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Estimating mutual information from i.i.d. samples drawn from an unknown jointdensity function is a basic statistical problem of broad interest withmultitudinous applications. The most popular estimator is one proposed byKraskov and St\"ogbauer and Grassberger (KSG) in 2004, and is nonparametric andbased on the distances of each sample to its $k^{\rm th}$ nearest neighboringsample, where $k$ is a fixed small integer. Despite its widespread use (part ofscientific software packages), theoretical properties of this estimator havebeen largely unexplored. In this paper we demonstrate that the estimator isconsistent and also identify an upper bound on the rate of convergence of thebias as a function of number of samples. We argue that the superior performancebenefits of the KSG estimator stems from a curious "correlation boosting"effect and build on this intuition to modify the KSG estimator in novel ways toconstruct a superior estimator. As a byproduct of our investigations, we obtainnearly tight rates of convergence of the $\ell_2$ error of the well known fixed$k$ nearest neighbor estimator of differential entropy by Kozachenko andLeonenko.

Demystifying Fixed k-Nearest Neighbor Information Estimators