Sufficient burn-in for Gibbs samplers for a hierarchical random effects model

Annals of Statistics

Volumn 32, Issue 2, 2004, Pages 784-817

  Jones, Galin L ^a   Hubert, James P ^b,c  

^a UNIVERSITY OF MINNESOTA   (United States)

^b UNIVERSITY OF FLORIDA   (United States)

^c Ford Hall 224 ^*  (United States)

Author keywords

Block Gibbs sampler; Burn in; Convergence rate; Drift condition; Geometric ergodicity; Markov chain; Minorization condition; Monte Carlo; Total variation distance

Indexed keywords

EID: 24344493048     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/009053604000000184     Document Type: Article

References (24)

1. COWLES, M. K. and CARLIN, B. P. (1996). Markov chain Monte Carlo convergence diagnostics: A comparative review. J. Amer. Statist. Assoc. 91 883-904.
2. COWLES, M. K., ROBERTS, G. O. and ROSENTHAL, J. S. (1999). Possible biases induced by MCMC convergence diagnostics. J. Statist. Comput. Simulation 64 87-104.
3. COWLES, M. K. and ROSENTHAL, J. S. (1998). A simulation approach to convergence rates for Markov chain Monte Carlo algorithms. Statist. Comput. 8 115-124.
4. DOUE, R., MOULINES, E. and ROSENTHAL, J. S. (2002). Quantitative bounds for geometric convergence rates of Markov chains. Technical report, Dept. Statistics, Univ. Toronto.
5. 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. J. Amer. Statist. Assoc. 85 972-985.
6. GELFAND, A. E. and SMITH, A. F. M. (1990). Sampling-based approaches to calculating marginal densities. J. Amer. Statist. Assoc. 85 398-409.
7. HUBERT, J. P. (2001). Discussion of "The art of data augmentation," by D. A. van Dyk and X.-L. Meng. J. Comput. Graph. Statist. 10 59-68.
8. HUBERT, J. P. and GEYER, C. J. (1998). Geometric ergodicity of Gibbs and block Gibbs samplers for a hierarchical random effects model. J. Multivariate Anal. 67 414-430.
9. JONES, G. L. (2001). Convergence rates and Monte Carlo standard errors for Markov chain Monte Carlo algorithms. Ph.D. dissertation, Univ. Florida.
10. JONES, G. L. and HUBERT, J. P. (2001). Honest exploration of intractable probability distributions via Markov chain Monte Carlo. Statist. Sci. 16 312-334.
11. MARCHEV, D. and HUBERT, J. P. (2004). Geometric ergodicity of van Dyk and Meng's algorithm for the multivariate Student's t model. J. Amer. Statist. Assoc. 99 228-238.
12. MENG, X.-L. and VAN DYK, D. A. (1999). Seeking efficient data augmentation schemes via conditional and marginal augmentation. Biometrika 86 301-320.
13. MEYN, S. P. and TWEEDIE, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.
14. MEYN, S. P. and TWEEDIE, R. L. (1994). Computable bounds for geometric convergence rates of Markov chains. Ann. Appl. Probab. 4 981-1011.
15. NATARAJAN, R. and MCCULLOCH, C. E. (1998). Gibbs sampling with diffuse proper priors: A valid approach to data-driven inference? J. Comput. Graph. Statist. 7 267-277.
16. NUMMELIN, E. (1984). General Irreducible Markov Chains and Non-negative Operators. Cambridge Univ. Press.
17. ROBERTS, G. O. and TWEEDIE, R. L. (1999). Bounds on regeneration times and convergence rates for Markov chains. Stochastic Process. Appl 80 211-229.
18. ROBERTS, G. O. and TWEEDIE, R. L. (2001). Corrigendum to "Bounds on regeneration times and convergence rates for Markov chains." Stochastic Process. Appl. 91 337-338.
19. ROSENTHAL, J. S. (1995a). Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 90 558-566.
20. ROSENTHAL, J. S. (1995b). Rates of convergence for Gibbs sampling for variance component models. Ann. Statist. 23 740-761.
21. ROSENTHAL, J. S. (1996). Analysis of the Gibbs sampler for a model related to James-Stein estimators. Statist. Comput. 6 269-275.
22. TANNER, M. A. and WONG, W. H. (1987). The calculation of posterior distributions by data augmentation (with discussion). J. Amer. Statist. Assoc. 82 528-550.
23. TIERNEY, L. (1994). Markov chains for exploring posterior distributions (with discussion). Ann. Statist. 22 1701-1762.
24. VAN DYK, D. A. and MENG, X.-L. (2001). The art of data augmentation (with discussion). J. Comput. Graph. Statist. 10 1-111.