Łatuszyński, Krzysztof and Rosenthal, Jeffrey S. (Jeffrey Seth) (2010) Adaptive Gibbs samplers. Working Paper. University of Warwick. Centre for Research in Statistical Methodology, Coventry.

Preview

PDF WRAP_latuszynski_10-02w.pdf - Published Version - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader Download (531Kb)

Abstract

We consider various versions of adaptive Gibbs and Metropolis- within-Gibbs samplers, which update their selection probabilities (and perhaps also their proposal distributions) on the fly during a run, by learning as they go in an attempt to optimise the algorithm. We present a cautionary example of how even a simple-seeming adaptive Gibbs sampler may fail to converge. We then present various positive results guaranteeing convergence of adaptive Gibbs samplers under certain conditions.

Item Type:	Working or Discussion Paper (Working Paper)
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH):	Sampling (Statistics)
Series Name:	Working papers
Publisher:	University of Warwick. Centre for Research in Statistical Methodology
Place of Publication:	Coventry
Date:	2010
Volume:	Vol.2010
Number:	No.2
Number of Pages:	30
Status:	Not Peer Reviewed
Access rights to Published version:	Open Access
Funder:	Natural Sciences and Engineering Research Council of Canada (NSERC)
References:	[1] C. Andrieu and E. Moulines (2006): On the ergodicity properties of some adaptive Markov Chain Monte Carlo algorithms. Ann. Appl. Probab. 16(3), 1462{1505. [2] Y. Atchade and G. Fort (2008): Limit Theorems for some adaptive MCMC algorithms with sub-geometric kernels. Bernoulli, to appear. [3] Y. Atchade, G. Fort, E. Moulines, and P. Priouret (2009): Adaptive Markov Chain Monte Carlo: Theory and Methods. Preprint. [4] Y.F. Athade, G.O. Roberts, and J.S. Rosenthal, (2009): Optimal Scaling of Metropolis-coupled Markov Chain Monte Carlo. Preprint. [5] Y.F. Atchade and J.S. Rosenthal (2005): On Adaptive Markov Chain Monte Carlo Algorithms. Bernoulli 11, 815{828. [6] Y. Bai (2009): Simultaneous drift conditions for Adaptive Markov Chain Monte Carlo algorithms. Preprint. [7] Y. Bai (2009): An Adaptive Directional Metropolis-within-Gibbs algorithm. Preprint. [8] Y. Bai, G.O. Roberts, J.S. Rosenthal (2009): On the Containment Condition for Adaptive Markov Chain Monte Carlo Algorithms. Preprint. [9] M. Bedard (2007): Weak Convergence of Metropolis Algorithms for Non-iid Target Distributions. Ann. Appl. Probab. 17, 1222{44. [10] M. Bedard (2008): Optimal acceptance rates for Metropolis algorithms: Moving beyond 0.234. Stochastic Processes and their Applications, 118(12), 2198{2222. [11] W. Bednorz, R. Lata la and K. Latuszynski (2008): A Regeneration Proof of the Central Limit Theorem for Uniformly Ergodic Markov Chains. Elect. Comm. in Probab. 13, 85{98. [12] L. Bottolo, S. Richardson, and J.S. Rosenthal (2010): Bayesian models for sparse regression analysis of high dimensional data. In preparation. [13] A.E. Brockwell and J.B. Kadane (2005): Identification of Regeneration Times in MCMC Simulation, with Application to Adaptive Schemes. Jour- nal of Computational and Graphical Statistics, 14, 436{458. [14] R.V. Craiu, J.S. Rosenthal, and C. Yang (2008): Learn From Thy Neighbor: Parallel-Chain and Regional Adaptive MCMC. J. Amer. Stat. Assoc., to appear. [15] P. Diaconis, K. Khare, and L. Salo-Coste (2008): Gibbs sampling, exponential families and orthogonal polynomials (with discussion and rejoinder). Statistical Science 23(2), 151{178. [16] W.R. Gilks, G.O. Roberts, and S.K. Sahu (1998): Adaptive Markov chain Monte Carlo through regeneration. J. Amer. Statist. Assoc. 93(443), 1045{ 1054. [17] H. Haario, E. Saksman, and J. Tamminen (2001): An adaptive Metropolis algorithm. Bernoulli 7, 223{242. [18] H. Haario, E. Saksman, and J. Tamminen (2005): Componentwise adaptation for high dimensional MCMC. Computational Statistics 20, 265{273. [19] O. Haggstrom and J.S. Rosenthal (2007): On Variance Conditions for Markov Chain CLTs. Elect. Comm. in Probab. 