Given two conditionals, does there exist a joint distribution?

Question

Let $q(X|Y)$ and $r(Y|X)$ be two conditional distributions. Under what conditions, there exists a joint distribution $p(X,Y)$ such that $p(X|Y) = q(X|Y)$ and $p(Y|X) = r(Y|X)$.

I am sure people must have solved this problem. Can anybody point me to a suitable reference?

Answer 1

I found the reference. https://projecteuclid.org/download/pdf_1/euclid.ss/1009213728

In order for $q(X|Y)$ and $r(Y|X)$ to be compatible, $\frac{q(X|Y)}{r(Y/X)}$ needs to decompose into a term corresponding to $X$ and another term corresponding to $Y$.

--edit-- To clarify, we should be able to write $\frac{q(X|Y)}{r(Y/X)}$ as a product of a function of $X$ alone and another function of $Y$ alone.