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Multilevel linear models, Gibbs Samplers and multigrid decompositions (with discussion)
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Zanella, Giacomo and Roberts, Gareth (2021) Multilevel linear models, Gibbs Samplers and multigrid decompositions (with discussion). Bayesian Analysis, 16 (4). pp. 1309-1391. doi:10.1214/20-BA1242 ISSN 1936-0975.
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Official URL: http://dx.doi.org/10.1214/20-BA1242
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.
Item Type: | Journal Article | |||||||||||||||||||||
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Subjects: | Q Science > QA Mathematics | |||||||||||||||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | |||||||||||||||||||||
Library of Congress Subject Headings (LCSH): | Sampling (Statistics), Monte Carlo method, Markov processes, Bayesian statistical decision theory, Convergence | |||||||||||||||||||||
Journal or Publication Title: | Bayesian Analysis | |||||||||||||||||||||
Publisher: | International Society for Bayesian Analysis | |||||||||||||||||||||
ISSN: | 1936-0975 | |||||||||||||||||||||
Official Date: | December 2021 | |||||||||||||||||||||
Dates: |
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Volume: | 16 | |||||||||||||||||||||
Number: | 4 | |||||||||||||||||||||
Page Range: | pp. 1309-1391 | |||||||||||||||||||||
DOI: | 10.1214/20-BA1242 | |||||||||||||||||||||
Status: | Peer Reviewed | |||||||||||||||||||||
Publication Status: | Published | |||||||||||||||||||||
Access rights to Published version: | Open Access (Creative Commons) | |||||||||||||||||||||
Date of first compliant deposit: | 14 March 2022 | |||||||||||||||||||||
Date of first compliant Open Access: | 14 March 2022 | |||||||||||||||||||||
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