# Multilevel linear models, Gibbs samplers and multigrid decompositions

## TL;DR

This paper analyzes the convergence of Gibbs samplers in multilevel Bayesian models, providing explicit formulas and guidelines for optimizing their implementation across various hierarchical structures.

## Contribution

It introduces a multigrid approach to derive convergence rates for Gibbs samplers in complex multilevel models, extending analysis beyond two-level hierarchies.

## Key findings

- Explicit convergence rate formulas for multilevel models
- Guidelines for parametrization and identifiability in Gibbs sampling
- Simulation results indicating broader applicability to non-Gaussian and gradient-based MCMC

## Abstract

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.