"Given a probability distribution $P(x_1, \dots, x_n)$ and any ordering d of the variables, the DAG(directed acyclic graph) created by designating as parents of $X_i$ any minimal set П$_{X_i}$ of predecessors satisfying $$P(x_i|П_{X_i}) = P(x_i|x_1, \dots, x_{i-1}), П_{X_i}\subseteq \{X_1, \dots, X_{i-1}\}$$ is a Bayesian network of P."
This text is a Corollary 3 in Chapter 3 from the book "Probabilistic Reasoning in Intelligent Systems".
I do not see how it is correct. Consider a simple network $X_1 \to X_2$ and ordering $d=\{X_2, X_1\}$. This ordering will induce a network that consist from two disconnected nodes $X_1$ and $X_2$. From this network follows that $X_1$ and $X_2$ are independent, even though they are.
Is this Corollary wrong or I am missing something?
PS Essential property of a Bayesian network for a probability distribution P is that it reflects independent relations (I-map of P). If two sets X, Y of variables are d-separable given set Z then X, Y are independent according to P given Z. In my example with two variables d-separable is just whether there is a path between variables or not (Z is empty set).
This Corollary is for Theorem 9 of the book:
[Verma 1986] Let M be any semi-graphoid. If D is a boundary DAG of M relative to ANY ordering d, then D is a minimal I-map of M.
Note that any probability distribution is a semi-graphoid.
