Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 219 / 3

For a fixed observed colour ${\tilde{c}}_s$, Bayes theorem givesCompleting the square shows that this is a truncated normal distribution with variance ${\sigma }_c^2$, lower endpoint zero, and untruncated locationHenceConditioning first on $E_s$ and using the law of total expectation yields the dusty colour-magnitude relationAs ${\tilde{c}}_s\to -\infty$, the Inverse Mills ratio implies ${\overline{E}}_s\to 0$ with vanishing derivative, so the asymptotic slope is $\beta$. As ${\tilde{c}}_s\to +\infty$, ${\overline{E}}_s\sim {\tilde{c}}_s-c_0-{\sigma }_c^2/\tau$, so the asymptotic slope is $R$.

Let $c_s={\tilde{c}}_s-E_s$ and $M_s={\tilde{M}}_s-R{E}_s$. With the stated improper hyperpriors, the unnormalized posterior density isIts Directed acyclic graph has, for each star, the arrowsfollowed by the deterministic observation arrowsThe latent variables $(c_s,M_s,E_s)$ are repeated inside a plate indexed by $s=1,\dots ,N$; all hyperparameters lie outside that plate.

A Gibbs sampler draws each variable from its full conditional distribution, so every proposed update is accepted. At the start of a sweep define $c_s={\tilde{c}}_s-E_s$, $y_s={\tilde{M}}_s-R{E}_s$, andFirst update the reddenings independently asRecompute $c_s$ and $y_s$, then make the Gaussian updatesandWiththe remaining full conditionals, in the parameterization given in the question, are the scaled inverse chi-squared lawsRepeating these updates in the displayed order gives a complete Gibbs sweep. The formulas assume $N>2$ and nondegenerate sampled predictors; posterior propriety must be checked because the hyperpriors are improper.