I was requested to find a general algorithm which, given a Bayesian network graph $\mathcal{G}$ over a set of random variables $\mathcal{X}$ and a node to remove $X\in \mathcal{X}$, builds a new graph $\mathcal{G'}$ which is a minimal I- Map for the distribution over $\mathcal{X}\backslash X$ obtained by marginalizing over $X$
Given a set of independence assertions, building an I-map is easy enough, but how to read off the independencies in the marginalized distribution?
All independence assertions are of the form $$A\perp B\mid C$$ For $A,B,C\subseteq\mathcal{X}$
I figure that if $X$ is not in neither $A, B$ or $C$ marginalizing over it should not affect the independence, if $X\in A$ then we should have $$A\backslash\{X\} \perp B\mid C$$ But how do I handle the case where $X\in C$?