Cross Validated Asked by quant_dev on December 10, 2020

In Spirtes (1995) there is an example (Fig. 4 on page 495, reproduced below) of a directed **cyclic** graph with non-linear structural equations in which $d$-separation of $X$ and $Y$ given ${Z, W}$ does not lead to conditional independence of $X$ and $Y$ given ${Z, W}$. I have a problem understanding the first part: why do we say that $X$ and $Y$ are $d$-separated given ${Z, W}?$ Both $Z$ and $W$ are colliders, and we include them both in the conditioning set.

Here is my explanation. I believe the author is right. It comes down to this: for a double arrow relationship $Wlongleftrightarrow Z,$ neither $W$ nor $Z$ is considered a descendant of the other (unless you have other edges relating them). That is, $W$ is not a descendant of $Z,$ nor is $Z$ a descendant of $W.$ So let us consider your graph, but only one direction at a time:

Here, conditioning on the set ${W,Z}$ does open up the collider at $Z$. However, the path from $X$ to $Y$ is still blocked by the chain at $W,$ since $W$ is in the conditioning set. Similarly, if we consider the other "half" of the graph,

the same conditioning set opens the collider at $W$ but closes the chain at $Z.$

In either setting, causal information cannot flow from $X$ to $Y,$ hence ${W,Z}$ $d$-separates $X$ and $Y.$

References: *Causality: Models, Reasoning, and Inference, 2nd Ed.,* by Judea Pearl, pp. 17-18. Note that in the example of Fig. 1.3(a), Pearl has to resort to the path $Z_3to Z_2to Z_1$ to show that $Z_1$ is a descendant of $Z_3;$ he does not use what would be the obvious $Z_1longleftrightarrow Z_3$ relationship.

Correct answer by Adrian Keister on December 10, 2020

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