I think I am misunderstanding the notion of mutual information of continuous variables. Could anyone help me clear up the following?
Let $X \sim N(0, \sigma^2) $ and $Y \sim N(0, \sigma^2) $ denote Gaussian random variables. If $X$ and $Y$ are correlated with a coefficient $\rho$, then the mutual information between $X$ and $Y$ is given by (reference: https://en.wikipedia.org/wiki/Mutual_information).
\begin{equation} I(X; Y) = -\frac{1}{2} \log (1-\rho^2). \end{equation}
Here, I thought $I(X; Y) \rightarrow \infty$ when $\rho \rightarrow 1$ (for $X = Y$, $\rho = 1$). I considered this another way.
I considered $Y = X$. In this case, I would obtain $ I (X; Y) = H(X) - H(Y|X) = H(X) $.
For the Gaussian random variable $X$, $H(X)$ is bounded as follows (reference: https://en.wikipedia.org/wiki/Differential_entropy): \begin{equation} H(X) \leq \frac{1}{2} \log ( 2 \pi e \sigma^2). \end{equation}
Thus, $ I (X; Y) \leq \frac{1}{2} \log ( 2 \pi e \sigma^2)$.
Here is my question. I obtained two different results on $ I (X; Y)$ for $X = Y$. What could be some mistakes in my understanding?
Thank you in advance.
Differential entropy can actually be negative, and thus the upper bound on your information is not correct. Indeed, if they are the same random variable on a continuous domain, then you would hope that the mutual information between them would be infinite (and if they are the same Gaussian, indeed that is the case).
EDIT: I guess I should have clarified: In differential entropy sense, H(Y | X) is not 0; it is negative infinity if X = Y. Any singularity in differential entropy has negative infinite relative uncertainty to any quantized uniform distribution.