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We study the convergence properties of the Gibbs Sampler in the context ofposterior distributions arising from Bayesian analysis of conditionallyGaussian hierarchical models. We develop a multigrid approach to deriveanalytic expressions for the convergence rates of the algorithm for variouswidely used model structures, including nested and crossed random effects. Ourresults apply to multilevel models with an arbitrary number of layers in thehierarchy, while most previous work was limited to the two-level nested case.The theoretical results provide explicit and easy-to-implement guidelines tooptimize practical implementations of the Gibbs Sampler, such as indications onwhich parametrization to choose (e.g. centred and non-centred), whichconstraint to impose to guarantee statistical identifiability, and whichparameters to monitor in the diagnostic process. Simulations suggest that theresults are informative also in the context of non-Gaussian distributions andmore general MCMC schemes, such as gradient-based ones.implementation of Gibbssamplers on conditionally Gaussian hierarchical models.

Multilevel linear models, Gibbs samplers and multigrid decompositions