third case described in the introduction. The expression in (\[eq:mixed\_general\]) is very similar to the expression in (\[eq:MI\_cond\]). After plugging in KDEs for the corresponding densities and conditional densities, a nearly identical procedure to that in Section \[subsec:mixed\_conv\] can be followed to derive the bias and variance of the corresponding plug-in estimator.

Now consider case B. Denote the discrete and continuous components of $\mathbf{X}$ as $\mathbf{X}_{1}$ and $\mathbf{X}_{2}$, respectively. Then if $\mathbf{Y}_{1}$ is the discrete component of $\mathbf{Y}$, then the expression inside the $g$ functional in (\[eq:mixed\_general\]) includes $f_{X_{1}}(x_{1})f_{Y_{1}}(y_{1})/f_{X_{1}Y_{1}}(x_{1},y_{1})$. Thus the plug-in estimator must include estimators for $f_{X_{1}}(x_{1}),$ $f_{Y_{1}}(y_{1})$, and $f_{X_{1}Y_{1}}(x_{1},y_{1})$. Define $\mathbf{N}_{y_{1}}=\sum_{i=1}^{N}1_{\{\mathbf{Y}_{1,i}=y_{1}\}}$ where $\mathbf{Y}_{1,i}$ is the discrete component of $\mathbf{Y}_{i}$. Then the estimator we use for $f_{Y_{1}}(y_{1})$ is $\mathbf{N}_{y_{1}}/N$. The estimators for $f_{X_{1}}(x_{1})$ and $f_{X_{1}Y_{1}}(x_{1},y_{1})$ are defined similarly. The bias and variance expressions of this plug-in estimator can then be derived with some slight modifications of Theorems \[thm:bias\] and \[thm:variance\]. See Appendix \[subsec:generalCase\] for an expression for $G_{3B}(\mathbf{X};\mathbf{Y})$ in this case and a sketch of these modifications. Case C follows similarly as the expression inside the $g$ functional in (\[eq:mixed\_general\]) includes $f_{X_{1}}(x_{1})f_{Y}(y)/f_{X_{1}Y}(x_{1},y)$ where all the terms are probability mass functions.

The resulting bias and variance expressions in these settings 