Exercise 10.2 - Bayes Ball

For question (a), for node $C$, there is a path:

so $C$ is not independent from $A$ given $B$.

For node $D$, there is a path:

so $D$ is not independent from $A$ given $B$.

For node $E$, there is a path:

so $E$ is not independent from $A$ given $B$.

For node $F$, there is a path:

so $F$ is not independent from $A$ given $B$.

For node $G$, there is a path:

so $G$ is not independent from $A$ given $B$.

For node $H$, there is a path:

so $H$ is not independent from $A$ given $B$.

For node $I$, all pathes from $I$ to $A$ have to go through a collider whose intermedium is not shadowed ( $G$ and $H$), hence $I$ is independent from $A$ given $B$.

For question (b), for node $B$, there exist a path to $A$,

which contains a collider structure that is not blocked since $G$’s child, $J$, is shadowed. Hence $B$ is not independent from $A$ given $J$.

For node $C$, each of its pathes to $A$ must bypass $F\to I\to E$, which is a blocked collider, hence $C$ is conditionally independent from $A$.

For node $D$, the result is the same as $B$.

For $E$, pathes:

is unblocked, hence $E$ is not independent from $A$ given $J$.

For $F$, the result is the same as $C$.

For $G$, it is obviously not independent from $A$ given $J$.

For $H$, path:

is unblocked, hence $H$ is not independent from $A$ given $J$.

For $I$, the path:

is unblocked, hence $I$ is not independent from $A$ given $J$.