Consider three probability distributions in the form $p_1(y,z),p_2(x,z),p_3(x,y)$.

When does a global joint probability $p(x,y,z)$ (possibly not unique) exist?

The first compatibility condition to check is of course that the first order marginals check out: $p_2(x)=p_3(x)$, and so on. Is this the only condition, is it necessary and sufficient? Where can I find it?

Thanks!

PS. I would also be curious about what happens in any order, not just 2 and three...if it's possible!

Note: cross-posting from MSE.

Consider three probability distributions in the form $p_1(y,z),p_2(x,z),p_3(x,y)$.

When does a global joint probability $p(x,y,z)$ (possibly not unique) exist?

The first compatibility condition to check is of course that the first order marginals check out: $p_2(x)=p_3(x)$, and so on. Is this the only condition, is it necessary and sufficient? Where can I find it?

Thanks!

PS. I would also be curious about what happens in any order, not just 2 and three...if it's possible!

Note: cross-posting from MSE.