Optimal scalings for local metropolis-hastings chains on nonproduct targets in high dimensions

Annals of Applied Probability

Volumn 19, Issue 3, 2009, Pages 863-898

  Beskos, Alexandros ^a   Roberts, Gareth ^a   Stuart, Andrew ^a  

^a UNIVERSITY OF WARWICK   (United Kingdom)

Author keywords

Diffusion; Gaussian law on hilbert space; Karhunen lo ve; Langevin; Navier stokes pde; Random walk Metropolis; Squared jump distance

Indexed keywords

EID: 69149086344     PISSN: 10505164     EISSN: 10505164     Source Type: Journal    
DOI: 10.1214/08-AAP563     Document Type: Article

References (24)

1. Bédard, M. (2008). Optimal acceptance rates for Metropolis algorithms. Stochastic Process. Appl. 118 2198-2222.
2. Bédard, M. (2007). Weak convergence of Metropolis algorithms for non-i.i.d. target distributions. Ann. Appl. Probab. 17 1222-1244. MR2344305
3. Bédard, M. and Rosenthal, J. (2008). Optimal scaling of Metropolis algorithms: Heading toward general target distributions. Canad. J. Statist. 36 483-503.
4. Bennett, A. F. (2002). Inverse Modeling of the Ocean and Atmosphere. Cambridge Univ. Press, Cambridge. MR1920432
5. Beskos, A., Roberts, G., Stuart, A. and Voss, J. (2008). MCMC methods for diffusion bridges. Stoch. Dyn. 8 319-350.
6. Beskos, S. and Stuart, A. (2008). MCMC methods for sampling function space. In Plenary Lectures Volume, ICIAM 2007. To appear.
7. Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York. MR1700749
8. Breyer, L. A., Piccioni, M. and Scarlatti, S. (2004). Optimal scaling of MaLa for nonlinear regression. Ann. Appl. Probab. 14 1479-1505. MR2071431
9. Breyer, L. A. and Roberts, G. O. (2000). From Metropolis to diffusions: Gibbs states and optimal scaling. Stochastic Process. Appl. 90 181-206. MR1794535
10. 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
11. Cotter, S., Dashti, M., Robinson, J. and Stuart, A. (2008). Mathematical foundations of data assimilation problems in fluid mechanics. Unpublished manuscript.
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. Delyon, B. and Hu, Y. (2006). Simulation of conditioned diffusion and application to parameter estimation. Stochastic Process. Appl. 116 1660-1675. MR2269221
14. Geyer, C. J. and Thompson, E. A. (1992). Constrained Monte Carlo maximum likelihood for dependent data. J. R. Stat. Soc. Ser. B Stat. Methodol. 54 657-683. With discussion and a reply by the authors. MR1185217
15. Gilks, W. R., Richardson, S. and Spiegelhalter, D. J., eds. (1996). Markov Chain Monte Carlo in Practice. Chapman & Hall, London. MR1397966
16. Hairer, M., Stuart, A. M., Voss, J. and Wiberg, P. (2005). Analysis of SPDEs arising in path sampling. I. The Gaussian case. Commun. Math. Sci. 3 587-603. MR2188686
17. Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York. MR624435
18. Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrica 57 97-109.
19. Kalnay, E. (2003). Atmospheric Modeling, Data Assimilation and Predictability. Cambridge Univ. Press, Cambridge.
20. Neal, P. and Roberts, G. (2006). Optimal scaling for partially updating MCMC algorithms. Ann. Appl. Probab. 16 475-515. MR2244423
21. 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
22. 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
23. Robinson, J. C. (2001). Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors. Cambridge Univ. Press, Cambridge. MR1881888
24. Sherlock, C. and Roberts, G. (2008). Optimal scaling of the random walk Metropolis on elliptically symmetric unimodal targets. Bernoulli. To appear.