Distribution of Mutual Information from Complete and Incomplete Data

TL;DR

This paper derives analytical expressions for the distribution of mutual information under Bayesian inference, enabling reliable, quick approximations that improve applications like feature selection over traditional descriptive measures.

Contribution

It provides exact and approximate analytical formulas for the mean, variance, skewness, and kurtosis of mutual information, including for incomplete data, facilitating inductive analysis.

Findings

Analytical expressions for mutual information distribution are derived.

Approximations are accurate to order O(1/n^3).

Inductive mutual information improves feature selection performance.

Abstract

Mutual information is widely used, in a descriptive way, to measure the stochastic dependence of categorical random variables. In order to address questions such as the reliability of the descriptive value, one must consider sample-to-population inferential approaches. This paper deals with the posterior distribution of mutual information, as obtained in a Bayesian framework by a second-order Dirichlet prior distribution. The exact analytical expression for the mean, and analytical approximations for the variance, skewness and kurtosis are derived. These approximations have a guaranteed accuracy level of the order O(1/n^3), where n is the sample size. Leading order approximations for the mean and the variance are derived in the case of incomplete samples. The derived analytical expressions allow the distribution of mutual information to be approximated reliably and quickly. In fact, the derived expressions can be computed with the same order of complexity needed for descriptive mutual information. This makes the distribution of mutual information become a concrete alternative to descriptive mutual information in many applications which would benefit from moving to the inductive side. Some of these prospective applications are discussed, and one of them, namely feature selection, is shown to perform significantly better when inductive mutual information is used.