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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?

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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.

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  • $\begingroup$ Hmm, I knew that.. Guess my information-theory is a bit rusty. Well, this is now my favorite intuition for what mutual information is. Thanks! $\endgroup$ – H.Rappeport Apr 26 at 8:20
  • $\begingroup$ Followup question in the context of using the mutual information as an "explained variation" measure: As I mentioned, $\rho$ is monotonically increasing with the cardinality of $A$, but I'd expect its not like PCA, where you can sort the variables such that for any $n$ the first $n$ variables are the subset of size $n$ with the most explained variation. Are there sufficient conditions on the $X_i$ which allow for such an ordering? I think complete pairwise independence should be enough, but is there some weaker condition? $\endgroup$ – H.Rappeport Apr 26 at 8:29
  • $\begingroup$ @H.Rappeport. Glad to help. Regarding your followup question, I suspect you're right that such an ordering doesn't always exist. Off the top of my head, not sure about sufficient conditions--you might post this as another question. $\endgroup$ – user20160 Apr 26 at 19:14

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