The Library

On the containment condition for adaptive Markov Chain Monte Carlo algorithms

Tools

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, Vol.2009 (No.15).

Preview

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

Official URL: http://www2.warwick.ac.uk/fac/sci/statistics/crism...

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

Official Date:

2009

Dates:

Date	Event
2009	Published

Volume:

Vol.2009

Number:

No.15

Number of Pages:

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Not Peer Reviewed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.