Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 221 / 3 / iv / Solution

This is the covariate-conditional front-door adjustment. The paths from $A$ to $M$ have no unblocked backdoor path after conditioning on $X$, and all backdoor paths from $M$ to $Y$ are blocked by $(A,X)$. ThereforeThe inner sum identifies the effect of setting $M=m$ at covariate value $x$ by adjusting for $A$; the outer sum transports this through the mediator distribution generated by setting $A=a$ and then averages over $X$.