Consider $d$ random variables. For each set of $k$ variables, we are given a joint probability distribution. We want to know that whether these distributions correspond to a valid joint probability distribution of all $d$ variables. We can assume that each variable has a finite domain.

I think a necessary condition is that, all given distributions should agree with the same lower dimensional distributions when we integrates some variables out. But this seems not a sufficient condition.

Is there any simple necessary and sufficient condition? or can we find a simple but stronger necessary condition? or is the above necessary condition in fact sufficient? Thanks.