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!
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!