variables have been developed. A popular $k$-nn-based estimator was proposed in [@kraskov2004estimating] which is a modification of the entropy estimator derived in [@kozachenko1987sample]. However, these estimators only achieve the parametric convergence rate when the dimension of each of the random variables is less than 3 [@gao2016demystifying]. Similarly, the Rényi information estimator in [@pal2010estimation] does not achieve the parametric rate. Some other estimators are based on maximum likelihood estimation of the likelihood ratio [@suzuki2008approximating] and minimal spanning trees [@khan2007relative].

Recent work has focused on nonparametric divergence estimation for purely continuous random variables. One approach [@krishnamurthy2014divergence; @kandasamy2015nonparametric; @singh2014exponential; @singh2014renyi] uses an optimal kernel density estimator (KDE) to achieve the parametric convergence rate when the densities are at least $d$ [@singh2014exponential; @singh2014renyi] or $d/2$ [@krishnamurthy2014divergence; @kandasamy2015nonparametric] times differentiable where $d$ is the dimension of the data. These optimal KDEs require knowledge of the density support boundary and are difficult to construct near the boundary. Numerical integration may also be required for estimating some divergence functionals under this approach, which can be computationally expensive. In contrast, our approach to MI estimation does not require numerical integration and can be performed without knowledge of the support boundary.

More closely related work [@sricharan2013ensemble; @moon2014isit; @moon2014nips; @moon2016arxiv; 