  Estimating mutual information from observed samples is a basic primitive,
useful in several machine learning tasks including correlation mining,
information bottleneck clustering, learning a Chow-Liu tree, and conditional
independence testing in (causal) graphical models. While mutual information is
a well-defined quantity in general probability spaces, existing estimators can
only handle two special cases of purely discrete or purely continuous pairs of
random variables. The main challenge is that these methods first estimate the
(differential) entropies of X, Y and the pair (X;Y) and add them up with
appropriate signs to get an estimate of the mutual information. These
3H-estimators cannot be applied in general mixture spaces, where entropy is not
well-defined. In this paper, we design a novel estimator for mutual information
of discrete-continuous mixtures. We prove that the proposed estimator is
consistent. We provide numerical experiments suggesting superiority of the
proposed estimator compared to other heuristics of adding small continuous
noise to all the samples and applying standard estimators tailored for purely
continuous variables, and quantizing the samples and applying standard
estimators tailored for purely discrete variables. This significantly widens
the applicability of mutual information estimation in real-world applications,
where some variables are discrete, some continuous, and others are a mixture
between continuous and discrete components.
