Conditions for rapid mixing of parallel and simulated tempering on multimodal distributions

Annals of Applied Probability

Volumn 19, Issue 2, 2009, Pages 617-640

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

^a Duke University   (United States)

^b DUKE UNIVERSITY   (United States)

Author keywords

Markov chain Monte Carlo; Metropolis algorithm.; rapidly mixing Markov chains; Spectral gap; Tempering

Indexed keywords

EID: 66149093135     PISSN: 10505164     EISSN: 10505164     Source Type: Journal    
DOI: 10.1214/08-AAP555     Document Type: Article

References (29)

1. Bhatnagar, N. and Randall, D. (2004). Torpid mixing of simulated tempering on the Potts model. In Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms 478-487 (electronic). ACM, New York. MR2291087
2. Binder, K. and Heermann, D. W. (2002). Monte Carlo Simulation in Statistical Physics: An Introduction, 4th ed. Springer Series in Solid-State Sciences 80. Springer, Berlin. MR1949328
3. Caracciolo, S., Pelissetto, A. and Sokal, A. D. (1992). Two remarks on simulated tempering. Unpublished manuscript.
4. Diaconis, P. and Saloff-Coste, L. (1993). Comparison theorems for reversible Markov chains. Ann. Appl. Probab. 3 696-730. MR1233621
5. Diaconis, P. and Saloff-Coste, L. (1996). Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 695-750. MR1410112
6. Diaconis, P. and Stroock, D. (1991). Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Probab. 1 36-61. MR1097463
7. Geyer, C. J. (1991). Markov chain Monte Carlo maximum likelihood. In Computing Science and Statistics, Vol. 23: Proceedings of the 23rd Symposium on the Interface (E. Keramidas, ed.) 156-163. Interface Foundation of North America, Fairfax Station, VA.
8. 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.
9. Gilks, W. R., Richardson, S. and Spiegelhalter, D. J., eds. (1996). Markov Chain Monte Carlo in Practice: Interdisciplinary Statistics. Chapman & Hall, London. MR1397966
10. Jerrum, M. and Sinclair, A. (1996). The Markov Chain Monte Carlo Method: An Approach to Approximate Counting and Integration. PWS Publishing, Boston.
11. Kannan, R. and Li, G. (1996). Sampling according to the multivariate normal density. In 37th Annual Symposium on Foundations of Computer Science (Burlington, VT, 1996) 204-212. IEEE Comput. Soc., Los Alamitos, CA. MR1450618
12. 2 spectrum for Markov chains and Markov processes: A generalization of Cheeger's inequality. Trans. Amer. Math. Soc. 309 557-580. MR930082
13. Madras, N. and Piccioni, M. (1999). Importance sampling for families of distributions. Ann. Appl. Probab. 9 1202-1225. MR1728560
14. Madras, N. and Randall, D. (2002). Markov chain decomposition for convergence rate analysis. Ann. Appl. Probab. 12 581-606. MR1910641
15. Madras, N. and Slade, G. (1993). The Self-Avoiding Walk. BirkhÄuser, Boston. MR1197356
16. Madras, N. and Zheng, Z. (2003). On the swapping algorithm. Random Structures Algorithms 22 66-97. MR1943860
17. Marinari, E. and Parisi, G. (1992). Simulated tempering: A new Monte Carlo scheme. Europhys. Lett. EPL 19 451-458.
18. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953). Equation of state calculations by fast computing machines. J. Chem. Phys. 21 1087-1092.
19. 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.
20. Robert, C. P. and Casella, G. (1999). Monte Carlo Statistical Methods. Springer, New York. MR1707311
21. Roberts, G. O. and Rosenthal, J. S. (1997). Geometric ergodicity and hybrid Markov chains. Electron. Comm. Probab. 2 13-25 (electronic). MR1448322
22. Roberts, G. O. and Rosenthal, J. S. (2004). General state space Markov chains and MCMC algorithms. Probab. Surv. 1 20-71 (electronic). MR2095565
23. 1 convergence are equivalent for reversible Markov chains. J. Appl. Probab. 38A 37-41. MR1915532
24. Rosenthal, J. S. (1995). Rates of convergence for Gibbs sampling for variance components models. Ann. Statist. 23 740-761. MR1345197
25. Sinclair, A. (1992). Improved bounds for mixing rates of Markov chains and multicommodity flow. Combin. Probab. Comput. 1 351-370. MR1211324
26. Tierney, L. (1994). Markov chains for exploring posterior distributions. Ann. Statist. 22 1701-1762. With discussion and a rejoinder by the author. MR1329166
27. Woodard, D. B. (2007). Conditions for rapid and torpid mixing of parallel and simulated tempering on multimodal distributions. Ph.D. dissertation, Duke Univ.
28. Woodard, D. B., Schmidler, S. C. and Huber, M. (2008). Sufficient conditions for torpid mixing of parallel and simulated tempering. Stochastic Process. Appl. Submitted.
29. Zheng, Z. (2003). On swapping and simulated tempering algorithms. Stochastic Process. Appl. 104 131-154. MR1956476