Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 221 / 3

A minimal causal directed acyclic graph has edges

Yes. Every path from $D$ to $G$ is blocked by a collider, such as $D\to A\leftarrow G$ or $D\to C\to M\leftarrow A\leftarrow G$. Thus D-separation gives $D\perp G$.

No. Conditioning on the collider $A$ opens $D\to A\leftarrow Z$, so $D$ and $Z$ are not conditionally independent given $A$.

Yes. Conditioning on $A$ blocks the directed paths from $Z$ to $Y$. Although conditioning on $A$ opens $Z\to A\leftarrow G\to Y$ and $Z\to A\leftarrow D\to C\to M\to Y$, conditioning additionally on $G$ and $D$ blocks those paths. Hence $Z\perp Y\mid (A,G,D)$.

Yes. Conditioning on $D$ blocks $C\leftarrow D\to A$, while $C\to M\leftarrow A$ is blocked at the collider $M$. Therefore $C\perp A\mid D$.

The independence fails after conditioning on $S=1$. Since $S$ is a descendant of collider $M$, selection opens $D\to C\to M\leftarrow A\leftarrow G$.

The independence already failed, and it still fails because conditioning on $A$ opens $D\to A\leftarrow Z$.

The independence still holds. Every path from $Z$ starts through $A$; paths on which $A$ is a noncollider are blocked by conditioning on $A$, while paths opened at collider $A$ are blocked by the conditioned variables $G$ or $D$. Conditioning on $S$ creates no route avoiding these blocks.

The independence fails. Conditioning on $S$, a descendant of $M$, opens the collider path $C\to M\leftarrow A$ even after conditioning on $D$.

Each factor $f_{j\mid \mathrm{pa}(j)}$ depends only on node $j$ and its parents. Those vertices form a clique in the moral graph, because moralization joins every pair of parents and removes arrow directions. The positive joint density therefore factors into clique potentials of the moral graph. The Hammersley-Clifford theorem then implies the global Markov property for that undirected graph.