Convergence of adaptive and interacting markov chain monte carlo algorithms

Annals of Statistics

Volumn 39, Issue 6, 2011, Pages 3262-3289

  Fort, G ^a   Moulines, E ^a   Priouret, P ^b  

^a INSTITUT TELECOM   (France)

^b SORBONNE UNIVERSITÉ   (France)

Author keywords

Adaptive Metropolis; Adaptive Monte Carlo; Equi energy sampler; Ergodic theorems; Interacting tempering; Law of large numbers; Markov chain Monte Carlo; Markov chains; Parallel tempering

Indexed keywords

EID: 84864051612     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/11-AOS938     Document Type: Article

References (40)

1. ANDRIEU, C. and MOULINES, É. (2006). On the ergodicity properties of some adaptive MCMC algorithms. Ann. Appl. Probab. 16 1462-1505. MR2260070
2. ANDRIEU, C. and THOMS, J. (2008). A tutorial on adaptive MCMC. Stat. Comput. 18 343-373. MR2461882
3. ANDRIEU, C., JASRA, A., DOUCET, A. and DEL MORAL, P. (2007). Non-linear Markov chain Monte Carlo. In Conference Oxford sur les Méthodes de Monte Carlo Séquentielles. ESAIM Proceedings 19 79-84. EDP Sci., Les Ulis. MR2405652
4. ANDRIEU, C., JASRA, A., DOUCET, A. and DEL MORAL, P. (2008). A note on convergence of the equi-energy sampler. Stoch. Anal. Appl. 26 298-312. MR2399737
5. ANDRIEU, C., JASRA, A., DOUCET, A. and DEL MORAL, P. (2011). On non-linear Markov chain Monte Carlo. Bernoulli 17 987-1014.
6. ATCHADÉ, Y. F. (2010). A cautionary tale on the efficiency of some adaptive Monte Carlo schemes. Ann. Appl. Probab. 20 841-868. MR2680550
7. ATCHADÉ, Y. and FORT, G. (2010). Limit theorems for some adaptive MCMC algorithms with subgeometric kernels. Bernoulli 16 116-154. MR2648752
8. ATCHADÉ, Y. F. and LIU, J. S. (2006). Discussion of "Equi-energy sampler," by Kou, Zhou and Wong. Ann. Statist. 34 1620-1628.
9. ATCHADE, Y., ROBERTS, G. O. and ROSENTHAL, J. S. (2011). Towards optimal scaling of Metropolis-coupled Markov chain Monte Carlo. Stat. Comput. 21 555-568.
10. ATCHADÉ, Y. F. andROSENTHAL, J. S. (2005). On adaptiveMarkov chain Monte Carlo algorithms. Bernoulli 11 815-828. MR2172842
11. ATCHADÉ, Y., FORT, G., MOULINES, E. and PRIOURET, P. (2011). Adaptive Markov chain Monte Carlo: Theory and methods. In Bayesian Time Series Models (D. Barber, A. T. Cemgil and S. Chiappia, eds.) 33-53. Cambridge Univ. Press, Cambridge.
12. BAI, Y., ROBERTS, G. O. and ROSENTHAL, J. S. (2011). On the containement condition for adaptive Markov chain Monte Carlo algorithms. Adv. Appl. Stat. 21 1-54.
13. BAXENDALE, P. H. (2005). Renewal theory and computable convergence rates for geometrically ergodic Markov chains. Ann. Appl. Probab. 15 700-738. MR2114987
14. BERCU, B., DEL MORAL, P. and DOUCET, A. (2009). A functional central limit theorem for a class of interacting Markov chain Monte Carlo methods. Electron. J. Probab. 14 2130-2155. MR2550295
15. BILLINGSLEY, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York. MR1700749
16. BROCKWELL, A.,DEL MORAL, P. andDOUCET, A. (2010). Sequentially interacting Markov chain Monte Carlo methods. Ann. Statist. 38 3387-3411. MR2766856
17. DOUC, R., MOULINES, E. and ROSENTHAL, J. S. (2004). Quantitative bounds on convergence of time-inhomogeneous Markov chains. Ann. Appl. Probab. 14 1643-1665. MR2099647
