How to interpret Pearl's do notation? I'm going through the Dragonnet paper (slides available here), and the authors use Pearl's do notation to make this claim:

How can I interpret the do notation? Is the author claiming that the average treatment effect is the difference between the expectation of the outcome given that the entire population is treated and the same expectation given that the entire population is not treated?
If so, why the conditioning on confounding effects would remove this need?
 A: The proper interpretation of the do notation is that the expression $\operatorname{do}(X=1)$ means you are forcing $X$ to have the value $1.$ You are intervening to make that happen. In the Directed Acyclic Graph (DAG) context, the notation $\operatorname{do}(X=1)$ means you do three things:

*

*Delete all arrows going into $X.$

*Replace the node $X$ with the value $1.$

*Update the Structural Causal Model accordingly. This assumes you have a Structural Causal Model.

In an experimental context, it is commonplace to force variables (or factors) to be certain values in order to eliminate unwanted variation. One of the many benefits of Pearl introducing the $\operatorname{do}$ notation is that we can now codify experimental assumptions and procedures in a framework that's easy to analyze.
You wrote:

Is the author claiming that the average treatment effect is the difference between the expectation of the outcome given that the entire population is treated and the same expectation given that the entire population is not treated?

The answer is "yes", if you interpret your words "is treated" as $\operatorname{do}(X=1)$ and "not treated" as $\operatorname{do}(X=0).$ The main goal of the do-calculus, the back-door criterion, the front-door criterion, instrumental variables, etc., is to reduce causal expressions containing the do operator to probabilistic expressions that do NOT contain the do operator. That way, you can evaluate the result using your data.
The problem is that, in general, $P(Y|\operatorname{do}(X=x))\not=P(Y|X=x).$ So you have these procedures that Pearl and others have come up with for removing the do operator from your expression.
You wrote

If so, why the conditioning on confounding effects would remove this need?

Read The Book of Why or Causal Inference in Statistics: A Primer to find out why this works. Indeed, the fact that we really know what a confounding variable is (a variable that sets up a back-door path from the cause to the effect) is one of the supremely valuable things the New Causal Revolution has given us.
