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Langevin-Type Models II: Self-Targeting Candidates for MCMC Algorithms

Author

Listed:
  • O. Stramer

    (University of Iowa)

  • R. L. Tweedie

    (University of Minnesota)

Abstract

The Metropolis-Hastings algorithm for estimating a distribution π is based on choosing a candidate Markov chain and then accepting or rejecting moves of the candidate to produce a chain known to have π as the invariant measure. The traditional methods use candidates essentially unconnected to π. We show that the class of candidate distributions, developed in Part I (Stramer and Tweedie 1999), which “self-target” towards the high density areas of π, produce Metropolis-Hastings algorithms with convergence rates that appear to be considerably better than those known for the traditional candidate choices, such as random walk. We illustrate this behavior for examples with exponential and polynomial tails, and for a logistic regression model using a Gibbs sampling algorithm. The detailed results are given in one dimension but we indicate how they may extend successfully to higher dimensions.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:1:y:1999:i:3:d:10.1023_a:1010090512027
    DOI: 10.1023/A:1010090512027
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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.
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    Cited by:

    1. Junming Li & Xiulan Han & Xiao Li & Jianping Yang & Xuejiao Li, 2018. "Spatiotemporal Patterns of Ground Monitored PM 2.5 Concentrations in China in Recent Years," IJERPH, MDPI, vol. 15(1), pages 1-15, January.
    2. 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.
    3. O. F. Christensen & J. Møller & R. P. Waagepetersen, 2001. "Geometric Ergodicity of Metropolis-Hastings Algorithms for Conditional Simulation in Generalized Linear Mixed Models," Methodology and Computing in Applied Probability, Springer, vol. 3(3), pages 309-327, September.
    4. Casella, Bruno & Roberts, Gareth O. & Stramer, Osnat, 2011. "Stability of Partially Implicit Langevin Schemes and Their MCMC Variants," MPRA Paper 95220, University Library of Munich, Germany.
    5. 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.
    6. 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.
    7. Yves F. Atchadé, 2006. "An Adaptive Version for the Metropolis Adjusted Langevin Algorithm with a Truncated Drift," Methodology and Computing in Applied Probability, Springer, vol. 8(2), pages 235-254, June.
    8. 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.
    9. G. K. Basak & Arunangshu Biswas, 2016. "Langevin type limiting processes for adaptive MCMC," Indian Journal of Pure and Applied Mathematics, Springer, vol. 47(2), pages 301-328, June.
    10. Bruno Casella & Gareth Roberts & Osnat Stramer, 2011. "Stability of Partially Implicit Langevin Schemes and Their MCMC Variants," Methodology and Computing in Applied Probability, Springer, vol. 13(4), pages 835-854, December.

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