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Bobertron comments on Causal Inference Sequence Part 1: Basic Terminology and the Assumptions of Causal Inference - Less Wrong Discussion
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The "A=a" stands for the event that the random variable A takes on the value a. It's another notation for the set {ω ∈ Ω | A(ω) = a}, where Ω is your probability space and A is a random variable (a mapping from Ω to something else, often R^n).
Okay, maybe you know that, but I just want to point out that there is nothing vague about the "A=a" notation. It's entirely rigorous.
I think the grandparent refers to the fact that in the context of causality (not ordinary probability theory) there is a distinction between ordinary mathematical equality and imperative assignment. That is, when I write a structural equation model:
Y = f(A, M, epsilon(y))
M = g(A, epsilon(m))
A = h(epsilon(a))
and then I use p(A = a) or p(Y = y | do(A = a)) to talk about this model, one could imagine getting confused because the symbol "=" is used in two different ways. Especially for p(Y = y | do(A = a)). This is read as: "the probability of Y being equal to y given that I performed an imperative assignment on the variable A in the above three line program, and set it to value a." Both senses of "=" are used in the same expression -- it is quite confusing!