18. DUDLEY, R. M. (2002). Real Analysis and Probability. Cambridge Studies in Advanced Mathematics 74. Cambridge Univ. Press, Cambridge. MR1932358
19. FORT, G. andMOULINES, E. (2003). Polynomial ergodicity of Markov transition kernels. Stochastic Process. Appl. 103 57-99. MR1947960
20. FORT, G.,MOULINES, E. and PRIOURET, P. (2011). Supplement to "Convergence of adaptive and interacting Markov chain Monte Carlo algorithms." DOI:10.1214/11-AOS938SUPP.
21. GELMAN, A., ROBERTS, G. O. and GILKS, W. R. (1996). Efficient Metropolis jumping rules. In Bayesian Statistics, 5 (Alicante, 1994) 599-607. Oxford Univ. Press, New York. MR1425429
22. GEYER, C. J. (1991). Markov chain Monte Carlo maximum likelihood. In Computing Science and Statistics: Proc. 23rd Symposium on the Interface, Interface Foundation, Fairfax Station, VA 156-163. Amer. Statist. Assoc., Alexandria, VA.
23. HAARIO, H., SAKSMAN, E. and TAMMINEN, J. (1999). Adaptive proposal distribution for random walk Metropolis algorithm. Comput. Statist. 14 375-395.
24. HAARIO, H., SAKSMAN, E. and TAMMINEN, J. (2001). An adaptive Metropolis algorithm. Bernoulli 7 223-242. MR1828504
25. HAARIO, H., LAINE, M., LEHTINEN, M., SAKSMAN, E. and TAMMINEN, J. (2004).Markov chain Monte Carlo methods for high dimensional inversion in remote sensing. J. R. Stat. Soc. Ser. B Stat. Methodol. 66 591-607. MR2088292
26. HAARIO, H., LAINE, M.,MIRA, A. and SAKSMAN, E. (2006). DRAM: Efficient adaptive MCMC. Stat. Comput. 16 339-354. MR2297535
27. HALL, P. and HEYDE, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York. MR0624435
28. JARNER, S. F. and HANSEN, E. (2000). Geometric ergodicity of Metropolis algorithms. Stochastic Process. Appl. 85 341-361. MR1731030
29. KATO, T. (1980). Perturbation Theory for Linear Operators. Springer, Berlin.
30. KOU, S. C., ZHOU, Q. and WONG, W. H. (2006). Equi-energy sampler with applications in statistical inference and statistical mechanics. Ann. Statist. 34 1581-1615.
31. LAINE, M. and TAMMINEN, J. (2008). Aerosol model selection and uncertainty modelling by adaptive MCMC technique. Atmospheric and Chemistry Physics 8 7697-7707.
32. MENGERSEN, K. L. and TWEEDIE, R. L. (1996). Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist. 24 101-121. MR1389882
33. 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.
34. MEYN, S. and TWEEDIE, R. L. (2009). Markov Chains and Stochastic Stability. Cambridge Univ. Press, London.
35. ROBERT, C. P. and CASELLA, G. (2004). Monte Carlo Statistical Methods, 2nd ed. Springer, New York. MR2080278
36. ROBERTS, G. O. and ROSENTHAL, J. S. (2004). General state space Markov chains and MCMC algorithms. Probab. Surv. 1 20-71. MR2095565
37. ROBERTS, G. O. and ROSENTHAL, J. S. (2007). Coupling and ergodicity of adaptive Markov chain Monte Carlo algorithms. J. Appl. Probab. 44 458-475. MR2340211
38. ROBERTS, G. O. and TWEEDIE, R. L. (1996). Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika 83 95-110. MR1399158
39. ROSENTHAL, J. S. (2009). Optimal proposal distributions and adaptive MCMC. In Handbook of MCMC. Chapman & Hall/CRC Press, Boca Raton, FL.
40. SAKSMAN, E. and VIHOLA, M. (2010). On the ergodicity of the adaptive Metropolis algorithm on unbounded domains. Ann. Appl. Probab. 20 2178-2203. MR2759732