Why is a' used in the proof of the front-door adjustment formula by Hernán and Robins in their book Causal inference What If Please can I confirm what the meaning of a' is in the proof of the front-door adjustment formula. I assume it represents all other treatment levels not equal to the level of interest (a) but I am unclear how it appears in the proof.
Thanks in advance
Edit 1: I have provided a picture of the proof from the book below and have underlined the speenter image description herecific line. I think I understand why the conditional exchangeability is relevant - it means we can express the counterfactual probability $Pr(Y^{m} = 1)$ via observational quantities. However, I would have written it as follows $Pr(Y^{m} = 1) = \sum_{a \in A} Pr(Y=1|M=m, A=a)Pr(A=a)$ i.e. I am unclear why a' has been used and why the notation I wrote was not.



The notation you used and the notation Robins used is $100\%$ equivalent. The variables $a'$ or $a$ in your formula are both dummy variables - variables used in a summation. It literally does not matter what is used, with one important caveat: a dummy variables needs to not inadvertently coincide with another variable being used, particularly in the summation. But it can help with clarity if it doesn't coincide with any variable being used anywhere else. I'm not saying, of course, that the dummy variable shouldn't show up in the summation expression: it can. But using a completely different dummy variable can have the virtue of increasing clarity by essentially saying this: "Look, I'm completely different from any variable that's defined elsewhere." Another way of saying this is that the dummy variable only has scope in the summation and nowhere else.
