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Optimal scaling of random-walk metropolis algorithms on general target distributions

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  • Yang, Jun
  • Roberts, Gareth O.
  • Rosenthal, Jeffrey S.

Abstract

One main limitation of the existing optimal scaling results for Metropolis–Hastings algorithms is that the assumptions on the target distribution are unrealistic. In this paper, we consider optimal scaling of random-walk Metropolis algorithms on general target distributions in high dimensions arising from practical MCMC models from Bayesian statistics. For optimal scaling by maximizing expected squared jumping distance (ESJD), we show the asymptotically optimal acceptance rate 0.234 can be obtained under general realistic sufficient conditions on the target distribution. The new sufficient conditions are easy to be verified and may hold for some general classes of MCMC models arising from Bayesian statistics applications, which substantially generalize the product i.i.d. condition required in most existing literature of optimal scaling. Furthermore, we show one-dimensional diffusion limits can be obtained under slightly stronger conditions, which still allow dependent coordinates of the target distribution. We also connect the new diffusion limit results to complexity bounds of Metropolis algorithms in high dimensions.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:10:p:6094-6132
    DOI: 10.1016/j.spa.2020.05.004
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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. 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. Roberts, G. O. & Tweedie, R. L., 1999. "Bounds on regeneration times and convergence rates for Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 80(2), pages 211-229, April.
    4. Arnak S. Dalalyan, 2017. "Theoretical guarantees for approximate sampling from smooth and log-concave densities," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(3), pages 651-676, June.
    5. Peter Neal & Gareth Roberts, 2008. "Optimal Scaling for Random Walk Metropolis on Spherically Constrained Target Densities," Methodology and Computing in Applied Probability, Springer, vol. 10(2), pages 277-297, June.
    6. 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.
    7. 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.
    8. 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.
    9. Peter Neal & Gareth Roberts, 2011. "Optimal Scaling of Random Walk Metropolis Algorithms with Non-Gaussian Proposals," Methodology and Computing in Applied Probability, Springer, vol. 13(3), pages 583-601, September.
    10. Roberts, G. O. & Smith, A. F. M., 1994. "Simple conditions for the convergence of the Gibbs sampler and Metropolis-Hastings algorithms," Stochastic Processes and their Applications, Elsevier, vol. 49(2), pages 207-216, February.
    11. 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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