Langevin-Type Models I: Diffusions with Given Stationary Distributions and their Discretizations* Article

DOI :
10.1023/A:1010086427957

Cite this article as: Stramer, O. & Tweedie, R.L. Methodology and Computing in Applied Probability (1999) 1: 283. doi:10.1023/A:1010086427957 Abstract 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.

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

References 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.

Google Scholar 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.

Google Scholar 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.

Google Scholar D. Down, S. P. Meyn, and R. L. Tweedie, “Exponential and uniform ergodicity of Markov processes,”

Ann. Probab. vol. 23 pp. 1671–1691, 1995.

Google Scholar S. Duane, A. D. Kennedy, B. J. Pendleton, and D. Roweth, “Hybrid Monte Carlo,”

Physics Letters B vol. 195 pp. 216–222, 1987.

Google Scholar 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.

Google Scholar C. R. Hwang, S. Y. Hwang-Ma, and S. J. Sheu, “Accelerating Gaussian diffusions,”

Ann. Appl. Probab. vol. 3 pp. 897–913, 1993.

Google Scholar Ioannis Karatzas and Steven E. Shreve,

Brownian Motion and Stochastic Calculus , Springer-Verlag: New York, 1991.

Google Scholar J. Kent, “Time-revesible diffusions,”

Adv. Appl. Probab. vol. 10 pp. 819–835, 1978.

Google Scholar P. E. Kloeden and E. Platen,

Numerical solution of stochastic differential equations , Springer-Verlag: Berlin, 1992.

Google Scholar 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.

Google Scholar S. P. Meyn and R. L. Tweedie,

Markov Chains and Stochastic Stability , Springer-Verlag: London, 1993.

Google Scholar 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.

Google Scholar S. P. Meyn and R. L. Tweedie, “Stability of Markovian processes III: Foster-Lyapunov criteria for continuous time processes,”

Adv. Appl. Probab. vol. 25 pp. 518–548, 1993.

Google Scholar 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.

Google Scholar 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.

Google Scholar G. O. Roberts and R. L. Tweedie, “Exponential convergence of Langevin diffusions and their discrete approximations,”

Bernoulli vol. 2 pp. 341–364, 1996.

Google Scholar G. O. Roberts and R. L. Tweedie, “Geometric convergence and central limit theorems for multi-dimensional Hastings and Metropolis algorithms,”

Biometrika vol. 83 pp. 95–110, 1996.

Google Scholar I. Shoji,

Approximation of continuous time stochastic processes by a local linearization method , Technical report, The Institute of Statistical Mathematics, Tokyo, 1995.

Google Scholar 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.

Google Scholar 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.

Google Scholar O. Stramer and R. L. Tweedie,

Langevin-type models II: Self-targeting candidates for MCMC algorithms, Methodology and Computing in Applied Probability vol. 1 pp. 307–328, 1999.

Google Scholar O. Stramer and R. L. Tweedie, “Existence and stability of weak solutions to stochastic differential equations with non-smooth coefficients,”

Statistica Sinica vol. 7 pp. 577–593, 1997.

Google Scholar D. W. Stroock and S. R. S. Varadhan,

Multidimensional Diffusion Processes , Springer-Verlag: Berlin, 1979.

Google Scholar L. Tierney, “Markov chains for exploring posterior distributions (with discussion),”

Ann. Statist. vol. 22 pp. 1701–1762, 1994.

Google Scholar D. Toussaint, “Introduction to algorithms for Monte Carlo simulations and their applications to QCD,”

Computer Physics Communications vol. 56 pp. 69–92, 1989.

Google Scholar P. Tuominen and R. L. Tweedie, “Subgeometric rates of convergence of f-ergodic Markov chains,”

Adv. Appl. Probab. vol. 26 pp. 775–798, 1994.

Google Scholar © Kluwer Academic Publishers 1999

Authors and Affiliations 1. Department of Statistics and Actuarial Science University of Iowa Iowa City USA 2. Division of Biostatistics University of Minnesota Minneapolis USA