Marshall, Tristan and Roberts, Gareth O. (2009) An ergodicity result for adaptive Langevin algorithms. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology. (Working papers).

Preview

PDF WRAP_Marshall_09-04w.pdf - Published Version - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader Download (3827Kb)

Abstract

We consider a class of adaptive MCMC algorithms using a Langevin-type proposal density. We prove that these are algorithms are ergodic when the target density has exponential tail behaviour. Unlike previous results, our approach does not require bounding the drift function.

Item Type:	Working or Discussion Paper (Working Paper)
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH):	Monte Carlo method, Markov processes, Langevin equations
Series Name:	Working papers
Publisher:	University of Warwick. Centre for Research in Statistical Methodology
Place of Publication:	Coventry
Date:	March 2009
Volume:	Vol.2009
Number:	No.4
Number of Pages:	34
Status:	Not Peer Reviewed
Access rights to Published version:	Open Access
References:	[1] Jesper Mø ller Anders Brix. Space-time multi type log gaussian cox processes with a view to modelling weeds. Scandinavian Journal of Statistics, 28(3):471–488, 2001. [2] Peter J. Diggle Anders Brix. Spatiotemporal prediction for log-gaussian cox processes. Journal of the Royal Statistical Society: Series B (Statis- tical Methodology), 63(4):823–841, 2001. [3] C. Andrieu and E. Moulines. On the ergodicity properties of some adaptive markov chain monte carlo algorithms. Ann. Appl. Probab., 16(3):1462–1505, 2006. [4] Yves Atchad´e. An adaptive version for the metropolis adjusted langevin algorithm with a truncated drift. Methodology and Computing in Applied Probability, 2005. [5] Yves Atchad´e and Jeffrey Rosenthal. On adaptive markov chain monte carlo algorithms. Bernoulli, 11(5):815–828, 2005. [6] Myl`ene B´edard. On the Robustness of Optimal Scaling for Random Walk Metropolis Algorithms. PhD thesis, University of Toronto, 2006. [7] V. Benes, K. Bodlak, J. Møller, and R.P. Waagepetersen. A case study on point process modelling in disease mapping. Image Analysis and Stereology, 24:159–168, 2005. [8] A. E. Brockwell and J. B. Kadane. Identification of regeneration times in mcmc simulation, with aplication to adaptive schemes. J. Comp. Graph. Stat., 14:436–458, 2005. [9] P. Coles and B. Jones. A lognormal model for the cosmological mass distribution. Royal Astronomical Society, Monthly Notices, 248:1–13, January 1991. [10] Walter R. Gilks, Gareth O. Roberts, and Sujit K. Sahu. Adaptive markov chain monte carlo through regeneration. Journal of the Ameri- can Statistical Association, 93(443):1045–, 1998. [11] H. Haario, E. Saksman, and J. Tamminen. An adaptive metropolis algorithm. Bernoulli, 7:223–242, 2001. [12] Peter Hall and C. C. Heyde. Martingale Limit Theory and its Applica- tion. Academic Press, 1980. [13] J. F. C. Kingman. Poisson processes, volume 3 of Oxford Studies in Probability. The Clarendon Press Oxford University Press, New York, 1993. Oxford Science Publications. [14] Jesper Møller, Anne Randi Syversveen, and Rasmus Plenge Waagepetersen. Log gaussian cox processes. Scandinavian Journal of Statistics, 25(3):451–482, 1998. [15] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E. Teller. Equation of state calculations by fast computing machines. J. Chem Phys., 21:1087–1092, 1953. [16] S.P. Meyn and R.L. Tweedie. Markov Chains and Stochastic Stability. Springer-Verlag, London, 1993. Available at probability.ca/MT. [17] Gareth Roberts and Jeffrey Rosenthal. Optimal scaling of discrete approximations to langevin diffusions. Journal of the Royal Statistical Society. Series B, 60(1):255–268, 1995. [18] Gareth Roberts and Jeffrey Rosenthal. Optimal scaling for various metropolis-hastings algorithms. Statistical Science, 15(4):351–367, 2001. [19] Gareth Roberts and Jeffrey Rosenthal. Coupling and ergodicity of adaptive mcmc. J. Appl. Prob., 44(2):458–477, 2007. [20] Gareth Roberts and Richard Tweedie. Exponential convergence of langevin diffusions and their discrete approximations. Bernoulli, 2(4):341–363, 1996. [21] G.O. Roberts, A. Gelman, and W.R. Gilks. Weak convergence and optimal scaling of random walk metropolis algorithms. Ann. Appl. Prob., 7(1):110–120, 1996. [22] G.O. Roberts and A.F.M. Smith. Simple conditions for the convergence of the gibbs sampler and metropolis-hastings algorithms. Stochastic Processes and their Applications, 49:207–216, 1994. [23] G.O. Roberts and R.L. Tweedie. Geometric convergence and central limit theorems for multidimensional hastings and metropolis algorithms. Biometrika, 83(1):95–110, 1996. [24] Rasmus Waagepetersen. Convergence of posteriors for discretized log gaussian cox processes. Statistics and Probability Letters, 66(3):229 – 235, 2004.
URI:	http://wrap.warwick.ac.uk/id/eprint/35196

Request changes to a record

Actions (login required)

View Item