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Consider discrete random variables $X_1,\cdots, X_n$, and let $D$ be their joint distribution. For each subset $S\subseteq[n]$ let $D_S$ be the marginal distribution $(X_i)_{i\in S}$.

Fix $k<n$. Suppose you are given, for every $S$ of size $k$, a distribution $D_S$. Under what conditions do the distributions $\{D_S\}_{S\subseteq[n],|S|=k}$ correspond to marginals of some distribution joint $D$?

In the case $k=1$, the solution is trivial: every set of distributions $(D_{\{i\}})_{i\in[n]}$ is the set of marginal distributions of the product distribution $D_{\{1\}}\times\cdots\times D_{\{n\}}$.

For $k>1$, a necessary condition is that the $D_S$ are consistent: for any two sets $S_0,S_1$ of size $k$, let $T=S_0\cap S_1$. Then we require that the marginal distribution of $D_{S_0}$ restricted to $T$ is identical to the marginal distribution of $D_{S_1}$ restricted to $T$.

Is consistency sufficient for the extistence of a joint distribution $D$? If not, are there any simple conditions under which we can conclude that a joint distribution $D$ exists? Note that I only care about existence, and do not care if the joint distribution is unique or how to actually compute the joint distribution.

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It turns out that consistency is not sufficient, even in the case $k=2$. Suppose the domain is just $\{0,1\}$. Consider the case where are the pairwise marginal distributions are uniform over pairs of distinct elements, that is uniform over $\{01,10\}$. Then all the variables $X_i$ are just uniform, giving consistency. However, any joint distribution $D$ with the given marginals would have to have all the $X_i$ be distinct. This is clearly impossible for $n\geq 3$.

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