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(.).
Comment author:dogirardo
01 August 2014 05:00:23PM
*
3 points
[-]
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.
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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.