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Langevin Diffusions and Metropolis-Hastings Algorithms

Author

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  • G. O. Roberts

    (Lancaster University)

  • O. Stramer

    (University of Iowa)

Abstract

We consider a class of Langevin diffusions with state-dependent volatility. The volatility of the diffusion is chosen so as to make the stationary distribution of the diffusion with respect to its natural clock, a heated version of the stationary density of interest. The motivation behind this construction is the desire to construct uniformly ergodic diffusions with required stationary densities. Discrete time algorithms constructed by Hastings accept reject mechanisms are constructed from discretisations of the algorithms, and the properties of these algorithms are investigated.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:4:y:2002:i:4:d:10.1023_a:1023562417138
    DOI: 10.1023/A:1023562417138
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    References listed on IDEAS

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    1. 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.
    2. Ganidis, H. & Roynette, B. & Simonot, F., 1999. "Convergence rate of some semi-groups to their invariant probability," Stochastic Processes and their Applications, Elsevier, vol. 79(2), pages 243-263, February.
    3. 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.
    4. Gareth O. Roberts & Jeffrey S. Rosenthal, 1998. "Optimal scaling of discrete approximations to Langevin diffusions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(1), pages 255-268.
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    Cited by:

    1. Ruben Loaiza-Maya & Didier Nibbering & Dan Zhu, 2023. "Hybrid unadjusted Langevin methods for high-dimensional latent variable models," Papers 2306.14445, arXiv.org.
    2. Gunawan, David & Dang, Khue-Dung & Quiroz, Matias & Kohn, Robert & Tran, Minh-Ngoc, 2019. "Subsampling Sequential Monte Carlo for Static Bayesian Models," Working Paper Series 371, Sveriges Riksbank (Central Bank of Sweden).
    3. Robert D. Skeel & Carsten Hartmann, 2021. "Choice of damping coefficient in Langevin dynamics," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 94(9), pages 1-13, September.
    4. 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.
    5. Radu Herbei & Rajib Paul & L Mark Berliner, 2017. "Applying diffusion-based Markov chain Monte Carlo," PLOS ONE, Public Library of Science, vol. 12(3), pages 1-14, March.
    6. Dang, Khue-Dung & Quiroz, Matias & Kohn, Robert & Tran, Minh-Ngoc & Villani, Mattias, 2019. "Hamiltonian Monte Carlo with Energy Conserving Subsampling," Working Paper Series 372, Sveriges Riksbank (Central Bank of Sweden).
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    8. Kengo Kamatani, 2009. "Metropolis–Hastings Algorithms with acceptance ratios of nearly 1," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(4), pages 949-967, December.

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    Keywords

    MCMC; Langevin diffusions and algorithms;

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