What is $\;\mathbb{E}[p(X,Y)/p(X)p(Y)]\;$? Let $X,Y$ be two random variables. Let $p(x,y)$ be the joint p.d.f. and $p(x),p(y)$ be the marginal p.d.fs. My question is: What is
$$\mathbb{E}_{X,Y}\!\left[\frac{p(X,Y)}{p(X)p(Y)}\right] = \iint \frac{p(x,y)^2}{p(x)p(y)} dx dy$$
What is the meaning of this quantity? And is there any work that studies this?
Background: I am working on a course project about estimating a certain quantity, and after some computation that quantity is equivalent to this quantity, so I am wondering what this quantity means.
Update: This is an information theory course, so probably this has something to do with information theory. If I am not mistaken, $\mathbb{E}_{X,Y}\!\left[\text{ln} \frac{p(X,Y)}{p(X)p(Y)}\right]$ is the mutual information between $X$ and $Y$.

The quantity $r(x,y)=\frac{p(x,y)}{p(x)p(y)}$ is the (Radon-Nikodym) derivative of $(X,Y)$ with respect to $X\otimes Y$, same as in the general definition of the mutual information.
The quantity $\mathbb E[r(X,Y)]$ you're looking at is the expected derivative under $(X,Y)$.
Recall the $\chi^2$-divergence $\chi^2(U\|V)$ with $U=(X,Y)$ and $V=X\otimes Y$, and notice that
\begin{align*}
\mathbb E[r(X,Y)]&=\mathbb E[r(U)]=\mathbb E[r(U)+r(U)^{-1}-2]+1=\mathbb E[r(U)^{-1}(r(U)-1)^2]\\
&=\mathbb E[(r(V)-1)^2]+1
=\chi^2(U\|V)+1,
\end{align*}
using that $u\mapsto r(u)^{-1}$ is the derivative of $V$ with respect to $U$. Notice that $\chi^2(U\|V)=\chi^2((X,Y)\|X\otimes Y)$ is the mutual information induced by the $\chi^2$-divergence, analogously to the mutual information $I(X,Y)=D((X,Y)\|X\otimes Y)$ induced by the relative entropy.
Thank you, stochasticboy321, for pointing out the $\chi^2$-divergence!
