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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:

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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?

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2 Answers 2

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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 $Z_2$.

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  • $\begingroup$ 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 $Z$. $\endgroup$ Commented Dec 3, 2021 at 2:28
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    $\begingroup$ 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 $Z_1$. $\endgroup$
    – Noah
    Commented Dec 3, 2021 at 14:39
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Your answer lies exactly on the next paragraph, on the same page of the book.

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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.

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    $\begingroup$ 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. $\endgroup$ Commented Dec 11, 2021 at 3:16
  • $\begingroup$ 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 😁 $\endgroup$ Commented Dec 11, 2021 at 10:33

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