12, 454-464. [20] C. Kipnis, S.R.S. Varadhan, (1986): Central Limit Theorem for Additive Functionals of Reversible Markov Processes and Applications to Simple Exclusions. Commun. Math. Phys. 104, 1{19. [21] K. Latuszynski (2008): Regeneration and Fixed-Width Analysis of Markov Chain Monte Carlo Algorithms. PhD Dissertation. Available at: arXiv:0907.4716v1 [22] R.A. Levine (2005): A note on Markov chain Monte Carlo sweep strategies. Journal of Statistical Computation and Simulation 75(4), 253{262. [23] R.A. Levine, Z. Yu, W.G. Hanley, and J.A. Nitao (2005): Implementing Random Scan Gibbs Samplers. Computational Statistics 20, 177{196. [24] R.A. Levine and G. Casella (2006): Optimizing random scan Gibbs samplers. Journal of Multivariate Analysis 97, 2071{2100. [25] J.S. Liu (2001): Monte Carlo Strategies in Scientific Computing. Springer, New York. [26] J.S. Liu, W.H. Wong, and A. Kong (1995): Covariance Structure and Convergence Rate of the Gibbs Sampler with Various Scans. J. Roy. Stat. Soc. B 57(1), 157{169. [27] K.L. Mengersen and R.L. Tweedie (1996): Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist. 24, 1, 101{121. [28] N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller (1953), Equations of state calculations by fast computing machines. J. Chem. Phys. 21, 1087{1091. [29] S.P. Meyn and R.L. Tweedie (1993): Markov Chains and Stochastic Stabil- ity. Springer-Verlag, London. Available at: probability.ca/MT [30] O. Papaspiliopoulos and G.O. Roberts (2008): Stability of the Gibbs sampler for Bayesian hierarchical models. Annals of Statistics 36(1), 95{117. [31] C.P. Robert and G. Casella (2004): Monte Carlo Statistical Methods. Springer-Verlag, New York. [32] G.O. Roberts, A. Gelman, and W.R. Gilks (1997): Weak convergence and optimal scaling of random walk Metropolis algorithms. Ann. Appl. Prob. 7, 110{120. [33] G.O. Roberts and N.G. Polson (1994): On the geometric convergence of the Gibbs sampler. J. R. Statist. Soc. B 56(2), 377{384. [34] G.O. Roberts and J.S. Rosenthal (1997): Geometric ergodicity and hybrid Markov chains. Elec. Comm. Prob. 2 (2). [35] G.O. Roberts and J.S. Rosenthal (1998): Two convergence properties of hybrid samplers. Ann. Appl. Prob. 8(2), 397{407. [36] G.O. Roberts and J.S. Rosenthal (1998): Optimal scaling of discrete approximations to Langevin diffusions. J. Roy. Stat. Soc. B 60, 255268. [37] G.O. Roberts and J.S. Rosenthal (2001): Optimal scaling for various Metropolis-Hastings algorithms. Stat. Sci. 16, 351{367. [38] G.O. Roberts and J.S. Rosenthal (2004): General state space Markov chains and MCMC algorithms. Probability Surveys 1, 20{71. [39] G.O. Roberts and J.S. Rosenthal (2007): Coupling and ergodicity of adaptive Markov chain Monte Carlo algorithms. J. Appl. Prob., 44, 458{475. [40] G.O. Roberts and J.S. Rosenthal (2006): Examples of Adaptive MCMC J. Comp. Graph. Stat. 18(2), 349{367. [41] J.S. Rosenthal (2008): Optimal Proposal Distributions and Adaptive MCMC. Preprint. [42] E. Saksman and M. Vihola (2008): On the Ergodicity of the Adaptive Metropolis Algorithm on Unbounded Domains. Preprint. [43] E. Turro, N. Bochkina, A.M.K. Hein, and S. Richardson (2007): BGX: a Bioconductor package for the Bayesian integrated analysis of Affymetrix GeneChips. BMC Bioinformatics 8, 439{448. Available at: http://www.biomedcentral.com/1471-2105/8/439 [44] M. Vihola (2009): On the Stability and Ergodicity of an Adaptive Scaling Metropolis Algorithm. Preprint. [45] C. Yang (2008): On The Weak Law Of Large Numbers For Unbounded Functionals For Adaptive MCMC. Preprint. [46] C. Yang (2008): Recurrent and Ergodic Properties of Adaptive MCMC. Preprint.
URI:	http://wrap.warwick.ac.uk/id/eprint/35066

Request changes to a record

Actions (login required)

View Item