= Solution
The <probabilistic graphical model> is the following <Directed acyclic graph>. The shared <hyperparameters> lie outside the plate; the plate repeats the local variables for $s=1,\ldots,N$. The observed colour is shaded, and the known error <variance> $r_s$ is a square.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-219.png]
{title=Graphical model for intrinsic supernova colour, dust reddening and measurement noise}
{description=Arrows from $\mu_C,v$ to $C_s$, from $\tau$ to $E_s$, and from $C_s,E_s,r_s$ to the observed $\widehat O_s$ encode the conditional factorization. The plate encloses $N$ local supernova models.}
Thus the parent sets are $\boxed{\operatorname{pa}(C_s)=\{\mu_C,v\},\quad\operatorname{pa}(E_s)=\{\tau\},\quad\operatorname{pa}(o_s)=\{C_s,E_s,r_s\}}$. Independent <hyperparameter> prior factors are represented by the three root nodes. With the printed improper priors this is a factorization diagram; it becomes a normalized generative model after proper priors are supplied.
Back to article page
