# Why are $X$ and $Y$ in this graph d-separated given $Z_2$?

I am learning the book "Causality" by Judea Pearl. In Chapter 1, page 18, the d-separation examples, there is a thing I do not understand. Consider this graph (a) on Page 18:

It says that $$X$$ and $$Y$$ are d-separated given $$Z = \{Z_2\}$$. However, I check all criteria of d-separation and I cannot find any criterion that satisfies it, as follows:

There is no chain $$X => ... => Y$$ and no fork $$X <= ... <= Z_2 => ... => Y$$ in the graph, thus the first criterion is eliminated. Thus, we have the collider left.

Consider the collider $$X => Z_1 <==> Z_3 <= Y$$, the second criterion says that "the path contains a collider $$i => m <= j$$ such that $$m$$ is not in $$Z$$ and no descendant of $$m$$ is in $$Z$$". The collider we consider has $$Z_2$$ as a descendant of $$Z_3$$ through the path $$Z_3 => Z_2$$, thus this path is not blocked by $$Z_2$$.

Moreover, consider the path $$X => Z_1 <= Z_2 <= Z_3 <= Y$$, this path is also not blocked because $$Z_2$$ is a collision node.

What details am I missing here?

In (a), $$X$$ and $$Y$$ are unconditionally d-separated because of the unconditioned-upon collider $$Z_1$$, which blocks all paths from $$X$$ to $$Y$$. conditioning on $$Z_2$$ does not change this fact, and so $$X$$ and $$Y$$ remain d-separated after conditioning on $$Z_2$$.

• Thanks for your answer, but could you elaborate more for me? It seems like I have not understood very well: the path $X => Z_1 <= Z_2 <= Z_3 <= Y$ could be my mistake: $Z_3$ is not a collision node, thus this path is still blocked by $Z_1$, but what about $X => Z_1 <==> Z_3 <= Y$? If the arrow points to $Z_1$ through $Z_3$, this path is blocked, but if it is vice versa: $X => Z_1 => Z_3 <= Y$, then $Z_2$ which is a descendant of $Z_3$, is in $Z$. Commented Dec 3, 2021 at 2:28
• A double-headed arrow doesn't mean its nodes point at each other; it means there is a hidden third variable pointing at both of them. $Z_1 <==> Z_3$ is the same as $Z_1 <== U ==> Z_3$. No matter what, no arrows emanate from $Z_1$.
– Noah
Commented Dec 3, 2021 at 14:39

Your answer lies exactly on the next paragraph, on the same page of the book.

In the graph (a), X is d-separated from Y with an empty separation set or even with Z2 on it. Carlos Cinelli and colleagues have published a very interesting paper regarding what variables to control for. You can read it clicking here. Sometimes, you don't need to adjust for any variable to obtain d-separation, however, by doing so you can get better precision.

• Thanks for your point to that paper! Yeah I did read that paragraph, but it was just because I misunderstood the definition of d-separated so I didn't understand why it was separated given Z2. Commented Dec 11, 2021 at 3:16
• You're welcome, @AerysS. This has already happened to me too. Sometimes I give a day or two and come back to read the text again and things are so much clearer haha 😁 Commented Dec 11, 2021 at 10:33