Spirtes' example of d-separation not leading to independence in a directed cyclic graph with non-linear structural equations 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.

 A: Here is my explanation. I believe the author is right. It comes down to this: for a double arrow relationship $W\longleftrightarrow Z,$ 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 $Z_3\to Z_2\to Z_1$ to show that $Z_1$ is a descendant of $Z_3;$ he does not use what would be the obvious $Z_1\longleftrightarrow Z_3$ relationship.
