Limit theorems for some adaptive MCMC algorithms with subgeometric kernels. II. (English) Zbl 1244.60072

Atchadé, Yves F.; Fort, Gersende

Bernoulli 18, No. 3, 975-1001 (2012).

Summary: We prove a central limit theorem for a general class of adaptive Markov Chain Monte Carlo algorithms driven by sub-geometrically ergodic Markov kernels. We discuss in detail the special case of stochastic approximation. We use the result to analyze the asymptotic behavior of an adaptive version of the Metropolis Adjusted Langevin algorithm with a heavy tailed target density.
For part I, cf. [ibid. 16, No. 1, 116–154 (2010; Zbl 1215.60046)].

MSC:

60J22	Computational methods in Markov chains
65C05	Monte Carlo methods
65C40	Numerical analysis or methods applied to Markov chains

Keywords:

adaptive Markov chain Monte Carlo; Markov chain; metropolis adjusted Langevin algorithms; stochastic approximations; subgeometric ergodicity

Citations:

Zbl 1215.60046

References:

[1]	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
[2]	Andrieu, C., Moulines, É. and Priouret, P. (2005). Stability of stochastic approximation under verifiable conditions. SIAM J. Control Optim. 44 283-312. · Zbl 1083.62073 · doi:10.1137/S0363012902417267
[3]	Andrieu, C. and Thoms, J. (2008). A tutorial on adaptive MCMC. Stat. Comput. 18 343-373. · doi:10.1007/s11222-008-9110-y
[4]	Atchadé, Y. and Fort, G. (2010). Limit theorems for some adaptive MCMC algorithms with subgeometric kernels. Bernoulli 16 116-154. · Zbl 1215.60046 · doi:10.3150/09-BEJ199
[5]	Atchade, Y.F. and Fort, G. (2009). Limit theorems for some adaptive mcmc algorithms with sub-geometric kernels: Part ii. Technical report. Univ. Michigan. Available at . 0911.0221
[6]	Atchade, Y.F., Fort, G., Moulines, E. and Priouret, P. (2009). Adaptive Markov chain Monte Carlo: Theory and methods. Technical report, Univ. Michigan. · doi:10.1017/CBO9780511984679.003
[7]	Atchadé, Y.F. and Rosenthal, J.S. (2005). On adaptive Markov chain Monte Carlo algorithms. Bernoulli 11 815-828. · Zbl 1085.62097 · doi:10.3150/bj/1130077595
[8]	Bai, Y. (2008). The simultaneous drift conditions for Adaptive Markov Chain Monte Carlo algorithms. Technical report, Univ. Toronto. (Personal communication).
[9]	Baxendale, P.H. (2005). Renewal theory and computable convergence rates for geometrically ergodic Markov chains. Ann. Appl. Probab. 15 700-738. · Zbl 1070.60061 · doi:10.1214/105051604000000710
[10]	Hall, P. and Heyde, C.C. (1980). Martingale Limit Theory and Its Application : Probability and Mathematical Statistics . New York: Academic Press [Harcourt Brace Jovanovich Publishers]. · Zbl 0462.60045
[11]	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
[12]	Jarner, S.F. and Roberts, G.O. (2007). Convergence of heavy-tailed Monte Carlo Markov chain algorithms. Scand. J. Statist. 34 781-815. · Zbl 1164.65004
[13]	Kamatani, K. (2009). Metropolis-Hastings algorithms with acceptance ratios of nearly 1. Ann. Inst. Statist. Math. 61 949-967. · Zbl 1332.60114 · doi:10.1007/s10463-008-0180-6
[14]	Kipnis, C. and Varadhan, S.R.S. (1986). Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 1-19. · Zbl 0588.60058 · doi:10.1007/BF01210789
[15]	Kushner, H.J. and Yin, G.G. (2003). Stochastic Approximation and Recursive Algorithms and Applications , 2nd ed. Applications of Mathematics ( New York ) 35 . New York: Springer. · Zbl 1026.62084
[16]	Maxwell, M. and Woodroofe, M. (2000). Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28 713-724. · Zbl 1044.60014 · doi:10.1214/aop/1019160258
[17]	Roberts, G.O. and Rosenthal, J.S. (2001). Optimal scaling for various Metropolis-Hastings algorithms. Statist. Sci. 16 351-367. · Zbl 1127.65305 · doi:10.1214/ss/1015346320
[18]	Roberts, G.O. and Rosenthal, J.S. (2007). Coupling and ergodicity of adaptive Markov chain Monte Carlo algorithms. J. Appl. Probab. 44 458-475. · Zbl 1137.62015 · doi:10.1239/jap/1183667414
[19]	Roberts, G.O. and Tweedie, R.L. (1996). Exponential convergence of Langevin distributions and their discrete approximations. Bernoulli 2 341-363. · Zbl 0870.60027 · doi:10.2307/3318418
[20]	Saksman, E. and Vihola, M. (2010). On the ergodicity of the adaptive Metropolis algorithm on unbounded domains. Ann. Appl. Probab. 20 2178-2203. · Zbl 1209.65004 · doi:10.1214/10-AAP682