# covariance to correlation is like mutual information to --?

It is well known that correlation is the normalized covariance, i.e. $\ Cor(X, Y) = Cov(X, Y)/\sqrt{Var(X)Var(Y)}$.

These two related measures describe the linear relationships in the data.

Is there a similar intuitive relationship to describe mutual information, capturing the non-linear relationships in the data?

I think the most obvious way to do this is to treat entropy as the analogue to variance, if mutual information is the analogue to covariance. Notice that one similarity is that $$\mathbb{I}[X;X] = \mathbb{H}[X]$$ (as for variance vs covariance) one difference is that (unlike covariance) we have $$\mathbb{I}[X;Y] \geq 0$$. Then: $$\mathbb{I}_N[X;Y] = \frac{\mathbb{I}[X;Y]}{\sqrt{\mathbb{H}[X]\phantom{|}}\sqrt{\mathbb{H}[Y]\phantom{|}}}$$
Also note that the following hold: \begin{align} \mathbb{I}[X;Y] &= \mathbb{H}[X] + \mathbb{H}[Y] - \mathbb{H}[X,Y] \\ \mathbb{H}[X] + \mathbb{H}[Y] &\geq \mathbb{H}[X,Y] \geq 0 \\ \mathbb{H}[X,Y] &= \mathbb{H}[X|Y] + \mathbb{H}[Y] \end{align}
Since Shannon entropy and mutual information are non-negative, so is $$\mathbb{I}_N[X;Y]$$. However, if $$X=Y$$, $$\frac{\mathbb{H}[X] + \mathbb{H}[X] - \mathbb{H}[X,X]}{\sqrt{\mathbb{H}[X]\phantom{|}}\sqrt{\mathbb{H}[X]\phantom{|}}} = 1$$
Another natural normalization is: $$\widetilde{\mathbb{I}}_N[X;Y] = \frac{\mathbb{I}[X;Y]}{{\mathbb{H}[X]}+{\mathbb{H}[Y]}}$$