Consider 2 random variables, $X$ and $Y$. $X$ is discrete and $Y$ is continuous.
In particular, we have a Gaussian distribution $Y$ with mean $X$ and variance $\sigma = 1$, and
$P(X=1) = \frac{1}{4}$,
$P(X=2) = \frac{1}{4}$,
$P(X=4) = \frac{1}{2}$
What is the mutual information between $X$ and $Y$? Is there an analytical solution to the problem? Otherwise, how to solve it numerically, given that the result is very sensitive to the binning choice?
