Estimating Mutual Information

TL;DR

This paper introduces two new classes of mutual information estimators based on k-nearest neighbor entropy estimates, which are data-efficient, adaptive, and unbiased, especially effective for high-dimensional and small-scale structures.

Contribution

The authors develop novel mutual information estimators that outperform traditional binning methods by being more accurate, adaptive, and unbiased, with extensions to multiple variables and practical applications in ICA.

Findings

Estimators become exact for independent distributions, vanishing when variables are independent.

The methods are effective across various marginal distributions and dimensions.

Applications include assessing independence in ICA and improving blind source separation.

Abstract

We present two classes of improved estimators for mutual information $M(X,Y)$, from samples of random points distributed according to some joint probability density $\mu(x,y)$. In contrast to conventional estimators based on binnings, they are based on entropy estimates from $k$-nearest neighbour distances. This means that they are data efficient (with $k=1$ we resolve structures down to the smallest possible scales), adaptive (the resolution is higher where data are more numerous), and have minimal bias. Indeed, the bias of the underlying entropy estimates is mainly due to non-uniformity of the density at the smallest resolved scale, giving typically systematic errors which scale as functions of $k/N$ for $N$ points. Numerically, we find that both families become {\it exact} for independent distributions, i.e. the estimator $\hat M(X,Y)$ vanishes (up to statistical fluctuations) if $\mu(x,y) = \mu(x) \mu(y)$. This holds for all tested marginal distributions and for all dimensions of $x$ and $y$. In addition, we give estimators for redundancies between more than 2 random variables. We compare our algorithms in detail with existing algorithms. Finally, we demonstrate the usefulness of our estimators for assessing the actual independence of components obtained from independent component analysis (ICA), for improving ICA, and for estimating the reliability of blind source separation.