LangevinType Models I: Diffusions with Given Stationary Distributions and their Discretizations*
 O. Stramer,
 R. L. Tweedie
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We describe algorithms for estimating a given measure π known up to a constant of proportionality, based on a large class of diffusions (extending the Langevin model) for which π is invariant. We show that under weak conditions one can choose from this class in such a way that the diffusions converge at exponential rate to π, and one can even ensure that convergence is independent of the starting point of the algorithm. When convergence is less than exponential we show that it is often polynomial at verifiable rates. We then consider methods of discretizing the diffusion in time, and find methods which inherit the convergence rates of the continuous time process. These contrast with the behavior of the naive or Euler discretization, which can behave badly even in simple cases. Our results are described in detail in one dimension only, although extensions to higher dimensions are also briefly described.
 J. E. Besag and P. J. Green, “Spatial statistics and Bayesian computation (with discussion),” J. Roy. Statist. Soc. Ser. B vol. 55 pp. 25–38, 1993.
 J. E. Besag, P. J. Green, D. Higdon, and K. L. Mengersen, “Bayesian computation and stochastic systems (with discussion),” Statistical Science vol. 10 pp. 3–66, 1995.
 J. D. Doll, P. J. Rossky, and H. L. Friedman, “Brownian dynamics as smart Monte Carlo simulation,” Journal of Chemical Physics vol. 69 pp. 4628–4633, 1978.
 D. Down, S. P. Meyn, and R. L. Tweedie, “Exponential and uniform ergodicity of Markov processes,” Ann. Probab. vol. 23 pp. 1671–1691, 1995.
 S. Duane, A. D. Kennedy, B. J. Pendleton, and D. Roweth, “Hybrid Monte Carlo,” Physics Letters B vol. 195 pp. 216–222, 1987.
 U. Grenander and M. I. Miller, “Representations of knowledge in complex systems (with discussion),” J. Roy. Statist. Soc. Ser. B vol. 56 pp. 549–603, 1994.
 C. R. Hwang, S. Y. HwangMa, and S. J. Sheu, “Accelerating Gaussian diffusions,” Ann. Appl. Probab. vol. 3 pp. 897–913, 1993.
 Ioannis Karatzas and Steven E. Shreve, Brownian Motion and Stochastic Calculus, SpringerVerlag: New York, 1991.
 J. Kent, “Timerevesible diffusions,” Adv. Appl. Probab. vol. 10 pp. 819–835, 1978.
 P. E. Kloeden and E. Platen, Numerical solution of stochastic differential equations, SpringerVerlag: Berlin, 1992.
 K. L. Mengersen and R. L. Tweedie, “Rates of convergence of the Hastings and Metropolis algorithms,” Annals of Statistics vol. 24 pp. 101–121, 1996.
 S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, SpringerVerlag: London, 1993.
 S. P. Meyn and R. L. Tweedie, “Stability of Markovian processes II: Continuous time processes and sampled chains,” Adv. Appl. Probab. vol. 25 pp. 487–517, 1993.
 S. P. Meyn and R. L. Tweedie, “Stability of Markovian processes III: FosterLyapunov criteria for continuous time processes,” Adv. Appl. Probab. vol. 25 pp. 518–548, 1993.
 T. Ozaki, “A bridge between nonlinear time series models and nonlinear stochastic dynamical systems: A local linearization approach,” Statistica Sinica vol. 2 pp. 113–135, 1992.
 M. Pollak and D. Siegmund, “A diffusion process and its applications to detecting a change in the drift of Brownian motion,” Biometrika vol. 72 pp. 207–216, 1985.
 G. O. Roberts and R. L. Tweedie, “Exponential convergence of Langevin diffusions and their discrete approximations,” Bernoulli vol. 2 pp. 341–364, 1996.
 G. O. Roberts and R. L. Tweedie, “Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms,” Biometrika vol. 83 pp. 95–110, 1996.
 I. Shoji, Approximation of continuous time stochastic processes by a local linearization method, Technical report, The Institute of Statistical Mathematics, Tokyo, 1995.
 I. Shoji and T. Ozaki, “A statistical method of estimation and simulation for systems of stochastic differential equations,” Biometrika vol. 85 pp. 240–243, 1998.
 A. F. M. Smith and G. O. Roberts, “Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion),” J. Roy. Statist. Soc. Ser. B vol. 55 pp. 3–24, 1993.
 O. Stramer and R. L. Tweedie, Langevintype models II: Selftargeting candidates for MCMC algorithms, Methodology and Computing in Applied Probability vol. 1 pp. 307–328, 1999.
 O. Stramer and R. L. Tweedie, “Existence and stability of weak solutions to stochastic differential equations with nonsmooth coefficients,” Statistica Sinica vol. 7 pp. 577–593, 1997.
 D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, SpringerVerlag: Berlin, 1979.
 L. Tierney, “Markov chains for exploring posterior distributions (with discussion),” Ann. Statist. vol. 22 pp. 1701–1762, 1994.
 D. Toussaint, “Introduction to algorithms for Monte Carlo simulations and their applications to QCD,” Computer Physics Communications vol. 56 pp. 69–92, 1989.
 P. Tuominen and R. L. Tweedie, “Subgeometric rates of convergence of fergodic Markov chains,” Adv. Appl. Probab. vol. 26 pp. 775–798, 1994.
 Title
 LangevinType Models I: Diffusions with Given Stationary Distributions and their Discretizations*
 Journal

Methodology And Computing In Applied Probability
Volume 1, Issue 3 , pp 283306
 Cover Date
 19991001
 DOI
 10.1023/A:1010086427957
 Print ISSN
 13875841
 Online ISSN
 15737713
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 Markov chain Monte Carlo
 diffusions
 Langevin models
 posterior distributions
 irreducible Markov processes
 exponential ergodicity
 uniform ergodicity
 Euler schemes
 Authors

 O. Stramer ^{(1)}
 R. L. Tweedie ^{(2)}
 Author Affiliations

 1. Department of Statistics and Actuarial Science, University of Iowa, Iowa City, IA, 52242, USA
 2. Division of Biostatistics, University of Minnesota, Minneapolis, MN, 55455, USA