Limit theorems for some adaptive MCMC algorithms with subgeometric kernels

Bernoulli

Volumn 16, Issue 1, 2010, Pages 116-154

  Atchadé, Yves ^a   Fort, Gersende ^b  

^a UNIVERSITY OF MICHIGAN   (United States)

^b CNRS   (France)

Author keywords

Adaptive Markov chain Monte Carlo; Markov chain; Subgeometric ergodicity

Indexed keywords

EID: 77951137712     PISSN: 13507265     EISSN: None     Source Type: Journal    
DOI: 10.3150/09-BEJ199     Document Type: Article

References (34)

1. Andrieu, C. and Atchade, Y. F. (2007). On the efficiency of adaptive MCMC algorithms. Electron. Commun. Probab. 12 336-349. MR2350572
2. Andrieu, C. and Fort, G. (2005). Explicit control of subgeometric ergodicity. Technical report, Univ. Bristol, 05: 17. Available at http://www.tsi.enst.fr/~gfort/biblio.html.
3. Andrieu, C. and Moulines, É. (2006). On the ergodicity properties of some adaptive MCMC algorithms. Ann. Appl. Probab. 16 1462-1505. MR2260070 (Pubitemid 44662199)
4. Andrieu, C. and Robert, C. P. (2001). Controlled MCMC for optimal sampling. Technical report, Univ. Paris Dauphine, Ceremade 0125.
5. Andrieu, C. and Tadic, V. (2008). General result for the stability of controlled MCMC. Technical report, Bristol University. (Personal communication).
6. Atchade, Y. and Fort, G. (2008). Limit theorems for some adaptive MCMC algorithms with subgeometric kernels (II). Technical report. To appear.
7. Atchade, Y. F. (2006). An adaptive version for the Metropolis adjusted Langevin algorithm with a truncated drift. Methodol. Comput. Appl. Probab. 8 235-254. MR2324873
8. Atchade, Y. F. (2009). A strong law of large numbers for martingale arrays. Technical report, Univ. of Michigan. Available at http://www.stat.lsa. umich.edu/~yvesa/.
9. Atchade, Y. F. and Rosenthal, J. S. (2005). On adaptive Markov chain Monte Carlo algorithm. Bernoulli 11 815-828. MR2172842
10. Bai, Y. (2008). The simultaneous drift conditions for Adaptive Markov Chain Monte carlo algorithms. Technical report, Univ. Toronto. (Personal communication).
11. Benveniste, A., Métivier, M. and Priouret, P. (1987). Adaptive Algorithms and Stochastic Approximations. Springer.
12. Chen, H., Guo, L. and Gao, A. (1988). Convergence and robustness of the Robbins-Monro algorithm truncated at randomly varying bounds. Stochastic Process. Appl. 27 217-231. MR0931029
13. Chen, H. and Zhu, Y. (1986). Stochastic approximation procedures with randomly varying truncations. Sci. Sinica. Ser. A 29 914-926. MR0869196
14. Douc, R., Fort, G., Moulines, E. and Soulier, P. (2004). Practical drift conditions for subgeometric rates of convergence. Ann. Appl. Probab. 14 1353-1377. MR2071426
15. Douc, R., Moulines, E. and Soulier, P. (2007). Computable convergence rates for sub-geometric ergodic Markov chains. Bernoulli 13 831-848. MR2348753
16. Fort, G. and Moulines, E. (2000). V-subgeometric ergodicity for a Hastings-Metropolis algorithm. Statist. Probab. Lett. 49 401-410. MR1796485
17. Fort, G. and Moulines, E. (2003). Polynomial ergodicity of Markov transition kernels. Stochastic Process. Appl. 103 57-99. MR1947960
18. Gilks, W. R., Roberts, G. O. and Sahu, S. K. (1998). Adaptive Markov chain Monte Carlo through regeneration. J. Amer. Statist. Assoc. 93 1045-1054. MR1649199
19. Haario, H., Saksman, E. and Tamminen, J. (2001). An adaptive Metropolis algorithm. Bernoulli 7 223-242. MR 1828504
20. Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. New York: Academic Press. MR0624435
21. Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their application. Biometrika 57 97-109.
22. Holden, L. (1998). Adaptive chains. Technical report. Norwegian Computing Center.
23. Jarner, S. F. and Hansen, E. (2000). Geometric ergodicity of Metropolis algorithms. Stochastic Process. Appl. 85 341-361. MR1731030
24. Jarner, S. F. and Roberts, G. O. (2002). Polynomial convergence rates of Markov chains. Ann. Appl. Probab. 12 224-247. MR 1890063
25. Maxwell, M. and Woodroofe, M. (2000). Central limit theorems for additive functional of Markov chains. Ann. Probab. 28 713-724. MR 1782272
26. Merlevede, F., Peligrad, M. and Utev, S. (2006). Recent advances in invariances principles for stationary sequences. Probab. Surv. 3 1-36. MR2206313
27. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953). Equations of state calculations by fast computing machines. J. Chem. Phys. 21 1087-1092.
28. Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. London: Springer. MR 1287609
29. Roberts, G. and Rosenthal, J. (2004). General state space Markov chains and MCMC algorithms. Probab. Surv. 1 20-71. MR2095565
30. Roberts, G. O. and Rosenthal, J. S. (2001). Optimal scaling of various Metropolis-Hastings algorithms. Statist. Sci. 16 351-367. MR 1888450
31. Roberts, G. O. and Rosenthal, J. S. (2007). Coupling and ergodicity of adaptive MCMC. J. Appl. Probab. 44 458-475. MR2340211
32. Roberts, G. O. and Tweedie, R. L. (1996). Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika 83 95-110. MR 1399158
33. Winkler, G. (2003). Image Analysis, Random Fields and Markov Chain Monte Carlo Methods, 2nd ed. Applications of Mathematics (New York) 27. Berlin: Springer. MR 1950762
34. Yang, C. (2007). Recurrent and ergodic properties of Adaptive MCMC. Technical report, Univ. Toronto, Canada. Available at http://probability.ca/ jeff/ftpdir/chao3.pdf.