Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 207 / 2 / b / ii / Solution

The original PDF shows the chain $A\to C\to B$, not a collider. Its Bayesian network factorization is $p(a,b,c)=p(a)p(c\mid a)p(b\mid c)$. For any $c$ with positive marginal probability,Integrating or summing over $a$ also gives the same conditional marginal $p(b\mid c)$. ThusThis is conditional independence; the graph does not generally imply marginal independence, since summing $p(a)p(c\mid a)p(b\mid c)$ over $c$ can transmit dependence from $A$ to $B$. Special statistical parameter choices can make marginal independence hold as well, so the claim is that only the conditional statement is guaranteed by the graph. Conditional distributions on null values of $C$ are immaterial to this assertion.