TITLE: Can conditional distributions determine the joint distribution?
QUESTION [7 upvotes]: Can conditional distributions determine the joint distribution?
For example, let $X_1, \dots, X_n$ be random variables. 
Can their joint distribution be determined from the conditional distribution of $X_i$ given others, $i=1, \dots, n$?
Can their joint distribution be determined from other types of conditional distributions, such as the type of $P(X_i|X_j)$, and/or the type of $P(X_i, X_j | \text{others})$, and/or other types?
Thanks!

REPLY [4 votes]: This can in fact be true, and is very important and relevant in the context of Gibbs sampling as mentioned. If a joint distribution $P_{XY}$ has two conditional distributions $P_{X \mid Y}$ and $P_{Y \mid X}$, then knowledge of the conditional distributions alone is often sufficient determine $P_{XY}$. See Arnold and Press, 1989.