Optimal scaling and diffusion limits for the Langevin algorithm in high dimensions

Annals of Applied Probability

Volumn 22, Issue 6, 2012, Pages 2320-2356

  Pillai, Natesh S ^a   Stuart, Andrew M ^b   Thiéry, Alexandre H ^b  

^a HARVARD UNIVERSITY   (United States)

^b UNIVERSITY OF WARWICK   (United Kingdom)

Author keywords

Diffusion approximation; Markov chain Monte Carlo; Metropolis adjusted Langevin algorithm; Scaling limit

Indexed keywords

EID: 84879614507     PISSN: 10505164     EISSN: None     Source Type: Journal    
DOI: 10.1214/11-AAP828     Document Type: Article

References (27)

1. BÉDARD, M. (2007).Weak convergence of Metropolis algorithms for non-i.i.d. target distributions. Ann. Appl. Probab. 17 1222-1244. MR2344305
2. BÉDARD, M. (2009). On the optimal scaling problem of Metropolis algorithms for hierarchical target distributions. Preprint.
3. BERGER, E. (1986). Asymptotic behaviour of a class of stochastic approximation procedures. Probab. Theory Relat. Fields 71 517-552. MR0833268
4. BESKOS, A., PILLAI, N., ROBERTS, G. O., SANZ-SERNA, J.M. and STUART, A. M. (2012). Optimal tuning of the hybrid Monte-Carlo algorithm. Bernoulli. To appear.
5. BESKOS, A., PINSKI, F. J., SANZ-SERNA, J. M. and STUART, A. M. (2011). Hybrid Monte Carlo on Hilbert spaces. Stochastic Process. Appl. 121 2201-2230. MR2822774
6. BESKOS, A., ROBERTS, G. and STUART, A. (2009). Optimal scalings for local Metropolis-Hastings chains on nonproduct targets in high dimensions. Ann. Appl. Probab. 19 863-898. MR2537193
7. BESKOS, A., ROBERTS, G., STUART, A. and VOSS, J. (2008). MCMC methods for diffusion bridges. Stoch. Dyn. 8 319-350. MR2444507
8. BESKOS, A. and STUART, A. (2009). MCMC methods for sampling function space. In ICIAM 07 - 6th International Congress on Industrial and Applied Mathematics 337-364. Eur. Math. Soc., Zürich. MR2588600
9. BREYER, L. A., PICCIONI, M. and SCARLATTI, S. (2004). Optimal scaling of MaLa for nonlinear regression. Ann. Appl. Probab. 14 1479-1505. MR2071431
10. BREYER, L. A. and ROBERTS, G. O. (2000). From Metropolis to diffusions: Gibbs states and optimal scaling. Stochastic Process. Appl. 90 181-206. MR1794535
11. CHRISTENSEN, O. F., ROBERTS, G. O. and ROSENTHAL, J. S. (2005). Scaling limits for the transient phase of local Metropolis-Hastings algorithms. J. R. Stat. Soc. Ser. B Stat. Methodol. 67 253-268. MR2137324
12. DA PRATO, G. and ZABCZYK, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge. MR1207136
13. DASHTI, M.,HARRIS, S. and STUART, A. M. (2012). Besov priors for Bayesian inverse problems. Inverse Probl. Imaging. To appear. Available at http://arxiv.org/abs/1105.0889.
14. ETHIER, S. N. and KURTZ, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York. MR0838085
15. HAIRER, M., STUART, A. M. and VOSS, J. (2011). Signal processing problems on function space: Bayesian formulation, stochastic PDEs and effectiveMCMmethods. In The Oxford Handbook of Nonlinear Filtering (D. Crisan and B. Rozovsky, eds.) 833-873. Oxford Univ. Press, Oxford. MR2884617
16. HAIRER, M., STUART, A. M. and VOSS, J. (2007). Analysis of SPDEs arising in path sampling. II. The nonlinear case. Ann. Appl. Probab. 17 1657-1706. MR2358638
17. HAIRER, M., STUART, A. M., VOSS, J. andWIBERG, P. (2005). Analysis of SPDEs arising in path sampling. I. The Gaussian case. Commun. Math. Sci. 3 587-603. MR2188686
18. LASSAS, M., SAKSMAN, E. and SILTANEN, S. (2009). Discretization-invariant Bayesian inversion and Besov space priors. Inverse Probl. Imaging 3 87-122. MR2558305
19. MATTINGLY, J. C., PILLAI, N. S. and STUART, A. M. (2012). Diffusion limits of the random walk Metropolis algorithm in high dimensions. Ann. Appl. Probab. 22 881-930.
20. METROPOLIS, N., ROSENBLUTH, A.W., TELLER, M. N. and TELLER, E. (1953). Equations of state calculations by fast computing machines. J. Chem. Phys. 21 1087-1092.
21. ROBERT, C. P. and CASELLA, G. (2004). Monte Carlo Statistical Methods, 2nd ed. Springer, New York. MR2080278
22. ROBERTS, G. O., GELMAN, A. and GILKS, W. R. (1997). Weak convergence and optimal scaling of random walk Metropolis algorithms. Ann. Appl. Probab. 7 110-120. MR1428751
23. ROBERTS, G. O. and ROSENTHAL, J. S. (1998). Optimal scaling of discrete approximations to Langevin diffusions. J. R. Stat. Soc. Ser. B Stat. Methodol. 60 255-268. MR1625691
24. ROBERTS, G. O. and ROSENTHAL, J. S. (2001). Optimal scaling for various Metropolis-Hastings algorithms. Statist. Sci. 16 351-367. MR1888450
25. SCHWAB, C. and STUART, A. M. (2012). Sparse deterministic approximation of Bayesian inverse problems. Inverse Problems 28 045003.
26. SHERLOCK, C., FEARNHEAD, P. and ROBERTS, G. O. (2010). The random walk Metropolis: Linking theory and practice through a case study. Statist. Sci. 25 172-190. MR2789988
27. STUART, A. M. (2010). Inverse problems: A Bayesian perspective. Acta Numer. 19 451-559. MR2652785