# 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.

• Good question, given that $X$ and $Y$ are $d$-separated given the empty set $\varnothing.$ I can think of one possible explanation, but I have to see if it's correct. Commented Aug 17, 2020 at 14:12

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.

• 1/2 Spirtes ignores decades of theory of causal inference and graphical representation of complex causal systems, e.g., Levins, R. (1974). The Qualitative Analysis of Partially Specified Systems. Annals of the New York Academy of Sciences, 231, 123–138; Sugihara, G., & May, R. M. (1990). Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series. Nature, 344(6268), 734–741. Commented Aug 17, 2020 at 16:34
• 2/2 While Spirtes references some influential papers from the 40s and 50s (a period when cybernetics and feedback in electronic circuits were being tackled as practical problems), he leaves untouched work, especially in ecosystem sciences which made both theoretical and applied advances in the interim. That said, the formal graphical conventions of Levins, and also those of Sugihara are different than Spirtes' DCGs (in how time fits into the graph). Commented Aug 17, 2020 at 16:36
• Alexis: Sorry, I haven't got a foggy clue what you're really saying. Could you please be more specific and use less jargon? Commented Aug 17, 2020 at 20:52
• 1/2 It is hard for me to lend credence to your answer or to call it flawed, since Spirtes does not, to my read, really land on the issue of time in DCGs.(My comment expresses a dissatisfaction with Spirtes.) In the quoted DCG it is unclear whether $W$ is measured at a single time, or whether it is (as in the case of signed digraph representations of causal feedback elsewhere) intended to represent the variable $W$ at all times during which the system exists. Commented Aug 17, 2020 at 23:11
• 2/2 Your answer's analysis is trying to have it both ways in that "one direction at a time" treats $W$ and $Z$ as single fixed points in time, however, the original DCG implies $W$ and $Z$ are both "ancestors" which implies these nodes represent variables at more than one point in time. Contrast the OP's DCG with a DAG of $X_0 \to W_1 \to Z_2$ and $Y_0 \to Z_1 \to W_2$ (Possibly with $X_0 \to W_2$ and $Y_0 \to Z_2$ also) where the subscripts explicitly indicate time (or at least temporal order). Commented Aug 17, 2020 at 23:13