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A Dirichlet form approach to MCMC optimal scaling

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  • Zanella, Giacomo
  • Bédard, Mylène
  • Kendall, Wilfrid S.

Abstract

This paper shows how the theory of Dirichlet forms can be used to deliver proofs of optimal scaling results for Markov chain Monte Carlo algorithms (specifically, Metropolis–Hastings random walk samplers) under regularity conditions which are substantially weaker than those required by the original approach (based on the use of infinitesimal generators). The Dirichlet form methods have the added advantage of providing an explicit construction of the underlying infinite-dimensional context. In particular, this enables us directly to establish weak convergence to the relevant infinite-dimensional distributions.

Suggested Citation

  • Zanella, Giacomo & Bédard, Mylène & Kendall, Wilfrid S., 2017. "A Dirichlet form approach to MCMC optimal scaling," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 4053-4082.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:12:p:4053-4082
    DOI: 10.1016/j.spa.2017.03.021
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    References listed on IDEAS

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    1. Ole F. Christensen & Gareth O. Roberts & Jeffrey S. Rosenthal, 2005. "Scaling limits for the transient phase of local Metropolis–Hastings algorithms," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(2), pages 253-268, April.
    2. Bédard, Mylène & Douc, Randal & Moulines, Eric, 2012. "Scaling analysis of multiple-try MCMC methods," Stochastic Processes and their Applications, Elsevier, vol. 122(3), pages 758-786.
    3. 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.
    4. Breyer, L. A. & Roberts, G. O., 2000. "From metropolis to diffusions: Gibbs states and optimal scaling," Stochastic Processes and their Applications, Elsevier, vol. 90(2), pages 181-206, December.
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    Cited by:

    1. Jure Vogrinc & Samuel Livingstone & Giacomo Zanella, 2023. "Optimal design of the Barker proposal and other locally balanced Metropolis–Hastings algorithms," Biometrika, Biometrika Trust, vol. 110(3), pages 579-595.
    2. Yang, Jun & Roberts, Gareth O. & Rosenthal, Jeffrey S., 2020. "Optimal scaling of random-walk metropolis algorithms on general target distributions," Stochastic Processes and their Applications, Elsevier, vol. 130(10), pages 6094-6132.

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