Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 219 / 3 / e

Solution

Up to normalization, the joint posterior is

p (M_{1 : N}, M_{0}, τ^{2} ∣ D) \propto (τ^{2})^{k} \prod_{s = 1}^{N} exp [- \frac{( D _{s} - M _{s} ) ^{2}}{2 σ ^{2}} - \frac{( M _{s} - M _{0} ) ^{2}}{2 τ ^{2}}] (τ^{2})^{- N /2} .

A Gibbs sampler cycle consists of:

independently draw every $M_{s}$ from the normal conditional in part a;
draw $M_{0} ∣ M_{1 : N}, τ^{2} \sim N (\overset{ˉ}{M}, τ^{2} / N)$ ;
draw
$τ^{2} ∣ M_{1 : N}, M_{0} \sim Inv - Gamma (\frac{N}{2} - k - 1, \frac{1}{2} \sum_{s} (M_{s} - M_{0})^{2}) .$
(2)