Hi, i've been trying to learn d-separation and came across this figure in figure 4 (page 9) of Spirtes paper and i am a bit confuse. In the paper, Spirtes says
X4 is not d-separated from X1 given its parents X2 and X3 (see figure 4).
I can see why X4 is not d-separated from X1 given X3. But I don’t see why conditioning on X2 breaks the D-separation between X1 and X3 (or why conditioning on X2 is even necessary?). What am I missing? Any help would greatly be appreciated
Not sure which paper this is from (although Spirtes used a nearly identical graph in Figure 2. of Spirtes, P. (1995). Directed Cyclic Graphical Representations of Feedback. In P. Besnard & S. Hanks (Eds.), Eleventh Conference on Uncertainty in Artificial Intelligence. Morgan Kaufmann Publishers, Inc.).
The key point is that the Figure 4 which you excerpted is not a Directed Acyclic Graph. The value of D separation in linking causal beliefs to statistical dependencies in order to inform counterfactual formal causal inference depends strongly on the "acyclical" portion of "DAG". For causal inference (and deduction) around cyclic graphs (i.e. there is a cycle of length two in Figure 4, made up of $X_3$ and $X_4$) you need to look to other causal formalisms.
Your confusion is understandable: I find that how time is (and is not) represented in DAGs is often glossed over. Bear the following in mind:
DAGs imply some temporal ordering among variables: $A \to B$ in a DAG means that $A$ occurred before $B$: causal ordering implies temporal ordering. Likewise $A \to B \to C$ means that both $A$ and $B$ occurred before $C$, as well as $A$ occurring before $B$.
DAGs do not necessarily imply temporal ordering among all variables: In the DAG where $X \to Y$ and $Z \to Y$ (I just ran out of MathJax/LaTeX chops, sorry ;) both $X$ and $Z$ occur before $Y$, but the DAG (and the logic that follows from it a la D separation, etc.) says nothing about the temporal ordering of $X$ and $Z$ with respect to one another: either may occur first, or they may occur simultaneously, and the DAG is still valid.
DAGs do not imply quantitative timing: The causal logic of D separation (or lack thereof) of the variables appearing in a DAG applies regardless of when on the clock or calendar variable phenomena occurred (temporal ordering aside).
Directed cyclic graphs represent time differently: In the causal formalisms using directed cyclic graphs which I am familiar with, the causal graph represents variables at all times during which the DCG is valid/exists. (I think of it as looking 'head on' at the arrow of time.) For example, Figure 4 implies (among other things) that $X_{3,t-1} \to X_{4,t}$, $X_{3,t} \to X_{4,t+1}$, $X_{3,t+1} \to X_{4,t+2}$, etc. You could re-represent Figure 4 as a DAG, but then:
(i) you have to make decisions about things like lags (i.e. does $X_{3,t-1} \to X_{4,t+2}$?),
(ii) you have to make a decision about whether you will capture some event like 'the beginning of time' where $t=1$ in a physical/natural sense, and possibly the end of time as well,
(iii) you gonna need many variables and arrows and your calculations are going to get combinatorically challenging. (You might get by using a "probabilistic graph template," which some of the folks doing data processing with Bayesean probabilistic causal graphs are familiar with.)
This point about "all times" applies to non-looped variables in DCGs also: Figure 4 implies $X_{1,t-1} \to X_{3,t}$, $X_{1,t} \to X_{3,t+1}$, etc. Same point about lags here as well.
This should give you a good insight as to why DCGs break D separation in the way Spites, Pearl, etc. talk about it (pick some point in time and draw a conclusion about D separated paths… then realize that the variables from 1 step in time backward from where your analysis was make for as yet unaccounted for open backdoor paths, etc.)
