I have been trying to get my head around using do-calculus in causal inference, and I've run into a little toy problem that's confused me and my classmates.
Imagine you have a DAG for some causal model that is defined by $(X \rightarrow M, M \rightarrow Y, U \rightarrow X, U \rightarrow Y)$, where $M$ is a mediator, $X$ is a treatment, $Y$ is an outcome and $U$ is unobserved confounders. I want to calculate $P(Y|do(M=m))$.
As I understand it, this amounts to cutting the link $ X \rightarrow M$, so the new DAG lacks that chain. Then there are no backdoor paths from $M \rightarrow Y$, so you can get the effect of the intervention trivially easily with $P(Y|do(M=m)) = P(Y|M=m)$. My classmate thinks that you still need to account for values of $X$ though, so you need to adjust for those and it should actually be $P(Y|do(M=m))=\Sigma_{X'} P(Y | M=m, X')P(X')$.
Which of us is right? My answer seems too simple, but I just don't understand why you need to condition for $X$ if there are no backdoor paths from $M$ to $Y$ via $X$?
