family of estimators based on a pairwise distance function between mixture components, and show that this estimator class has many attractive properties. For many distributions of interest, the proposed estimators are efficient to compute, differentiable in the mixture parameters, and become exact when the mixture components are clustered. We prove this family includes lower and upper bounds on the mixture entropy. The Chernoff $\alpha$-divergence gives a lower bound when chosen as the distance function, with the Bhattacharyya distance providing the tightest lower bound for components that are symmetric and members of a location family. The Kullback-Leibler divergence gives an upper bound when used as the distance function. We provide closed-form expressions of these bounds for mixtures of Gaussians, and discuss their applications to the estimation of mutual information. We then demonstrate that our bounds are significantly tighter than well-known existing bounds using numeric simulations. This estimator class is very useful in optimization problems involving maximization/minimization of entropy and mutual information, such as MaxEnt and rate distortion problems.