  Estimation of mutual information between (multidimensional) real-valued
variables is used in analysis of complex systems, biological systems, and
recently also quantum systems. This estimation is a hard problem, and
universally good estimators provably do not exist. Kraskov et al. (PRE, 2004)
introduced a successful mutual information estimation approach based on the
statistics of distances between neighboring data points, which empirically
works for a wide class of underlying probability distributions. Here we improve
this estimator by (i) expanding its range of applicability, and by providing
(ii) a self-consistent way of verifying the absence of bias, (iii) a method for
estimation of its variance, and (iv) a criterion for choosing the values of the
free parameter of the estimator. We demonstrate the performance of our
estimator on synthetic data sets, as well as on neurophysiological and systems
biology data sets.
