Bernoulli

  • Bernoulli
  • Volume 23, Number 1 (2017), 459-478.

Convergence analysis of block Gibbs samplers for Bayesian linear mixed models with $p>N$

Tavis Abrahamsen and James P. Hobert

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Exploration of the intractable posterior distributions associated with Bayesian versions of the general linear mixed model is often performed using Markov chain Monte Carlo. In particular, if a conditionally conjugate prior is used, then there is a simple two-block Gibbs sampler available. Román and Hobert [Linear Algebra Appl. 473 (2015) 54–77] showed that, when the priors are proper and the $X$ matrix has full column rank, the Markov chains underlying these Gibbs samplers are nearly always geometrically ergodic. In this paper, Román and Hobert’s (2015) result is extended by allowing improper priors on the variance components, and, more importantly, by removing all assumptions on the $X$ matrix. So, not only is $X$ allowed to be (column) rank deficient, which provides additional flexibility in parameterizing the fixed effects, it is also allowed to have more columns than rows, which is necessary in the increasingly important situation where $p>N$. The full rank assumption on $X$ is at the heart of Román and Hobert’s (2015) proof. Consequently, the extension to unrestricted $X$ requires a substantially different analysis.

Article information

Source
Bernoulli Volume 23, Number 1 (2017), 459-478.

Dates
Received: February 2015
First available in Project Euclid: 27 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.bj/1475001361

Digital Object Identifier
doi:10.3150/15-BEJ749

Mathematical Reviews number (MathSciNet)
MR3556779

Zentralblatt MATH identifier
06673484

Keywords
conditionally conjugate prior convergence rate geometric drift condition Markov chain matrix inequality Monte Carlo

Citation

Abrahamsen, Tavis; Hobert, James P. Convergence analysis of block Gibbs samplers for Bayesian linear mixed models with $p>N$. Bernoulli 23 (2017), no. 1, 459--478. doi:10.3150/15-BEJ749. https://projecteuclid.org/euclid.bj/1475001361.


Export citation

References

  • [1] Abrahamsen, T. (2015). Convergence analysis of Gibbs samplers for Bayesian linear mixed models for large $p$, small $n$ problems. Ph.D. thesis, Dept. Statistics, Univ. Florida.
  • [2] Diaconis, P., Khare, K. and Saloff-Coste, L. (2008). Gibbs sampling, exponential families and orthogonal polynomials. Statist. Sci. 23 151–178.
  • [3] Flegal, J.M., Haran, M. and Jones, G.L. (2008). Markov chain Monte Carlo: Can we trust the third significant figure? Statist. Sci. 23 250–260.
  • [4] Gelman, A. (2006). Prior distributions for variance parameters in hierarchical models. Bayesian Anal. 1 515–534.
  • [5] Johnson, A.A. and Jones, G.L. (2010). Gibbs sampling for a Bayesian hierarchical general linear model. Electron. J. Stat. 4 313–333.
  • [6] Jones, G.L. and Hobert, J.P. (2001). Honest exploration of intractable probability distributions via Markov chain Monte Carlo. Statist. Sci. 16 312–334.
  • [7] Khare, K. and Hobert, J.P. (2011). A spectral analytic comparison of trace-class data augmentation algorithms and their sandwich variants. Ann. Statist. 39 2585–2606.
  • [8] Meyn, S.P. and Tweedie, R.L. (1993). Markov Chains and Stochastic Stability. Communications and Control Engineering Series. London: Springer.
  • [9] Roberts, G.O. and Rosenthal, J.S. (1998). Markov-chain Monte Carlo: Some practical implications of theoretical results. Canad. J. Statist. 26 5–31.
  • [10] Roberts, G.O. and Rosenthal, J.S. (2001). Markov chains and de-initializing processes. Scand. J. Stat. 28 489–504.
  • [11] Román, J.C. (2012). Convergence analysis of block Gibbs samplers for Bayesian general linear mixed models. Ph.D. thesis, Dept. Statistics, Univ. Florida.
  • [12] Román, J.C. and Hobert, J.P. (2012). Convergence analysis of the Gibbs sampler for Bayesian general linear mixed models with improper priors. Ann. Statist. 40 2823–2849.
  • [13] Román, J.C. and Hobert, J.P. (2015). Geometric ergodicity of Gibbs samplers for Bayesian general linear mixed models with proper priors. Linear Algebra Appl. 473 54–77.
  • [14] Román, J.C., Hobert, J.P. and Presnell, B. (2014). On reparametrization and the Gibbs sampler. Statist. Probab. Lett. 91 110–116.
  • [15] Searle, S.R., Casella, G. and McCulloch, C.E. (1992). Variance Components. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. New York: Wiley.
  • [16] Sun, D., Tsutakawa, R.K. and He, Z. (2001). Propriety of posteriors with improper priors in hierarchical linear mixed models. Statist. Sinica 11 77–95.