The Annals of Applied Probability

Coordinate selection rules for Gibbs sampling

George S. Fishman

Full-text: Open access


This paper studies several different plans for selecting coordinates for updating via Gibbs sampling. It exploits the inherent features of the Gibbs sampling formulation, most notably its neighborhood structure, to characterize and compare the plans with regard to convergence to equilibrium and variance of the sample mean. Some of the plans rely completely or almost completely on random coordinate selection. Others use completely or almost completely deterministic coordinate selection rules. We show that neighborhood structure induces idempotency for the individual coordinate transition matrices and commutativity among subsets of these matrices. These properties lead to bounds on eigenvalues for the Gibbs sampling transition matrices corresponding to several of the plans. For a frequently encountered neighborhood structure, we give necessary and sufficient conditions for a commonly employed deterministic coordinate selection plan to induce faster convergence to equilibrium than the random coordinate selection plans. When these conditions hold, we also show that this deterministic selection rule achieves the same worst-case bound on the variance of the sample mean as that arising from the random selection rules when the number of coordinates grows without bound. This last result encourages the belief that faster convergence for the deterministic selection rule may also imply a variance of the sample mean no larger than that arising for random selection rules.

Article information

Ann. Appl. Probab. Volume 6, Number 2 (1996), 444-465.

First available in Project Euclid: 18 October 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J05: Discrete-time Markov processes on general state spaces 62E25

Coordinate selection Gibbs sampling Markov random field Monte Carlo


Fishman, George S. Coordinate selection rules for Gibbs sampling. Ann. Appl. Probab. 6 (1996), no. 2, 444--465. doi:10.1214/aoap/1034968139.

Export citation


  • Amit, Y. (1991). On rates of convergence of stochastic relaxation for Gaussian and non-Gaussian distributions. J. Multivariate Anal. 38 82-89.
  • Amit, Y. and Grenander, U. (1989). Comparing sweeping strategies for stochastic relaxation. J. Multivariate Anal. 37 197-222.
  • Barone, P. and Frigessi, A. (1990). Improving stochastic relaxation for Gaussian random fields. Problems in the Informational Sciences 4 369-389.
  • Fan, K. (1949). On a theorem of Wey l concerning eigenvalues of linear transformations. I. Proc. Nat. Acad. Sci. U.S.A. 35 652-655.
  • Fan, K. and Hoffman, A. (1955). Some metric inequalities in the space of matrices. Proc. Amer. Math. Soc. 6 111-116.
  • Fishman, G. S. (1994). Coordinate selection rules for Gibbs sampling. Report UNC/OR TR-92/15, Operations Research Dept., Univ. North Carolina at Chapel Hill.
  • Frigessi, A., Hwang, C.-R., Sheu, S.-J. and di Stefano, P. (1993). Convergence rates of the Gibbs sampler, Metropolis algorithm and other single-site updating dy namics. J. Roy. Statist. Soc. Ser. B 55 205-219.
  • Gelfand, A. E. and Smith, A. F. M. (1990). Sample-based approaches to calculating marginal densities. J. Amer. Statist. Assoc. 85 398-409.
  • Horn, R. A. and Johnson, C. A. (1985). Matrix Analy sis. Cambridge Univ. Press.
  • Johnson, V. (1989). On statistical image reconstruction. Ph.D. dissertation, Dept. Statistics, Univ. Chicago.
  • Kemeny, J. G. and Snell, J. L. (1960). Finite Markov Chains. Van Nostrand, New York.
  • Liu, J., Wong, W. H. and Kong, A. (1994). Covariance structure of the Gibbs sampler with applications to the comparison of estimators and augmentation schemes. Biometrika 81 27-40.
  • Liu, J., Wong, W. H. and Kong, A. (1995). Covariance structure and convergence rate of the Gibbs sampler with various scans. J. Roy. Statist. Soc. Ser. B 57 157-169.
  • MacEachern, S. and Berliner, L. M. (1994). Subsampling the Gibbs sampler. Amer. Statist. 48 188-190.
  • Marcus, M. and Minc, H. (1964). A Survey of Matrix Theory and Matrix Inequalities. Ally n and Bacon, Boston.
  • Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.
  • Roberts, G. O. and Polson, N. G. (1991). A note on the geometric convergence of the Gibbs sampler. Dept. Mathematics, Univ. Nottingham.
  • Roberts, G. O. and Smith, A. F. M. (1992). Simple conditions for the convergence of the Gibbs sampler and Metropolis-Hastings algorithms. Research Report 92-30, Statistical Laboratory, Univ. Cambridge.
  • Schervish, M. J. and Carlin, B. P. (1993). On the convergence of successive substitution sampling. Dept. Statistics, Carnegie Mellon Univ.