# Will the maximum entropy joint distribution given a known set of marginal distributions have the maximum plausible support?

Define $$[n] = \{1, 2, ..., n\}$$. Given a distribution $$P : \{0, 1\}^{[n]} \rightarrow [0, 1]$$ and a subset $$S \subseteq [n]$$, we can define the $$S$$-marginal of $$P$$, $$P_S : \{0, 1\}^S \rightarrow [0, 1]$$ as $$P_S(x) = \sum_{y \in \{0, 1\}^{[n] - S}} P(x \cup y)$$

Given a set of sets $$\Sigma \subseteq 2^{[n]}$$ indexing probability distributions $$\phi_S : \{0, 1\}^S \rightarrow [0, 1]$$, we can define $$P = argmax_{P : \forall S \in \Sigma, P_S = \phi_S, \sum_{x \in \{0, 1\}^{[n]}} P(x) = 1} H(P)$$ where $$H(P)$$ is the Shannon entropy of $$P$$.

In other words, we can define $$P$$ as the maximum entropy distribution with the marginals $$\{ \phi_S \mid S \in \Sigma \}$$. Define $$support(P) = \{ x \mid x \in \{0, 1\}^{[n]}, P(x) > 0 \}$$.

You'll notice each marginal constraints the support to a set:

$$supp_n(S) = \{ x \cup y \mid x \in \{0, 1\}^S, \phi_S(x) > 0, y \in \{0, 1\}^{[n] - S}\}$$

and its clear that $$support(P) \subseteq \bigcap_{S \in \Sigma} supp_n(S)$$.

My question is if $$support(P) = \bigcap_{S \in \Sigma} supp_n(S)$$ when $$P$$ is well-defined. My use case for this has to do with analyzing a particular model for distributions which compares in interesting ways to circuit-based models for distributions.