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What will happen if we remove one edge (new graph G') from a directed graph G? For example, does the probability distribution P that factorizes G also factorize G′? and what will happen if G and G′ were undirected graphs?

Any help will be appreciated!

probscie
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Think of minimal I-map. Minimal I-map is part of the definition of Bayesian Network. That will lead you in the right direction.