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 $(A,X)$. Therefore
$$
\mathbb P(Y(a)=y)
=\sum_{x,m}\mathbb P(x)\mathbb P(m\mid a,x)
\sum_{a'}\mathbb P(y\mid m,a',x)\mathbb P(a'\mid x).
$$
The 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 $X$.

Solution

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

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Front-door adjustment identifies an exposure effect through an observed mediator when the mediator intercepts every directed exposure-outcome path, the exposure-mediator relation has no unblocked backdoor path, and exposure blocks every backdoor path from mediator to outcome.


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 Incoming links: Front-door adjustment