Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 221 / 1

A path in a causal directed acyclic graph is active given a conditioning set $K$ when every noncollider on the path is outside $K$ and every collider has itself or a descendant in $K$. Two vertex sets are d-separated by $K$ when no path between them is active given $K$. The global Markov property of a directed acyclic graph then turns d-separation into conditional independence.

In the displayed graph, the edges areEvery observed pair except $(X_1,X_4)$ and $(X_2,X_4)$ is joined by a direct edge, which remains active under conditioning on any other observed variables. The pair $(X_2,X_4)$ is always joined by the fork $X_2\leftarrow U\to X_4$, because the unobserved noncollider $U$ cannot be conditioned on.

For $(X_1,X_4)$, if $X_3$ is not conditioned on, $X_1\to X_3\to X_4$ is active. If $X_3$ is conditioned on, the pathbecomes active because its collider $X_2$ has conditioned descendant $X_3$; conditioning on $X_2$ itself also opens it. Thus every observed pair is d-connected given every subset of the other observed variables. Any conditional independence between two nonempty observed subvectors would imply one between each selected pair, so the graph entails none.

The directed acyclic graph factorization isThe graph gives $U\perp X_1$ and $U\perp X_3\mid (X_1,X_2)$. Therefore Bayes theorem givesConsequentlywhich contains no $x_1$. This observed equality is a Verma constraint: it is implied by the latent-variable causal graph even though it is not a conditional independence.

WriteSince $X_1$ has no parents, its total effect has no backdoor path, and

For $X_2$, $X_3$ intercepts every directed path to $X_4$. Conditional on $X_1$, there is no unblocked backdoor path from $X_2$ to $X_3$, and $(X_1,X_2)$ blocks every backdoor path from $X_3$ to $X_4$. The conditional front-door adjustment therefore givesandPart ii shows that the inner sum, after also averaging $X_4$, does not actually depend on $x_1$.

For $X_3$, $(X_1,X_2)$ is a sufficient backdoor adjustment set. HenceThese three formulas identify the requested average treatment effects from the observed joint distribution.