The description of Pearl’s counterfactuals in this post isn’t entirely correct, in my opinion.
Your description of the “sunny day beachgoers” statement is describing an interventional distribution (in other words something of the form P(y|do(x))).
The key difference between interventions and counterfactuals is that the former are concerned with a single possible world (after an intervention), while the latter are concerned with multiple hypothetical worlds simultaneously.
Example of a statement about an intervention (causal effect): “I am about to take an aspirin. Will my headache go away?” This is just asking about P(headache|do(aspirin)).
Example of a counterfactual statement: “I took an aspirin an hour ago and my headache is gone. Would I still suffer a headache had I not taken an aspirin?” This is asking about a counterfactual distribution which is denoted in Pearl’s notation by P(headache (subscript) no aspirin | no headache, aspirin). Here the variable with a subscript (headache (subscript) no aspirin) corresponds to headache in the post-intervention world after do(no aspirin). This expression is a ‘true’ counterfactual since it contains a conflict between values observed in the ‘true world’ where events occurred (the values past the conditioning bar), and the ‘hypothetical world’ where we intervene.
The description of Pearl’s counterfactuals in this post isn’t entirely correct, in my opinion.
Your description of the “sunny day beachgoers” statement is describing an interventional distribution (in other words something of the form P(y|do(x))).
The key difference between interventions and counterfactuals is that the former are concerned with a single possible world (after an intervention), while the latter are concerned with multiple hypothetical worlds simultaneously.
Example of a statement about an intervention (causal effect): “I am about to take an aspirin. Will my headache go away?” This is just asking about P(headache|do(aspirin)).
Example of a counterfactual statement: “I took an aspirin an hour ago and my headache is gone. Would I still suffer a headache had I not taken an aspirin?” This is asking about a counterfactual distribution which is denoted in Pearl’s notation by P(headache (subscript) no aspirin | no headache, aspirin). Here the variable with a subscript (headache (subscript) no aspirin) corresponds to headache in the post-intervention world after do(no aspirin). This expression is a ‘true’ counterfactual since it contains a conflict between values observed in the ‘true world’ where events occurred (the values past the conditioning bar), and the ‘hypothetical world’ where we intervene.