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Let $G=(V, E)$ be a DAG. Let $\mathrm{dom}$ be a domain for each node in $V$ and $P$ be a joint probabiliy distribution over those domains, that factors as a product of conditional probability distributions in the standard way, giving us a Bayesian Network.

Let $P_O$ be the marginal distribution of some arbitrary subset $O \subset V$ of variables.

How do we know, purely from the graphical structure of $G$ (i.e. without knowing $P$), the minimal DAG structures that the marginal distribution $P_O$ factors over? Specifically, I would like a reference to a published result or textbook.

(Note that we cannot simply take the subgraph given by the nodes in $O$, since we often have to add arrows. For example, if G consists of $X \to Y \to Z$, and $O=\{X,Z\}$ then we must add the arrow $X\to Z$).

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Unfortunately I cannot comment. However, I can recommend the article "Foundations of Structural Causal Models with Cycles and Latent Variables" (https://arxiv.org/abs/1611.06221). It has some theoretical results about marginal SCM. This seems closely related to your question, but I cannot recall whether it has the precise answer that you are looking for. It definitely points in the right direction, though.

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