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!

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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!