Conditional density and variance of Nadaraya-Watson model

Question

Given $N$ data points $x$ and $N$ targets $t$, considering a new point $x$ and the corresponding new target $t$, what would be:

• The conditional density
• The conditional mean
• variance

of the Nadaraya-Watson model

$$p(x,t) = \frac{1}{N}\sum_{n=1}^Nf(x - x_t, t - t_n)$$

Where the kernel is an isotropic Gaussian with mean $(0,0)$ and covariance is $\sigma^2I$, in terms of $k(x, x_n)$?

How do you verify that the sum of the functions = 1, since it is a distribution?

Answer 1

Here you have well described the conditional density (with $y$ in the place of your $t$): http://en.wikipedia.org/wiki/Kernel_regression#Derivation

The conditional density is the $f(y|x) = \frac{f(x,y)}{f(x)}$.

The conditional mean would be the mean for that density $E(Y|X)$ and the same thing for variance $E(Y-E(Y|X)|X)$.

It is the integral of the conditional density $f(y|X)$ over all y that shall be equal to 1. That will lead you to an integral of all K functions.