×

zbMATH — the first resource for mathematics

Limit theorems for some adaptive MCMC algorithms with subgeometric kernels. (English) Zbl 1215.60046
The authors investigate the asymptotic behaviour of adaptive Markov chain Monte Carlo algorithms with Markov kernels that are subgeometrically ergodic. Thus, the class of adaptive MCMC algorithms that are accessible for rigorous analysis is enlarged. The contribution of the paper is twofold. On the one hand, it extends previous results to a larger class of kernels, and, on the other hand, this is achieved employing a more sophisticated technical machinery which allows to weaken certain assumptions.
Firstly, the authors present conditions implying the ergodicity of the algorithms (convergence of the marginals to the invariant distribution) irrespective of the initial distribution; secondly, introducing stronger assumptions, a strong law of large numbers is proved. The authors believe that their conditions implying ergodicity are close to optimal, assuming only the “diminishing adaptation condition” (shown to be necessary) and some “fairly weak additional assumptions”.
On a technical level, there are two contributions originating from the proofs of the two asymptotic results. The coupling technique used to study ergodicity is more careful and the strong law of large numbers is established using a resolvent kernel approach together with martingale theory. A discussion comparing their conditions and results to previous work in relation to the technical machinery is presented.
Finally, the authors illustrate the application of their limit theorems first on a toy example and secondly on a adaptive random walk Metropolis algorithm for distributions with subexponential tails.

MSC:
60J22 Computational methods in Markov chains
65C05 Monte Carlo methods
65C40 Numerical analysis or methods applied to Markov chains
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Andrieu, C. and Atchade, Y.F. (2007). On the efficiency of adaptive MCMC algorithms. Electron. Commun. Probab. 12 336-349. · Zbl 1129.65006 · eudml:128369
[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. · Zbl 1114.65001 · doi:10.1214/105051606000000286
[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. · Zbl 1215.60046 · doi:10.3150/09-BEJ199
[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. · Zbl 1104.65004 · doi:10.1007/s11009-006-8550-0
[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. · Zbl 1085.62097 · doi:10.3150/bj/1130077595
[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. · Zbl 0752.93073
[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. · Zbl 0632.62082 · doi:10.1016/0304-4149(87)90039-1
[13] Chen, H. and Zhu, Y. (1986). Stochastic approximation procedures with randomly varying truncations. Sci. Sinica. Ser. A 29 914-926. · Zbl 0613.62107
[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. · Zbl 1082.60062 · doi:10.1214/105051604000000323
[15] Douc, R., Moulines, E. and Soulier, P. (2007). Computable convergence rates for sub-geometric ergodic Markov chains. Bernoulli 13 831-848. · Zbl 1131.60065 · doi:10.3150/07-BEJ5162 · euclid:bj/1186503489
[16] Fort, G. and Moulines, E. (2000). V -subgeometric ergodicity for a Hastings-Metropolis algorithm. Statist. Probab. Lett. 49 401-410. · Zbl 0981.60032 · doi:10.1016/S0167-7152(00)00074-2
[17] Fort, G. and Moulines, E. (2003). Polynomial ergodicity of Markov transition kernels. Stochastic Process. Appl. 103 57-99. · Zbl 1075.60547 · doi:10.1016/S0304-4149(02)00182-5
[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. · Zbl 1064.65503 · doi:10.2307/2669848
[19] Haario, H., Saksman, E. and Tamminen, J. (2001). An adaptive Metropolis algorithm. Bernoulli 7 223-242. · Zbl 0989.65004 · doi:10.2307/3318737
[20] Hall, P. and Heyde, C.C. (1980). Martingale Limit Theory and Its Application . New York: Academic Press. · Zbl 0462.60045
[21] Hastings, W.K. (1970). Monte Carlo sampling methods using Markov chains and their application. Biometrika 57 97-109. · Zbl 0219.65008 · doi:10.1093/biomet/57.1.97
[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. · Zbl 0997.60070 · doi:10.1016/S0304-4149(99)00082-4
[24] Jarner, S.F. and Roberts, G.O. (2002). Polynomial convergence rates of Markov chains. Ann. Appl. Probab. 12 224-247. · Zbl 1012.60062 · doi:10.1214/aoap/1015961162
[25] Maxwell, M. and Woodroofe, M. (2000). Central limit theorems for additive functional of Markov chains. Ann. Probab. 28 713-724. · Zbl 1044.60014 · doi:10.1214/aop/1019160258
[26] Merlevede, F., Peligrad, M. and Utev, S. (2006). Recent advances in invariances principles for stationary sequences. Probab. Surv. 3 1-36. · Zbl 1189.60078 · doi:10.1214/154957806100000202 · eudml:231536
[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. · Zbl 0925.60001
[29] Roberts, G. and Rosenthal, J. (2004). General state space Markov chains and MCMC algorithms. Probab. Surv. 1 20-71. · Zbl 1189.60131 · doi:10.1214/154957804100000024 · eudml:224487
[30] Roberts, G.O. and Rosenthal, J.S. (2001). Optimal scaling of various Metropolis-Hastings algorithms. Statist. Sci. 16 351-367. · Zbl 1127.65305 · doi:10.1214/ss/1015346320
[31] Roberts, G.O. and Rosenthal, J.S. (2007). Coupling and ergodicity of adaptive MCMC. J. Appl. Probab. 44 458-475. · Zbl 1137.62015 · doi:10.1239/jap/1183667414
[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. · Zbl 0888.60064 · doi:10.1093/biomet/83.1.95 · www3.oup.co.uk
[33] Winkler, G. (2003). Image Analysis, Random Fields and Markov Chain Monte Carlo Methods , 2nd ed. Applications of Mathematics (New York) 27 . Berlin: Springer. · Zbl 1008.68147
[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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.