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This has probably been asked before, so if it has please provide a link to the original question and close this as a duplicate -- I was not able to find the original question myself.

Question: Let's say we are given a joint probability distribution $\mathbb{P}$ of some random variables $X_1, \dots, X_n$, and we know that there exists an undirected graph $G$ such that $\mathbb{P}$ is a distribution compatible with the Markov random field defined by $G$. Then does there exist a directed acyclic graph $N$ such that $\mathbb{P}$ is a distribution compatible with the Bayesian network defined by $N$?

And conversely, let's say that we know $\mathbb{P}$ is a distribution compatible with the Bayesian network defined by the directed acyclic graph $N$. Then does there exist an undirected graph $G$ for which $\mathbb{P}$ is compatible with the Markov random field defined by $G$?

I would also be interested in knowing whether this is true only when given some restrictions on $\mathbb{P}$, e.g. that it be strictly positive. I found some lecture slides talking about how we can triangulate a graph $G$ corresponding to a Markov random field and get something called an "I-map" that corresponds to a Bayesian network, and something similar by "moralizing" a Bayesian network to get a Markov random field. But I am interested in the statement at the level of specific, concrete probability distributions and am having difficulty translating it to that level.

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