Analysis of the Gibbs Sampler for Gaussian hierarchical models via multigrid decomposition

We study the convergence properties of the Gibbs Sampler in the context of posterior distributions arising from Bayesian analysis of Gaussian hierarchical models. We consider centred and non-centred parameterizations as well as their hybrids including the full family of partially non-centred parameterizations. We develop a novel methodology based on multi-grid decompositions to derive analytic expressions for the convergence rates of the algorithm for an arbitrary number of layers in the hierarchy, while previous work was typically limited to the two-level case. Our work gives a complete understanding for the three-level symmetric case and this gives rise to approximations for the non-symmetric case. We also give analogous, if less explicit, results for models of arbitrary level. This theory gives rise to simple and easy-to-implement guidelines for the practical implementation of Gibbs samplers on conditionally Gaussian hierarchical models.