Sufficient conditions for torpid mixing of parallel and simulated tempering

Electronic Journal of Probability

Volumn 14, Issue , 2009, Pages 780-804

  Woodard, Dawn B ^a   Schmidler, Scott C ^b   Huber, Mark ^c  

^a Cornell University ^*  (United States)

^b Duke University   (United States)

^c DUKE UNIVERSITY   (United States)

Author keywords

Markov chain; Metropolis algorithm; Rapid mixing; Spectral gap

Indexed keywords

EID: 65749095586     PISSN: None     EISSN: 10836489     Source Type: Journal    
DOI: 10.1214/EJP.v14-638     Document Type: Article

References (24)

1. Banerjee, S., Carlin, B. P., and Gelfand, A. E. (2004). Hierarchical Modeling and Analysis for Spatial Data. Chapman and Hall, Boca Raton, FL.
2. Bhatnagar, N. and Randall, D. (2004). Torpid mixing of simulated tempering on the Potts model. In Proceedings of the 15th ACM/SIAM Symposium on Discrete Algorithms. 478-487.
3. 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, 721-741.
4. GEYER, C. J. (1991). Markov chain Monte Carlo maximum likelihood. In Computing Science and Statistics, Volume 23: Proceedings of the 23rd Symposium on the Interface, E. Keramidas, Ed. Interface Foundation of North America, Fairfax Station, VA, 156-163.
5. Geyer, C. J. and Thompson, E. A. (1995). Annealing Markov chain Monte Carlo with applications to ancestral inference. J. Amer. Statist. Assoc. 90, 909-920.
6. Gore, V K. and Jerrum, M. R. (1999). The Swendsen-Wang process does not always mix rapidly. J. of Statist. Physics 97, 67-85.
7. Green, P J. and Richardson, S. (2002). Hidden Markov models and disease mapping. J. Amer. Statist. Assoc. 97, 1055-1070.
8. Kannan, R. and Li, G. (1996). Sampling according to the multivariate normal density. In Proceedings of the IEEE Symposium on Foundations of Computer Science. 204-213.
9. 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. Transactions of the American Mathematical Society 309, 557-580.
10. Madras, N. and Randall, D. (2002). Markov chain decomposition for convergence rate analysis. Annals of Applied Probability 12, 581-606.
11. Madras, N. and Slade, G. (1993). The Self-Avoiding Walk. Birkhauser, Boston.
12. Madras, N. and Zheng, Z. (2003). On the swapping algorithm. Random Structures and Algorithms 1, 66-97.
13. Marinari, E. and Parisi, G. (1992). Simulated tempering: a new Monte Carlo scheme. Euro-physics Letters 19, 451-458.
14. MATTHEWS, P (1993). A slowly mixing Markov chain with implications for Gibbs sampling. Statistics and Probability Letters 17, 231-236.
15. Predescu, C., Predescu, M., and Ciobanu, C. V. (2004). The incomplete beta function law for parallel tempering sampling of classical canonical systems. J. Chem. Phys. 120, 4119-4128.
16. Roberts, G. O. and Rosenthal, J. S. (2004). General state space Markov chains and MCMC algorithms. Probability Surveys 1, 20-71.
17. Roberts, G. O. and Tweedie, R. L. (2001). Geometric L2 and L1 convergence are equivalent for reversible Markov chains. Journal of Applied Probability 38A, 37-41.
18. Schmidler, S. C. and Woodard, D. B. (2008). Computational complexity and Bayesian analysis. in preparation.
19. TIERNEY, L. (1994). Markov chains for exploring posterior distributions. Annals of Statistics 22, 1701-1728.
20. WOODARD, D. B. (2007a). Conditions for rapid and torpid mixing of parallel and simulated tempering on multimodal distributions. Ph.D. thesis, Duke University.
21. WOODARD, D. B. (2007b). Detecting poor convergence of posterior samplers due to multimodality. Discussion Paper 2008-05, Duke University, Dept. of Statistical Science. http://ftp.isds.duke.edu/pub/WorkingPapers/08-05.html.
22. Woodard, D. B., Schmidler, S. C., and Huber, M. (2009). Conditions for rapid mixing of parallel and simulated tempering on multimodal distributions. In press, Annals of Applied Probability.
23. YUEN, W. K. (2001). Application of geometric bounds to convergence rates of Markov chains and Markov processes on Rn. Ph.D. thesis, University of Toronto.
24. ZHENG, Z. (2003). On swapping and simulated tempering algorithms. Stochastic Processes and their Applications 104, 131-154.