Entropy based measure of explained variation I have an observable $Y$ which is a function of some set of variables $\mathcal{X}=\left\{ X_{n}\right\} _{n=1}^{N}$. Now $Y$ is a deterministic function of the $X_n$s, but conditioned on only a subset of them, it is not completely determined  - and so a "proper" random variable. 
I want to express the degree to which $Y$ is dependent on any given subset $A\subseteq\mathcal{X}$, the amount of "Explained variation" - Something analogous to $R^2$ in the case where $Y$ is a linear function of the $X_n$s.
It seems that the conditional entropy is a natural such measure. i.e. for a subset  $A\subseteq\mathcal{X}$ the explained variation (lets denote it by $\rho$) would be
 $$\rho\left(A\right) = H\left(Y\right) - H\left(Y\mid A\right)$$ 
or perhaps normalized to range between 0 and 1 
$$\rho\left(A\right) = 1- \frac{H\left(Y\mid A\right)}{H\left(Y\right)}$$
which has a nice interpretation as the amount of surprise left in $Y$ after Iv'e observed the variables in $A$, as well as the properties one would expect from such a measure - $\rho\left(\emptyset\right)=0$, $\rho\left(\mathcal{X}\right)=1$ and $\rho$ is monotonically increasing with the cardinality of $A$
Question: 
Is this measure used in practice? If so, I'd appreciate a reference. If not, why not?
 A: The first quantity you proposed is actually the mutual information between $Y$ and $A$:
$$I(Y; A) = H(Y) - H(Y \mid A)$$
This is a central quantity in information theory, and a widely used measure of dependence between random variables. It measures the amount by which knowing $A$ reduces uncertainty about $Y$ (or the amount by which knowing $Y$ reduces uncertainty about $A$, since mutual information is symmetric). Equivalently, it can be seen as the KL divergence (a measure of dissimilarity) between the joint distribution and what it would be if $Y$ and $A$ were independent (the product of the marginals):
$$I(Y; A) = D_{KL} \big( p(Y,A) \parallel p(Y) p(A) \big)$$
A number of normalized variants of mutual information have also been proposed. Your second quantity is actually equal to the uncertainty coefficient $U(Y \mid A)$:
$$U(Y \mid A) = \frac{H(Y) - H(Y \mid A)}{H(Y)} = \frac{I(Y; A)}{H(Y)}$$
Also, be aware that estimating mutual information and entropy from data can be challenging, and sometimes even infeasible. This is particularly true with small datasets or in high dimensional settings. A number of estimators have been proposed in the literature; I'd recommend taking a look to get a good grasp of the underlying issues. For example, naive methods like plugging a density estimate in to the entropy formula can be badly biased.
