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.


University of Michigan - Ann Arbor

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 $see(x)$

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 is follows: if we do something that does not affect $Y$, then the probability distribution of $Y$ will not change.

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