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

A nonparametric structural equation model for the graph can be writtenThe bidirected edge $A\leftrightarrow Y$ permits dependence between $U_A$ and $U_Y$; apart from this pair the exogenous variables are mutually independent. The basic potential outcomes areand the natural nested outcome is $Y(a)=Y(M(a,X),X)$.

The exogenous independences imply that the whole family $\{M(a,x):a,x\}$ is independent of $\{X,A,Y(m,x):m,x\}$, while $X$ is independent of $A$ and the basic response potentials. The bidirected edge means that $A$ and $Y(m,x)$ need not be independent. Useful observed-counterfactual consequences include

In an acyclic directed mixed graph, a fixable vertex $v$ is one whose bidirected district contains no proper directed descendant of $v$:Here $A$ and $Y$ lie in the same district because $A\leftrightarrow Y$, and $Y$ is a directed descendant of $A$ along $A\to M\to Y$. Thus $A$ is not fixable.

The claim is not implied. On the path $A\leftrightarrow Y\leftarrow X$, $Y$ is a collider, and conditioning on $Y$ opens that path. Equivalently, conditioning on the common effect $Y$ can induce collider bias between $A$ and $X$.

The claim is not implied. In the graph for $Y(m)$ the pathis open after conditioning only on $X$. The latent common cause represented by the bidirected edge associates $A$ with $Y(m)$, and $A$ also causes $M$.

The claim is not implied. The intervention setting $A=a$ removes the outgoing dependence of descendants on the observed value of $A$, but the bidirected path $A\leftrightarrow Y(a)$ remains open. Conditioning on $X$ does not block latent confounding between $A$ and $Y$.

This conditional independence is implied. Fixing the mediator at $m$ removes the directed edge $M\to Y$ in the relevant counterfactual graph. Conditioning on $A$ blocks $M\leftarrow A\leftrightarrow Y(m)$, while conditioning on $X$ blocks $M\leftarrow X\to Y(m)$. Every remaining path is blocked by the m-separation criterion, so

This conditional independence is implied. In the graph for $M(a)$, the incoming causal value of $A$ has been fixed, so the path $A\to M$ no longer transmits association to $M(a)$. The only possible route through $Y$ contains a collider, and conditioning on $X$ blocks the common-cause path through $X$. Hence m-separation gives

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 $X$.