Exercise 10.9 - Moralization does not introduce new independence statements

To prove that:

it is sufficient to prove that:

where $\neg <!--\; FUNCTION\; APPLICATION\; -->\text{CI}(G)$ is the collection of denials of conditional independence from $G$.

A statement in $\neg <!--\; FUNCTION\; APPLICATION\; -->\text{CI}(G)$ taking form:

is tantamount to that: there exists one unblocked path, $p$, from $𝒜$to $B$. Consider this $p$ in $M$, where the directions on the edges have vanished. The only possibility that this path in $M$ is blocked is that $p$ encounters a collider structure in $G$ and the intermedium node $m$, or some of its descents is in $E$. In this case there exists an extra edge between the parents of $m$ in $M$, hence we replace the two edges connecting $m$ and $m$’s parents with the extra edge. This procedure ends up with another unblocked edge, hence we have the statement $\neg <!--\; FUNCTION\; APPLICATION\; -->\text{CI}(G)\subset \neg <!--\; FUNCTION\; APPLICATION\; -->\text{CI}(M)$ completed. This finishes the proof of the original proposition.