$X$ --> $Z$ <-- $Y$
When using Bayes ball to check the independence of $X$ and $Y$ given $Z$, it is known that $P(X,Y|Z)\neq P(X|Z)P(Y|Z)$ in the above graph.
(There is an example to help you to understand why. Considering that
- $X$: I am kidnapped.
- $Y$: I am sick.
- $Z$: I am late for the meeting.
I am late may be because of $X$ or $Y$. If you see me that I am late for the meeting and I am sick. Then $P(X|Z,Y)\ll P(X|Z)$. )
My question is:
There is another rule called Markov Blanket, which states that given the Markov blanket $B(X)$ of $X$, $X$ is conditional independent with others $\neg X$ given $B(X)$.
$B(X)$ is composed of $X$'s parents, children, children's parents.
In the above graph, $B(X)=\{Z\}$, however, as I stated before, $X\not\perp Y|Z$.
So, what's wrong?
In the Markov Blanket of X you are forgetting that Y should also be included. Y is one of X's children's parents (otherwise known as one of X's co-parents). So B(X) includes both Z and Y.