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IlyaShpitser comments on Causal Inference Sequence Part 1: Basic Terminology and the Assumptions of Causal Inference - Less Wrong
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I agree p(A = a) is imprecise.
Good notation for interventions has to permit easy nesting and conflicts for [ good reasons I don't want to get into right now ]. do(.) actually isn't very good for this reason (and I have deprecated it in my own work). I like various flavors of the potential outcome notation, e.g. Y(a) to mean "response Y under intervention do(a)". Ander uses Y^a (with a superscript) for the same thing.
With potential outcomes we can easily express things like "what would happen to Y if A were forced to a, and M were forced to whatever value M attained had A been instead forced to a' ": Y(a,M(a')). You can't even write this down with do(.).
The A=a notation always bugged me too. I like the above notation because it betrays morphism composition.
If we consider random variables as measure(able) spaces and conditional probabilities P(B | A) as stochastic maps B -> P(A), then every element 'a' of (a countably generated) A induces a point measure * -> A giving probability 1 to that event. This is the map named by do(a). But since we're composing maps, not elements, we can use an element a unambiguously to mean its point measure. Then a series of measures separated by ',' give the product measure. In the above example, let a : A (implicitly, * -> A), a' : B (implicitly, * -> B), M : B ~> C, Y : (A,C) ~> D, then Y(a,M(a')) is a stochastic map * ~> D given by composition
EDIT: How do I ascii art?
All of this is a fancy way of saying that "potential outcome" notation conveys exactly the right information to make probabilities behave nicely.
Yes, one of the reasons I am not very fond of subscript or superscript notation (that to be fair is very commonly used) is because it quickly becomes awkward to nest things, and I personally often end up nesting things many level deep. Parentheses is the only thing I found that works acceptably well.
If you think of interventions as a morphism, then it is indeed very natural to think in terms of arbitrary function composition, which leads one to the usual functional notation. The reason people in the causal inference community perhaps do not find this as natural as a mathematician would is because it is difficult to interpret things like Y(a,M(a')) as idealized experiments we could actually perform. There is a strong custom in the community (a healthy one in my opinion, because it grounds the discussion) to only consider quantities which can be so interpreted. See also this:
http://imai.princeton.edu/research/Design.html
Hunh, I never noticed that. TIL.