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.
References
Ba nuelos, R. and Carroll, T. (1994). Brownian motion and the fundamental frequency of a drum. Duke Math. J. 75 575-602. MR96m:31003 10.1215/S0012-7094-94-07517-0 euclid.dmj/1077287810
Ba nuelos, R. and Carroll, T. (1994). Brownian motion and the fundamental frequency of a drum. Duke Math. J. 75 575-602. MR96m:31003 10.1215/S0012-7094-94-07517-0 euclid.dmj/1077287810
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/. http://www.stats.bris.ac.uk/MCMC/.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/. 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. 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. 0702.62020 MR1141740 10.2307/2289776Gelfand, A. E. and Smith, A. F. M. (1990). Sampling based approaches to calculating marginal densities. J. Amer. Statist. Assoc. 85 398-409. 0702.62020 MR1141740 10.2307/2289776
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/. http://www.stats.bris.ac.uk/MCMC/.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/. 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. MR89h:60105 0716.60073 10.2307/2000925Lawler, 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. MR89h:60105 0716.60073 10.2307/2000925
Mey n, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, New York. MR95j:60103Mey n, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, New York. MR95j:60103
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. 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. MR87a:60074Nummelin, 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. MR87a:60074
Roberts, G. O. and Tweedie, R. L. (1996). Geometric convergence and central limit theorems for multidimensional-Hastings Metropolis algorithms. Biometrika 83 96-110. 0888.60064 MR1399158 10.1093/biomet/83.1.95 0006-3444%28199603%2983%3A1%3C95%3AGCACLT%3E2.0.CO%3B2-HRoberts, G. O. and Tweedie, R. L. (1996). Geometric convergence and central limit theorems for multidimensional-Hastings Metropolis algorithms. Biometrika 83 96-110. 0888.60064 MR1399158 10.1093/biomet/83.1.95 0006-3444%28199603%2983%3A1%3C95%3AGCACLT%3E2.0.CO%3B2-H
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. MR1210421 0035-9246%281993%2955%3A1%3C3%3ABCVTGS%3E2.0.CO%3B2-%23Smith, 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. MR1210421 0035-9246%281993%2955%3A1%3C3%3ABCVTGS%3E2.0.CO%3B2-%23
Tierney, L. (1994). Markov chains for exploring posterior distributions (with discussion). Ann. Statist. 22 1701-1762. MR96m:60150 0829.62080 10.1214/aos/1176325750 euclid.aos/1176325750
Tierney, L. (1994). Markov chains for exploring posterior distributions (with discussion). Ann. Statist. 22 1701-1762. MR96m:60150 0829.62080 10.1214/aos/1176325750 euclid.aos/1176325750