We can remove $$do(X)$$ from $$P(Y | do(X))$$ in any case where there are no causal paths from $$X$$ to $$Y$$. That is, $$P(Y | do(X)) = P(Y)$$ if there is no path from $$X$$ to $$Y$$ with only forward-directed arrows. We can paraphrase this rule as follows: if we do something that does not affect $$Y$$, then the probability distribution of $$Y$$ will not change.

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Pearl introduces an axiomatic system containing three rules to remove do operators in Chapter 7. 

Rule 1. Irrelevance
Rule 2. Conditional equivalence of *do(x)* and *see(x)*
Rule 3. Lack of causal paths

Generalizing Do-Calculus

Pearl, J., & Mackenzie, D. (2018). $$\mathit{The \ Book \ of \ Why: \ The \ New \ Science \ of \ Cause \ and \ Effect.}$$ Basic Books.

The Book of Why

A conventional measure of a treatment's effectiveness. It averages treatment efficacy over a population and can be found using a randomized control trial (RCT). It measures how much an increase in a single unit of treatment variable X will affect the outcome variable Y. When you have properly eliminated all confounding variables, you can obtain an estimate through running a regression.

Average Causal Effect (ACE)

When we observe a variable W that is irrelevant to Y (possibly conditional on other variables Z), then the probability distribution of Y will not change. 

This assertion of irrelevance translates into a symbolic manipulation:
$P(Y | do(X), Z, W) = P(Y | do(X), Z)$

The stated equation holds provided that the variable set $Z$ blocks all the paths from $W$ to $Y$ after we have deleted all the arrows leading into $X$.


Rule 1. Irrelevance

If a set Z of variables blocks all back-door paths from $X$ to $Y$, then conditional on $Z$, $do(X)$ is equivalent to $see(X)$. We can, therefore, write
$P(Y | do(X), Z) = P(Y | X, Z)$
if $Z$ satisfies the back-door criterion.

Rule 2: Conditional Equivalence of $do(x)$ and $see(x)$

Derivation of the Front-Door Adjustment Formula from the Rules of Do-Calculus

Rule 1 permits the addition or deletion of observations. 
Rule 2 permits the replacement of an intervention with an observation, or vice versa. 
Rule 3 permits the deletion or addition of interventions. 

All of these permits are issued under appropriate conditions, which have to be verified in any particular case from the causal diagram.


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