 Solution

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.