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

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  • O. Stramer

    (University of Iowa)

  • R. L. Tweedie

    (University of Minnesota)

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.

Suggested Citation

  • O. Stramer & R. L. Tweedie, 1999. "Langevin-Type Models I: Diffusions with Given Stationary Distributions and their Discretizations," Methodology and Computing in Applied Probability, Springer, vol. 1(3), pages 283-306, October.
    • Handle: RePEc:spr:metcap:v:1:y:1999:i:3:d:10.1023_a:1010086427957
    DOI: 10.1023/A:1010086427957
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    1. O. Stramer & R. L. Tweedie, 1999. "Langevin-Type Models II: Self-Targeting Candidates for MCMC Algorithms," Methodology and Computing in Applied Probability, Springer, vol. 1(3), pages 307-328, October.
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    Cited by:

    1. G. O. Roberts & O. Stramer, 2002. "Langevin Diffusions and Metropolis-Hastings Algorithms," Methodology and Computing in Applied Probability, Springer, vol. 4(4), pages 337-357, December.
    2. Douc, Randal & Fort, Gersende & Guillin, Arnaud, 2009. "Subgeometric rates of convergence of f-ergodic strong Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 119(3), pages 897-923, March.
    3. Masuda, Hiroki, 2007. "Ergodicity and exponential [beta]-mixing bounds for multidimensional diffusions with jumps," Stochastic Processes and their Applications, Elsevier, vol. 117(1), pages 35-56, January.
    4. Allassonnière, Stéphanie & Kuhn, Estelle, 2015. "Convergent stochastic Expectation Maximization algorithm with efficient sampling in high dimension. Application to deformable template model estimation," Computational Statistics & Data Analysis, Elsevier, vol. 91(C), pages 4-19.
    5. Dalalyan, Arnak S. & Karagulyan, Avetik, 2019. "User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient," Stochastic Processes and their Applications, Elsevier, vol. 129(12), pages 5278-5311.
    6. Fort, Gersende & Moulines, Eric, 2000. "V-Subgeometric ergodicity for a Hastings-Metropolis algorithm," Statistics & Probability Letters, Elsevier, vol. 49(4), pages 401-410, October.
    7. Fort, G. & Moulines, E., 2003. "Polynomial ergodicity of Markov transition kernels," Stochastic Processes and their Applications, Elsevier, vol. 103(1), pages 57-99, January.
    8. O. Stramer & R. L. Tweedie, 1999. "Langevin-Type Models II: Self-Targeting Candidates for MCMC Algorithms," Methodology and Computing in Applied Probability, Springer, vol. 1(3), pages 307-328, October.

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