Bai, Yan, Roberts, Gareth O. and Rosenthal, Jeffrey S. (Jeffrey Seth) (2009) On the containment condition for adaptive Markov Chain Monte Carlo algorithms. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology. (Working papers).

Preview

PDF WRAP_Bai_09-15w.pdf - Published Version - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader Download (489Kb)

Abstract

This paper considers ergodicity properties of certain adaptive Markov chain Monte Carlo (MCMC) algorithms for multidimensional target distributions, in particular Adaptive Metropolis and Adaptive Metropolis-within-Gibbs. It was previously shown (Roberts and Rosenthal [21]) that Diminishing Adaptation and Containment imply ergodicity of adaptive MCMC. We derive various sufficient conditions to ensure Containment, and connect the convergence rates of algorithms with the tail properties of the corresponding target distributions. An example is given to show that Diminishing Adaptation alone does not imply ergodicity. We also present a Summable Adaptive Condition which, when satisfied, proves ergodicity more easily.

Item Type:	Working or Discussion Paper (Working Paper)
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH):	Markov processes, Monte Carlo method, Ergodic theory
Series Name:	Working papers
Publisher:	University of Warwick. Centre for Research in Statistical Methodology
Place of Publication:	Coventry
Date:	2009
Volume:	Vol.2009
Number:	No.15
Number of Pages:	25
Status:	Not Peer Reviewed
Access rights to Published version:	Open Access
Funder:	Natural Sciences and Engineering Research Council of Canada (NSERC)
References:	[1] Andrieu, C and Moulines, E. On the ergodicity properties of some adaptive Markov Chain Monte Carlo algorithms. Ann. Appl. Probab., 16(3):1462–1505, 2006. [2] Andrieu, C and Robert, C.P. Controlled mcmc for optimal sampling. Preprint, 2002. [3] Atchad´e, Y.F. and Rosenthal, J.S. On Adaptive Markov Chain Monte Carlo Algorithms. Bernoulli, 11(5):815–828, 2005. [4] Bai, Y. Simultaneous drift conditions on adaptive Markov Chain Monte Carlo algorithms. Preprint, 2009. [5] Brockwell, A.E. and Kadane, J.B. Indentification of regeneration times in mcmc simulation, with application to adaptive schemes. J. Comp. Graph. Stat, 14:436–458, 2005. [6] Fort, G. and Moulines, E. Computable bounds for subgeometrical and geometrical ergodicity. 2000. [7] Fort, G. and Moulines, E. V-Subgeometric ergodicity for a Hastings-Metropolis algorithm. Statist. Prob. Lett., 49:401–410, 2000. [8] Fort, G. and Moulines, E. Polynomial ergodicity of Markov transition kernels. Stoch. Process. Appl., 103:57–99, 2003. [9] Fort, G., and Moulines, E., Roberts, G.O. and Rosenthal, J.S. On the geometric ergodicity of hybrid samplers. J. Appl. Prob., 40:123–146, 2003. [10] Gilks, W.R., and Roberts, G.O. and Sahu, S.K. Adaptive Markov chain Monte Carlo. J. Amer. Statist. Assoc., 93:1045–1054, 1998. [11] Haario, H., and Saksman, E., and Tamminen, J. An adaptive metropolis algorithm. Bernoulli, 7:223–242, 2001. [12] Jarner, S.F. and Hansen, E. Geometric ergodicity of Metropolis algorithms. Stoch. Process. Appl., 85:341–361, 2000. [13] Jarner, S.F. and Roberts, G.O. Polynomial convergence rates of markov chains. Ann. Appl. Probab., 12(1):224–247, 2002. [14] Mengersen, K.L. and Tweedie, R.L. Rate of convergences of the Hasting and Metropolis algorithms. Ann. Statist., 24(1):101–121, 1996. [15] Meyn, S.P. and Tweedie, R.L. Markov Chains and Stochastic Stability. London: Springer- Verlag, 1993. [16] Robbins, H. and Monro, S. A stochastic approximation method. Ann. Math. Stat., 22:400–407, 1951. [17] Roberts, G.O., and Gelman, A., and Gilks, W.R. Weak convergence and optimal scaling of random walk Metropolis algorithms. Ann. Appl. Prob., 7:110–120, 1997. [18] Roberts, G.O. and Rosenthal, J.S. Two convergence properties of hybrid samplers. Ann. Appl. Prob., 8:397-407, 1998. [19] Roberts, G.O. and Rosenthal, J.S. Optimal scaling for various Metropolis-Hastings algorithms. Stat. Sci., 16:351–367, 2001. [20] Roberts, G.O. and Rosenthal, J.S. Examples of Adaptive MCMC. Preprint, 2006. [21] Roberts, G.O. and Rosenthal, J.S. Coupling and Ergodicity of adaptive Markov chain Monte Carlo algorithms. J. Appl. Prob., 44:458–475, 2007. [22] Roberts, G.O. and Tweedie, R.L. Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika, 83:95–110, 1996. [23] Saksman, E. and Vihola, M. On the Ergodicity of the Adaptive Metropolis Algorithms on Unbounded Domains. Preprint, 2008. [24] Tuominen, P. and Tweedie, R.L. Subgeometric rates of convergence of f-ergodic Markov chains. Adv. Appl. Probab., 26(3):775–798, 1994. [25] Yang, C. On the weak law of large number for unbounded functionals for adaptive MCMC. Preprint, 2008.
URI:	http://wrap.warwick.ac.uk/id/eprint/35203

Request changes to a record

Actions (login required)

View Item