# Can conditional distributions determine the joint distribution?

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!

#### Solutions Collecting From Web of "Can conditional distributions determine the joint distribution?"

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.

Well, I guess you have to know at least one density function among all the random variables $X_i$. Recall the basic property of conditional distributions that $P(X_i\in \mathrm{d}t_i, X_j\in \mathrm{d}t_j) = P(X_i\in \mathrm{d}t_i) P(X_j\in \mathrm{d}t_j| X_i\in \mathrm{d}t_i)$. And by induction you can figure out the joint distributions based on conditional distributions among the variables.
Hope this helps!