Investigation of Alternative Measures for Mutual Information

TL;DR

This paper evaluates alternative divergence measures like Wasserstein and Jensen-Shannon as substitutes for mutual information in continuous variables, analyzing their estimation methods and effectiveness.

Contribution

It introduces and compares various divergence-based measures as alternatives to mutual information for continuous variables, with a focus on their estimation and performance.

Findings

Wasserstein distance provides a robust alternative to KL-divergence.

Jensen-Shannon divergence offers a bounded measure suitable for continuous data.

Different estimation methods vary in accuracy and computational efficiency.

Abstract

Mutual information $I(X;Y)$ is a useful definition in information theory to estimate how much information the random variable $Y$ holds about the random variable $X$. One way to define the mutual information is by comparing the joint distribution of $X$ and $Y$ with the product of the marginals through the KL-divergence. If the two distributions are close to each other there will be almost no leakage of $X$ from $Y$ since the two variables are close to being independent. In the discrete setting the mutual information has the nice interpretation of how many bits $Y$ reveals about $X$ and if $I(X;Y)=H(X)$ (the Shannon entropy of $X$) then $X$ is completely revealed. However, in the continuous case we do not have the same reasoning. For instance the mutual information can be infinite in the continuous case. This fact enables us to try different metrics or divergences to define the mutual information. In this paper, we are evaluating different metrics or divergences such as Kullback-Liebler (KL) divergence, Wasserstein distance, Jensen-Shannon divergence and total variation distance to form alternatives to the mutual information in the continuous case. We deploy different methods to estimate or bound these metrics and divergences and evaluate their performances.