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2021 Block Gibbs samplers for logistic mixed models: Convergence properties and a comparison with full Gibbs samplers
Yalin Rao, Vivekananda Roy
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Electron. J. Statist. 15(2): 5598-5625 (2021). DOI: 10.1214/21-EJS1930

Abstract

The logistic linear mixed model (LLMM) is one of the most widely used statistical models. Generally, Markov chain Monte Carlo algorithms are used to explore the posterior densities associated with the Bayesian LLMMs. Polson, Scott and Windle’s (2013) Pólya-Gamma data augmentation (DA) technique can be used to construct full Gibbs (FG) samplers for the LLMMs. Here, we develop efficient block Gibbs (BG) samplers for Bayesian LLMMs using the Pólya-Gamma DA method. We compare the FG and BG samplers in the context of a real data example, as the correlation between the fixed effects and the random effects changes as well as when the dimensions of the design matrices vary. These numerical examples demonstrate superior performance of the BG samplers over the FG samplers. We also derive conditions guaranteeing geometric ergodicity of the BG Markov chain when the popular improper uniform prior is assigned on the regression coefficients, and proper or improper priors are placed on the variance parameters of the random effects. This theoretical result has important practical implications as it justifies the use of asymptotically valid Monte Carlo standard errors for Markov chain based estimates of the posterior quantities.

Acknowledgments

The authors thank the editor and two anonymous reviewers for several helpful comments and suggestions that led to an improved revision of the paper.

Citation

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Yalin Rao. Vivekananda Roy. "Block Gibbs samplers for logistic mixed models: Convergence properties and a comparison with full Gibbs samplers." Electron. J. Statist. 15 (2) 5598 - 5625, 2021. https://doi.org/10.1214/21-EJS1930

Information

Received: 1 February 2021; Published: 2021
First available in Project Euclid: 27 December 2021

Digital Object Identifier: 10.1214/21-EJS1930

Subjects:
Primary: 60J05
Secondary: 62F15

Keywords: Data augmentation , drift condition , geometric ergodicity , GLMM , Markov chain CLT , MCMC , standard errors

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Vol.15 • No. 2 • 2021
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