  We study the convergence properties of the Gibbs Sampler in the context of
posterior distributions arising from Bayesian analysis of conditionally
Gaussian hierarchical models. We develop a multigrid approach to derive
analytic expressions for the convergence rates of the algorithm for various
widely used model structures, including nested and crossed random effects. Our
results apply to multilevel models with an arbitrary number of layers in the
hierarchy, while most previous work was limited to the two-level nested case.
The theoretical results provide explicit and easy-to-implement guidelines to
optimize practical implementations of the Gibbs Sampler, such as indications on
which parametrization to choose (e.g. centred and non-centred), which
constraint to impose to guarantee statistical identifiability, and which
parameters to monitor in the diagnostic process. Simulations suggest that the
results are informative also in the context of non-Gaussian distributions and
more general MCMC schemes, such as gradient-based ones.implementation of Gibbs
samplers on conditionally Gaussian hierarchical models.
