Recently, I was confused with the proofs of some conditional independent rules (decomposition, weak union, contraction, intersection), particularly the conditional independent rule of INTERSECTION.
In the book Probabilistic Graphical Models Principles and Techniques, Page25 mentioned that the STRICT POSITIVE DISTRIBUTION is necessary for INTERSECTION rule.
INTERSECTION rule states:
For positive distributions, and for mutually disjoint sets X, Y, Z, W: (X⊥Y|Z,W)&(X⊥W|Z,Y) ⇒ (X⊥Y,W|Z). (2.11)
But according to definition (2.3) in the same book on Page24, the right side of (2.11) is trivially established for some events {y1, w1, z1} which leads P(Y=y1, W=w1, Z=z1)=0, it seems STRICT POSITIVE DISTRIBUTION is not necessary for INTERSECTION rule.
Definition (2.3) states:
We say that an event a is conditionally independent of event b given event r in P, denoted P ⊨ (a⊥b|r), if P(a|b ∩ r) = P(a|r) or if P(b ∩ r) = 0.
So, why the book mentioned the STRICT POSITIVE DISTRIBUTION as necessary for INTERSECTION rule?
Suppose that $\mathbf Y, \mathbf Z$ are independent coinflips (equally likely to be $0$ or $1$), and $\mathbf W = \mathbf Y+ \mathbf Z \bmod 2$. Any two of $\mathbf Y, \mathbf Z,\mathbf W$ determine the third.
Then for absolutely any $\mathbf X$, we have $(\mathbf X \perp \mathbf Y \mid \mathbf Z, \mathbf W)$. That's just because, conditioned on any event of the form $\mathbf Z=z \land \mathbf W=w$, the value of $\mathbf Y$ is constant, and a constant random variable is independent from any other random variable.
Similarly, for absolutely any $\mathbf X$, we have $(\mathbf X \perp \mathbf W \mid \mathbf Z,\mathbf Y)$.
However, if we could conclude that $(\mathbf X \perp \mathbf Y, \mathbf W \mid \mathbf Z)$, then in particular we could conclude that $(\mathbf X \perp \mathbf Y \mid \mathbf Z)$. This is not true in general; in particular, it is false when $\mathbf X = \mathbf Y$.