nonparametric estimation of a large class of MI measures where we only have available a finite population of i.i.d. samples. **** We separately consider three cases: 1) $\X$ and $\Y$ are both continuous; 2) $\X$ is continuous and $\Y$ is discrete; 3) $\X$ and $\Y$ may have any mixture of discrete and continuous components. We focus primarily on the second case which includes the problem of feature selection in classification. We derive a MI estimator for this case that achieves the parametric MSE rate when the conditional densities of the continuous variables are sufficiently smooth. We also show how these estimators are extended to the first and third cases.

Our estimation method applies to other MI measures in addition to Shannon information, which have been the focus of much interest. The authors of [@torkkola2003feature] defined an information measure based on a quadratic divergence that could be estimated more efficiently than Shannon information. A MI measure based on the Pearson divergence was considered in [@sugiyama2012machine] for computational efficiency and numerical stability. The authors of [@costa2004geodesic] and [@pal2010estimation] used minimal spanning tree generalized nearest-neighbor graph approaches, respectively, to estimate Rényi information.

Related Work
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Many estimators for Shannon MI between continuous random 