A way I’d phrase John’s sibling comment, at least for the exact case: adding arrows to a DAG increases the set of probability distributions it can represent. This is because the fundamental rule of a Bayes net is that d-separation has to imply conditional independence—but you can have conditional independences in a distribution that aren’t represented by a network. When you add arrows, you can remove instances of d-separation, but you can’t add any (because nodes are d-separated when all paths between them satisfy some property, and (a) adding arrows can only increase the number of paths you have to worry about and (b) if you look at the definition of d-separation the relevant properties for paths get harder to satisfy when you have more arrows). Therefore, the more arrows a graph G has, the fewer constraints distribution P has to satisfy for P to be represented by G.
A way I’d phrase John’s sibling comment, at least for the exact case: adding arrows to a DAG increases the set of probability distributions it can represent. This is because the fundamental rule of a Bayes net is that d-separation has to imply conditional independence—but you can have conditional independences in a distribution that aren’t represented by a network. When you add arrows, you can remove instances of d-separation, but you can’t add any (because nodes are d-separated when all paths between them satisfy some property, and (a) adding arrows can only increase the number of paths you have to worry about and (b) if you look at the definition of d-separation the relevant properties for paths get harder to satisfy when you have more arrows). Therefore, the more arrows a graph G has, the fewer constraints distribution P has to satisfy for P to be represented by G.