Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 219 / 4 / d / Solution

Every move can be drawn from a full conditional distribution, producing a Gibbs sampler with acceptance probability one. Write $s_{x,i}^2={\sigma }_{x,i}^2$ and $s_{y,i}^2={\sigma }_{y,i}^2$. First update independentlyand thenLet $X$ have rows $(1,{\xi }_i)$ and $\eta =({\eta }_i)$. Update the linear regression coefficients jointly byand updateFinally, the flat positive variance priors give the following full conditionals, each an inverse-gamma distribution:A systematic sweep in the displayed order, using the newly sampled values immediately, defines the chain. The shapes are positive for $N>2$; full column rank of $X$ is also required.