Exponential convergence of Langevin distributions and their discrete approximations

Bernoulli

Volumn 2, Issue 4, 1996, Pages 341-363

  Roberts, Gareth O ^a   Tweedie, Richard L ^b  

^a UNIVERSITY OF CAMBRIDGE   (United Kingdom)

^b Veterinary Dentistry and Oral Surgery   (United States)

Author keywords

Diffusions; Discrete approximations; Geometric ergodicity; Hastings algorithms; Irreducible Markov processes; Langevin models; Markov chain Monte Carlo; Metropolis algorithms; Posterior distributions

Indexed keywords

EID: 85132364916     PISSN: 13507265     EISSN: None     Source Type: Journal    
DOI: 10.2307/3318418     Document Type: Article

References (26)

1. Besag, J.E. (1994) Comments on `Representations of knowledge in complex systems' by U. Grenander and M.I. Miller. J. Roy. Statist. Soc. Ser. B, 56, 591-592.
2. Besag, J.E. and Green, P.J. (1993) Spatial statistics and Bayesian computation (with discussion). J. Roy. Statist. Soc. Ser. B, 55, 25-38.
3. Besag, J.E., Green, P.J., Higdon, D. and Mengersen, K.L. (1995) Bayesian computation and stochastic systems (with discussion). Statist. Sci., 10, 3-66.
4. Bhattacharya, R.N. (1978) Criteria for recurrence and existence of invariant measures for multi-dimensional diffusions. Ann. Probab., 6, 541-553.
5. Davis, M.H.A. (1993) Markov Models and Optimizations. London: Chapman & Hall.
6. Down, D., Meyn, S.P. and Tweedie, R.L. (1995) Geometric and uniform ergodicity of Markov processes. Ann. Probab., 23, 1671-1691.
7. Grenander, U. and Miller, M.I. (1994) Representations of knowledge in complex systems (with discussion). J. Roy. Statist. Soc. Ser. B, 56, 549-603.
8. Hastings, W.K. (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57, 97-109.
9. Ikeda, N. and Watanabe, S. (1989) Stochastic Differential Equations and diffusion Processes. Amsterdam: Elsevier.
10. Lamperti, J. (1963) Criteria for stochastic processes II: Passage time moments. J. Math. Anal. Appl., 7, 127-145.
11. Lerche, H.R. (1986) Boundary Crossings of Brownian Motion. Berlin: Springer-Verlag.
12. Mengersen, K.L. and Tweedie, R.L. (1996) Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist., 24, 101-121.
13. Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A. and Teller, E. (1953) Equations of state calculations by fast computing machines. J. Chem. Phys., 21, 1087-1091.
14. Meyn, S.P. and Tweedie, R.L. (1993a) Markov Chains and Stochastic Stability. London: Springer-Verlag.
15. Meyn, S.P. and Tweedie, R.L. (1993b) Stability of Markovian processes II: Continuous time processes and sampled chains. Adv. Appl. Probab., 25, 487-517.
16. Neal, R. (1994) An improved acceptance procedure for the hybrid Monte Carlo algorithm. J. Comp. Phys., 11, 194-203.
17. Parisi, G. (1981) Correlation functions and computer simulations. Nuclear Phys. B, 180, 378-384.
18. Roberts, G.O. and Rosenthal, J.S. (1995a) Optimal scaling of discrete approximations to Langevin diffusions. Submitted for publication.
19. Roberts, G.O. and Rosenthal, J.S. (1995b) Shift-coupling and convergence rates of ergodic averages. Submitted for publication.
20. Roberts, G.O. and Smith, A.F.M. (1994) Simple conditions for the convergence of the Gibbs sampler and Hastings-Metropolis algorithms. Stochastic Process. Appl., 49, 207-216.
21. Roberts, G.O. and Tweedie, R.L. (1996) Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika, 83, 96-110.
22. Roberts, G.O., Gelman, A. and Gilks, W.R. (1994) Weak convergence and optimal scaling of Hastings-Metropolis algorithms. Ann. Appl. Prob. To appear.
23. Sheu, S.-J. (1992) Some estimates of the transition density of a non-degenerate diffusion Markov process. Ann. Probab., 19, 538-561.
24. Smith, A.F.M. and Roberts, G.O. (1993) Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion). J. Roy. Statist. Soc. Ser. B, 55, 3-24.
25. Tierney, L. (1994) Markov chains for exploring posterior distributions (with discussion). Ann. Statist., 22, 1701-1762.
26. Tweedie, R.L. (1994) Topological conditions enabling use of Harris methods in discrete and continuous time. Acta Appl. Math., 34, 175-188.