In Wasserman's All of Statistics chapter 18, he defines the following undirected graph:
Let $V$ be a set of random variables with distribution $\mathbb{P}$. Construct a graph with one vertex for each random variable in $V$. Omit the edge between a pair of variables $X,Y$ if they are independent given the rest of the variables: $X \perp Y | rest$ where this means $X$ and $Y$ are independent conditional on the rest of the variables.
He then states a theorem 18.1 that says for disjoint subsets $A,B,C$, that $A \perp B | C$ if the removal of $C$ disconnects the graph. In particular, if two variables $X,Y$ lie in separate connected components, then they are independent.
I don't believe this is true. For example, consider three random variables $X=Y=Z$. It is trivially true that $X \perp Y | Z$, so this graph would have no edges. Thus, the theorem would state that $X$ and $Y$ are independent, which is false.
I can't find any mention of this theorem in his errata. How should this theorem or exposition be corrected?
