Distribution of Mutual Information

TL;DR

This paper derives reliable, fast approximations for the distribution of mutual information between two variables, considering Bayesian priors, to improve inference beyond point estimates in statistical learning.

Contribution

It provides new methods to compute the distribution of mutual information under Bayesian priors, including mean, variance, skewness, and kurtosis, with practical approximations.

Findings

Derived accurate approximations for p(I|n)

Provided exact expression for the mean mutual information

Discussed numerical stability and applicability range

Abstract

The mutual information of two random variables i and j with joint probabilities t_ij is commonly used in learning Bayesian nets as well as in many other fields. The chances t_ij are usually estimated by the empirical sampling frequency n_ij/n leading to a point estimate I(n_ij/n) for the mutual information. To answer questions like "is I(n_ij/n) consistent with zero?" or "what is the probability that the true mutual information is much larger than the point estimate?" one has to go beyond the point estimate. In the Bayesian framework one can answer these questions by utilizing a (second order) prior distribution p(t) comprising prior information about t. From the prior p(t) one can compute the posterior p(t|n), from which the distribution p(I|n) of the mutual information can be calculated. We derive reliable and quickly computable approximations for p(I|n). We concentrate on the mean, variance, skewness, and kurtosis, and non-informative priors. For the mean we also give an exact expression. Numerical issues and the range of validity are discussed.