The Annals of Applied Probability

On convergence rates of Gibbs samplers for uniform distributions

Abstract

We consider a Gibbs sampler applied to the uniform distribution on a bounded region $R \subseteq \mathbf{R}^d$. We show that the convergence properties of the Gibbs sampler depend greatly on the smoothness of the boundary of R. Indeed, for sufficiently smooth boundaries the sampler is uniformly ergodic, while for jagged boundaries the sampler could fail to even be geometrically ergodic.

Article information

Source
Ann. Appl. Probab., Volume 8, Number 4 (1998), 1291-1302.

Dates
First available in Project Euclid: 9 August 2002

https://projecteuclid.org/euclid.aoap/1028903381

Digital Object Identifier
doi:10.1214/aoap/1028903381

Mathematical Reviews number (MathSciNet)
MR1661176

Zentralblatt MATH identifier
0935.60054

Subjects
Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 62M05: Markov processes: estimation

Citation

Roberts, Gareth O.; Rosenthal, Jeffrey S. On convergence rates of Gibbs samplers for uniform distributions. Ann. Appl. Probab. 8 (1998), no. 4, 1291--1302. doi:10.1214/aoap/1028903381. https://projecteuclid.org/euclid.aoap/1028903381

References

• Ba nuelos, R. and Carroll, T. (1994). Brownian motion and the fundamental frequency of a drum. Duke Math. J. 75 575-602.
• B´elisle, C. (1997). Slow convergence of the Gibbs sampler. Technical Report 172, Dept. Statistics, Univ. British Columbia. Available at http://www.stats.bris.ac.uk/MCMC/.
• Damien, P., Wakefield, J. C. and Walker, S. (1997). Gibbs sampling for Bayesian nonconjugate and hierarchical models using auxiliary variables. Imperial College, London. Preprint.
• Gelfand, A. E. and Smith, A. F. M. (1990). Sampling based approaches to calculating marginal densities. J. Amer. Statist. Assoc. 85 398-409.
• Higdon, D. M. (1997). Auxiliary variable methods for Markov chain Monte Carlo with applications. Preprint, Statistics and Decision Sciences, Duke Univ. Available at http://www.stats.bris.ac.uk/MCMC/.
• Lawler, G. F. and Sokal, A. D. (1988). Bounds on the L2 spectrum for Markov chains and Markov processes: a generalization of Cheeger's inequality. Trans. Amer. Math. Soc. 309 557-580.
• Mey n, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, New York.
• Mira, A. and Tierney, L. (1997). On the use of auxiliary variables in Markov chain Monte Carlo sampling. School of Statistics, Univ. Minnesota. Unpublished manuscript.
• Nummelin, E. (1984). General Irreducible Markov Chains and Non-negative Operators. Cambridge Univ. Press. Roberts, G. O. and Rosenthal, J. S. (1997a). Geometric ergodicity and hy brid Markov chains. Elec. Comm. Prob. 2 13-25. Roberts, G. O. and Rosenthal, J. S. (1997b). Convergence of slice sampler Markov chains. Unpublished manuscript.
• Roberts, G. O. and Tweedie, R. L. (1996). Geometric convergence and central limit theorems for multidimensional-Hastings Metropolis algorithms. Biometrika 83 96-110.
• Smith, A. F. M. and Roberts, G. O. (1993). Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion). J. Roy. Statist. Soc. Ser. B 55 3-24.
• Tierney, L. (1994). Markov chains for exploring posterior distributions (with discussion). Ann. Statist. 22 1701-1762.