Lower Bounds on Mutual Information

TL;DR

This paper revises previous claims about lower bounds on mutual information, emphasizing the dependence on marginal distributions and proposing bounds based on rank correlations, with practical applications demonstrated on gene expression data.

Contribution

It corrects earlier misconceptions about MI bounds, highlighting the importance of marginal distributions and introducing bounds using rank-based correlations like Spearman's.

Findings

Bounds are non-trivial for uniform marginals.

Rank-based bounds can be practically estimated.

Gene expression data illustrates the bounds' relevance.

Abstract

We correct claims about lower bounds on mutual information (MI) between real-valued random variables made in A. Kraskov {\it et al.}, Phys. Rev. E {\bf 69}, 066138 (2004). We show that non-trivial lower bounds on MI in terms of linear correlations depend on the marginal (single variable) distributions. This is so in spite of the invariance of MI under reparametrizations, because linear correlations are not invariant under them. The simplest bounds are obtained for Gaussians, but the most interesting ones for practical purposes are obtained for uniform marginal distributions. The latter can be enforced in general by using the ranks of the individual variables instead of their actual values, in which case one obtains bounds on MI in terms of Spearman correlation coefficients. We show with gene expression data that these bounds are in general non-trivial, and the degree of their (non-)saturation yields valuable insight.