Investigation of Alternative Measures for Mutual Information
This work addresses a foundational problem in information theory for researchers dealing with continuous data, but it appears incremental as it explores existing divergences rather than introducing a new paradigm.
The paper tackles the issue of mutual information being infinite in continuous cases by evaluating alternative metrics like KL divergence, Wasserstein distance, Jensen-Shannon divergence, and total variation distance to define mutual information, and it assesses their performance through estimation and bounding methods.
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.