SPDE limits of the random walk Metropolis algorithm in high dimensions
Mattingly, Jonathan C., Pillai, Natesh S., 1981- and Stuart, A. M. (2009) SPDE limits of the random walk Metropolis algorithm in high dimensions. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology. (Working papers).
WRAP_Mattingly_09-19w.pdf - Published Version - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader
Official URL: http://www2.warwick.ac.uk/fac/sci/statistics/crism...
Diffusion limits of MCMC methods in high dimensions provide a useful theoretical tool for studying efficiency. In particular they facilitate precise estimates of the number of steps required to explore the target measure, in stationarity, as a function of the dimension of the state space. However, to date such results have only been proved for target measures with a product structure, severely limiting their applicability to real applications. The purpose of this paper is to study diffusion limits for a class of naturally occuring high dimensional measures, found from the approximation of measures on a Hilbert space which are absolutely continuous with respect to a Gaussian reference measure. The diffusion limit to an infinite dimensional Hilbert space valued SDE (or SPDE) is proved.
|Item Type:||Working or Discussion Paper (Working Paper)|
|Subjects:||Q Science > QA Mathematics|
|Divisions:||Faculty of Science > Mathematics
Faculty of Science > Statistics
|Library of Congress Subject Headings (LCSH):||Random walks (Mathematics), Monte Carlo method, Markov processes, Sampling (Statistics)|
|Series Name:||Working papers|
|Publisher:||University of Warwick. Centre for Research in Statistical Methodology|
|Place of Publication:||Coventry|
|Number of Pages:||42|
|Status:||Not Peer Reviewed|
|Access rights to Published version:||Open Access|
|References:||B´edard, Myl`ene. 2007. Weak convergence of Metropolis algorithms for non-i.i.d. target distributions. Ann. Appl. Probab., 17(4), 1222–1244. Berger, Erich. 1986. Asymptotic behaviour of a class of stochastic approximation procedures. Probab. Theory Relat. Fields, 71(4), 517–552. Beskos, A., & Stuart, A.M. 2007. MCMC Methods for Sampling Function Space. ICIAM Plenary Lectures. To Appear. Beskos, A., Roberts, G.O., Stuart, A.M., & Voss, J. 2008. An MCMC Method for diffusion bridges. Stochastics and Dynamics, 8(3), 319–350. Chen, Xiaohong, & White, Halbert. 1998. Central limit and functional central limit theorems for Hilbert-valued dependent heterogeneous arrays with applications. Econometric Theory, 14(2), 260–284. Da Prato, Giuseppe, & Zabczyk, Jerzy. 1992. Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge: Cambridge University Press. Diaconis, Persi, & Holmes, Susan (eds). 2004. Stein’s method: expository lectures and applications. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 46. Beachwood, OH: Institute of Mathematical Statistics. Papers from the Workshop on Stein’s Method held at Stanford University, Stanford, CA, 1998. Ethier, Stewart N., & Kurtz, Thomas G. 1986. Markov processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. New York: John Wiley & Sons Inc. Characterization and convergence. Hairer, M., A.M.Stuart, Voss, J., & Wiberg, P. 2005. Analysis of SPDEs Arising in Path Sampling. Part 1: The Gaussian Case. Comm. Math. Sci., 587–603. Hairer, M., Stuart, A. M., & Voss, J. 2007. Analysis of SPDEs arising in path sampling. II. The nonlinear case. Ann. Appl. Probab., 17(5-6), 1657–1706. Hastings, W.K. 1970. Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 97–109. Liu, Jun S. 2008. Monte Carlo strategies in scientific computing. Springer Series in Statistics. New York: Springer. Ma, Zhi Ming, & R¨ockner, Michael. 1992. Introduction to the theory of (nonsymmetric) Dirichlet forms. Universitext. Berlin: Springer-Verlag. N. Metropolis, A.W. Rosenbluth, M.N. Teller, & Teller, E. 1953. Equations of state calculations by fast computing machines. J. Chem. Phys., 21, 1087–1092. Robert, Christian P., & Casella, George. 2004. Monte Carlo statistical methods. Second edn. Springer Texts in Statistics. New York: Springer-Verlag. Roberts, G. O., Gelman, A., & Gilks, W. R. 1997. Weak convergence and optimal scaling of random walk Metropolis algorithms. Ann. Appl. Probab., 7(1), 110–120. Roberts, Gareth O., & Rosenthal, Jeffrey S. 1998. Optimal scaling of discrete approximations to Langevin diffusions. J. R. Stat. Soc. Ser. B Stat. Methodol., 60(1), 255–268. Roberts, Gareth O., & Rosenthal, Jeffrey S. 2001. Optimal scaling for various Metropolis- Hastings algorithms. Statist. Sci., 16(4), 351–367. Walk, H. 1977. An invariance principle for the Robbins-Monro process in a Hilbert space.|
Actions (login required)