Given a Bayesian network $(p,\mathcal{G})$, where $p$ is our joint distribution, and $\mathcal{G}$ is a DAG.
Then by the d-separation theorem we can deduce conditional independence statements, in which contain all of the local independencies.
It can then be shown that all of these conditional independent statements hold for $p$, that can be factorised according to $\mathcal{G}$.
I was now wondering whether further conditional independence statements can be deduced from the DAG $\mathcal{G}$ and whether these are guaranteed to hold for $p$ that can be factorised according to $\mathcal{G}$.
