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Causal Inference In Statistics by Pearl, section 3.5, page 70 clearly mentions that -

This effect, written $P(Y=y|do(X=x),Z=z)$, measures the distribution of $Y$ in a subset of the population for which $Z$ achieves the value $z$ after the intervention.

In that context Rule 2 says -

$P(Y=y|do(X=x),Z=z)$
$= \sum_s {P(Y=y|X=x,S=s,Z=z)P(S=s|Z=z)}$

where $S \cup Z$ satisfies the backdoor criterion.

Query: Is $Z=z$ in the second expression pre- or post-intervention measure?

Note: In subsequent section where the effect of conditional intervention is evaluated, it says -

$P(Y=y|do(X=g(Z)))$
$=\sum_z{P(Y=y|do(X=g(Z)),Z=z)P(Z=z|do(X=g(Z)))}$
$=\sum_z{P(Y=y|do(X=g(z)),Z=z)P(Z=z)}$

The equality $P(Z=z|do(X=g(Z)))=P(Z=z)$ stems, of course, from the fact that $Z$ occurs before $X$; hence any control exerted on $X$ can have no effect on the distribution of $Z$.

Here it seems that $Z=z$ is a pre-intervention measure. Hence, in the above sections, when is $Z-z$ a pre-intervention measure and when is it a post-intervention measure?

Anirban Chakraborty
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    It's pre-intervention if either $Z\to X$ or there are a series of arrows from $Z$ to $X.$ It's post-intervention if either $X\to Z$ or there are a series of arrows from $X$ to $Z.$ – Adrian Keister Jan 23 '23 at 13:45
  • @AdrianKeister could you please provide some reference for the above claim? Besides, in $P(Y=y|do(X=x),Z=z)$, $Z=z$ may be post-intervention measure. But in the RHS expression $\sum_s {P(Y=y|X=x,S=s,Z=z)P(S=s|Z=z)}$, $Z=z$ needs to be observational/pre-intervention correct (as it's just a theoretically simulated intervention and we do not have access to post-intervention data)? – Anirban Chakraborty Jan 24 '23 at 22:25
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    Alas, I don't have a reference for it. I heard this concept in a presentation that was utilizing causality. But this idea makes sense: for something to be pre-intervention, it must be before $X;$ and for it to have any interest, it must be causally related to $X.$ Hence, the arrows go into $X$. You see it in passing here: https://stats.stackexchange.com/questions/598107/should-pre-intervention-outcome-variable-y0-be-excluded-from-counterfactual-mode – Adrian Keister Jan 25 '23 at 00:50

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