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On the Convergence Complexity of Gibbs Samplers for a Family of Simple Bayesian Random Effects Models

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  • Bryant Davis

    (Department of Statistics University of Florida)

  • James P. Hobert

    (Department of Statistics University of Florida)

Abstract

The emergence of big data has led to so-called convergence complexity analysis, which is the study of how Markov chain Monte Carlo (MCMC) algorithms behave as the sample size, n, and/or the number of parameters, p, in the underlying data set increase. This type of analysis is often quite challenging, in part because existing results for fixed n and p are simply not sharp enough to yield good asymptotic results. One of the first convergence complexity results for an MCMC algorithm on a continuous state space is due to Yang and Rosenthal (2019), who established a mixing time result for a Gibbs sampler (for a simple Bayesian random effects model) that was introduced and studied by Rosenthal (Stat Comput 6:269–275, 1996). The asymptotic behavior of the spectral gap of this Gibbs sampler is, however, still unknown. We use a recently developed simulation technique (Qin et al. Electron J Stat 13:1790–1812, 2019) to provide substantial numerical evidence that the gap is bounded away from 0 as n → ∞. We also establish a pair of rigorous convergence complexity results for two different Gibbs samplers associated with a generalization of the random effects model considered by Rosenthal (Stat Comput 6:269–275, 1996). Our results show that, under a strong growth condition, the spectral gaps of these Gibbs samplers converge to 1 as the sample size increases.

Suggested Citation

  • Bryant Davis & James P. Hobert, 2021. "On the Convergence Complexity of Gibbs Samplers for a Family of Simple Bayesian Random Effects Models," Methodology and Computing in Applied Probability, Springer, vol. 23(4), pages 1323-1351, December.
    • Handle: RePEc:spr:metcap:v:23:y:2021:i:4:d:10.1007_s11009-020-09808-8
    DOI: 10.1007/s11009-020-09808-8
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    References listed on IDEAS

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    1. Asmussen, Søren & Glynn, Peter W., 2011. "A new proof of convergence of MCMC via the ergodic theorem," Statistics & Probability Letters, Elsevier, vol. 81(10), pages 1482-1485, October.
    2. Román, Jorge Carlos & Hobert, James P. & Presnell, Brett, 2014. "On reparametrization and the Gibbs sampler," Statistics & Probability Letters, Elsevier, vol. 91(C), pages 110-116.
    3. Komorowski, Tomasz & Walczuk, Anna, 2012. "Central limit theorem for Markov processes with spectral gap in the Wasserstein metric," Stochastic Processes and their Applications, Elsevier, vol. 122(5), pages 2155-2184.
    4. Gareth O. Roberts & Jeffrey S. Rosenthal, 2001. "Markov Chains and De‐initializing Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 28(3), pages 489-504, September.
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