A simulation approach to convergence rates for Markov chain Monte Carlo algorithms

Statistics and Computing

Volumn 8, Issue 2, 1998, Pages 115-124

  Cowles, Mary Kathryn ^a   Rosenthal, Jeffrey S ^b  

^a HARVARD SCHOOL OF PUBLIC HEALTH   (United States)

^b UNIVERSITY OF TORONTO   (Canada)

Author keywords

Drift condition; Gibbs sampler; Markov chain Monte Carlo; Metropolis Hastings algorithm; Minorization condition; Ordinal probit; Variance components

Indexed keywords

EID: 0042376950     PISSN: 09603174     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1008982016666     Document Type: Article

References (32)

1. Albert, J. H. and Chib, S. (1993) Bayesian analysis of binary and polychotomous response data. Journal of the American Statistical Association, 88, 669-79.
2. Baxendale, P. H. (1994) Uniform Estimates for Geometric Ergodicity of Recurrent Markov Chains, Technical Report, Department of Mathematics, University of Southern California.
3. Box, G. E. P. and Tiao, G. C. (1973) Bayesian Inference in Statistical Analysis, New York: Wiley.
4. Carlin, B. P. and Polson, N. G. (1992) Monte Carlo Bayesian methods for discrete regression models and categorical time series. In J. M. Bernardo, J. Berger, A. P. Dawid and C. J. Geyer et al. (eds) Bayesian Statistics, Vol. 4 pp. 577-86, Oxford University Press.
5. Cowles, M. K. (1996) Accelerating Monte Carlo Markov chain convergence for cumulative-link generalized linear models. Statistics and Computing, 6, 101-11.
6. Cowles, M. K. and Carlin, B. P. (1996) Markov Chain Monte Carlo convergence diagnostics: a comparative review. Journal of the American Statistical Association, in press.
7. Davies, O. L. (1967) Statistical Methods in Research and Production, 3rd edn, London: Oliver and Boyd.
8. Frieze, A., Kannan, R. and Polson, N. G. (1993) Sampling from log-concave distributions. Annals of Applied Probability, 4, 812-37.
9. Frigessi, A., Hwang, C. -R., Sheu, S. J. and Di Stefano, P. (1993) Convergence rates of the Gibbs sampler, the Metropolis algorithm, and other single-site updating dynamics. Journal of the Royal Statistical Society, Series B, 55, 205-20.
10. Garren, S. T. and Smith, R. L. (1995) Estimating the Second Largest Eigenvalue of a Markov Transition Matrix, Research Report No. 95-18, Statistical Laboratory, University of Cambridge.
11. Gelfand, A. E., Hills, S. E., Racine-Poon, A. and Smith, A.F.M. (1990) Illustration of Bayesian inference in normal data models using Gibbs sampling. Journal of the American Statistical Association, 85, 972-85.
12. Gelman, A. E. and Rubin, D. B. (1992) Inference from iterative simulation using multiple sequences. Statistical Science, 7(4) 457-72.
13. Gelfand, A. E. and Smith, A. F. M. (1990) Sampling based approaches to calculating marginal densities. Journal of the American Statistical Association, 85, 398-409.
14. Geman, S. and Geman, D. (1984) Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721-41.
15. Geweke, J. (1992) Evaluating the accuracy of sampling based approaches to the calculation of posterior moments. In J. M. Bernardo, J. Berger, A. P. Dawid and A. F. M. Smith (eds) Bayesian Statistics, vol. 4, pp. 169-93, Oxford University Press.
16. Geyer, C. J. (1992) Practical Markov Chain Monte Carlo. Statistical Science, 7(4), 473-83.
17. Geyer, C. J. (1994) Geometric ergodicity of the block Gibbs sampler for a simple hierarchical model. Unpublished manuscript.
18. Hastings, W. K. (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57, 97-109.
19. Ingrassia, S. (1994) On the rate of convergence of the Metropolis algorithm and Gibbs sampler by geometric bounds. Annals of Applied Probability, 4, 347-89.
20. Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A. and Teller, E. (1953) Equations of state calculations by fast computing machines. Journal of Chemistry and Physics 21, 1087-91.
21. Meyn, S. P. and Tweedie, R. L. (1993) Markov Chains and Stochastic Stability. London: Springer-Verlag.
22. Meyn, S. P. and Tweedie, R. L. (1994) Computable bounds for convergence rates of Markov chains. Annals of Applied Probability, 4, 981-1011.
23. Morris, C. (1983) Parametric empirical Bayes confidence intervals. Scientific Inference, Data Analysis and Robustness, 25-50.
24. Raftery, A. E. and Lewis, S. (1992) How many iterations in the Gibbs sampler? In J. M. Bernardo, J. Berger, A. P. Dawid and A. F. M. Smith (eds), Bayesian Statistics, Vol. 4, pp. 763-73, Oxford University Press.
25. Roberts, G. O. (1992) Convergence diagnostics of the Gibbs sampler. In J. M. Bernardo, J. Berger, A. P. Dawid and A. F. M. Smith (eds), Bayesian Statistics, Vol. 4 pp. 777-84, Oxford University Press.
26. Roberts, G. O. and Tweedie, R. L. (1994) Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika, in press.
27. Rosenthal, J. S. (1995a) Rates of convergence for Gibbs sampling for variance components models. Annals of Statistics, 23, 740-61.
28. Rosenthal, J. S. (1995b) Minorization conditions and convergence rates for Markov Chain Monte Carlo. Journal of the American Statistical Association, 90, 558-66, correction, p. 1136.
29. Rosenthal, J. S. (1996) Analysis of the Gibbs sampler for a model related to James-Stein Estimators. Statistics and Computing, 6, 269-75.
30. SAS Institute (1990) The Logistic Procedure. SAS/STAT User's Guide, Vol. 2, Version 6, 4th edn, pp. 1071-126. Cary, NC: SAS Institute.
31. Smith, A. F. M. and Roberts, G. O (1993) Bayesian computation via the Gibbs sampler and related Markov Chain Monte Carlo methods (with discussion) Journal of the Royal Statistical Society, Series B, 55, 3-24.
32. Spivak, M. (1980) Calculus, 2nd edn. Wilmington, DE: Publish or Perish Inc.