"Information" Correlation (Let $X$ and $Y$ be random variables, sufficiently nice for my question to make sense.)
$$
\text{Correlation}
$$
$$
\rho(X, Y) = \dfrac{\text{cov}(X, Y)}{\sqrt{\text{var}(X)}\sqrt{\text{var}(Y)}}
$$
If we draw analogies between information theory and classical statistics, entropy is analogous to variance, and mutual information is analogous to covariance. Running with that idea, I should be able to make some kind of "information" correlation, $\rho_I$.
$$
\rho_I(X, Y) 
= \dfrac{\text{MI}(X, Y)}{\sqrt{\text{MI}(X, X)}\sqrt{\text{MI}(Y, Y)}}
= \dfrac{\text{MI}(X, Y)}{\sqrt{\text{H}(X)}\sqrt{\text{H}(Y)}}
$$
Does this make sense? Has any literature explored this idea? I am working on a project where it would be convenient to be able to say that some percentage of the entropy in $Y$ is explained by $X$ (something like the usual interpretation of $R^2$ for variance). Better yet, I would like to be able to say that $X$ contains some percentage of the "information" contained in $Y$.
 A: Taking a different angle from the information coefficient of correlation, I'll give background on the formula you gave.
It's called the normalized mutual information.
Unfortunately, lots of things are called the normalized mutual information. What you've shown uses the geometric mean of $H(X)$ and $H(Y)$ as the denominator. Other common choices are the arithmetic mean, the min, and the max. Any generalized mean will do. Another choice of denominator is the joint entropy $H(X, Y)$.
Because of this, it's important to give the formula you use for normalized mutual information.

That being said, there are big problems with the normalized mutual information. It suffers from what's called the finite size effect; the baseline keeps steadily rising, and you can get a high NMI in cases when one of the variables conveys nothing. It's better to use an adjusted-for-chance variant ("adjusted mutual information") whose expectation (rather than its minimum) is 0.
A: To answer your question "Has any literature explored this idea?". Linfoot (1957) introduced the informational coefficient of correlation, $\mathit{(IC)}$:
$$IC=\sqrt{1-e^{-2\cdot{I(X;Y)}}}$$
where $\mathit{I(X;Y)}$ is the mutual information.
While it does not seem to be a commonly used statistic in literature, in may satisfy your requirements. (Note: $\mathit{IC}$ is a measure of ${r}$, not ${r^2}$).
Reference: Linfoot, E.H. (1957). An informational measure of correlation. Information and Control, 1, 85-89.
