I'm working on a problem from "The Elements of Statistical Learning" (prob. 6.8):
Suppose that for continuous response $Y$ and predictor $X$, we model the joint density of $X, Y$ using a multivariate Gaussian kernel estimator. Note that the kernel in this case would be the product kernel $\phi_{\lambda}(X) \phi_{\lambda}(Y)$.
(a) Show that the conditional mean $E(Y|X)$ derived from this estimate is a Nadaraya-Watson estimator.
(b) Extend this result to classification by providing a suitable kernel for the estimation of the joint distribution of a continuous $X$ and discrete $Y$.
I know that the Nadaraya-Watson estimator is just the weighted average (equation 2.41 and 6.2 in ESL):
$$\hat f (x_0) = \frac{\sum_{i=0}^N K_{\lambda}(x_0, x_i) y_i}{\sum_{i=0}^N K_{\lambda}(x_0, x_i)}$$
Where $K$ in this case would be the multivariate Gaussian kernel function.
I can think about how to extend this to a classification problem, but am not sure how to approach the first part of this question.
Any pointers would be greatly appreciated!