Suppose $X$ and $Y$ are correlated random variables in a finite set ${\mathcal A}$, and let $f, g$ be functions that map elements from ${\mathcal A}$ to ${\mathcal B}$ for some finite set ${\mathcal B}$.
Assume the following:
- $f(X)$ is independent of $Y$
- $g(Y)$ is independent of $X$
Can we say that there exist independent variables $A, B, C$ and functions $h_1, h_2$ such that the joint distribution of f(X), g(Y), X, Y is same as $A$, $B$, $h_1(A, C)$ and $h_2(B, C)$?
If not, then is there a counter-example? Of course, coming up with a counter-example might be hard since one needs to show that it is not true for any choice $A, B, C$, $h_1$ and $h_2$, but maybe there is some intuitive reasoning why this is not true..