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Optimal Scaling of Random Walk Metropolis Algorithms with Non-Gaussian Proposals

Author

Listed:
  • Peter Neal

    (University of Manchester)

  • Gareth Roberts

    (University of Warwick)

Abstract

The asymptotic optimal scaling of random walk Metropolis (RWM) algorithms with Gaussian proposal distributions is well understood for certain specific classes of target distributions. These asymptotic results easily extend to any light tailed proposal distribution with finite fourth moment. However, heavy tailed proposal distributions such as the Cauchy distribution are known to have a number of desirable properties, and in many situations are preferable to light tailed proposal distributions. Therefore we consider the question of scaling for Cauchy distributed proposals for a wide range of independent and identically distributed (iid) product densities. The results are somewhat surprising as to when and when not Cauchy distributed proposals are preferable to Gaussian proposal distributions. This provides motivation for finding proposal distributions which improve on both Gaussian and Cauchy proposals, both for finite dimensional target distributions and asymptotically as the dimension of the target density, d → ∞. Throughout we seek the scaling of the proposal distribution which maximizes the expected squared jumping distance (ESJD).

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:13:y:2011:i:3:d:10.1007_s11009-010-9176-9
    DOI: 10.1007/s11009-010-9176-9
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    References listed on IDEAS

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    1. 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.
    2. Søren F. Jarner & Gareth O. Roberts, 2007. "Convergence of Heavy‐tailed Monte Carlo Markov Chain Algorithms," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 34(4), pages 781-815, December.
    3. 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. Kamatani, Kengo, 2020. "Random walk Metropolis algorithm in high dimension with non-Gaussian target distributions," Stochastic Processes and their Applications, Elsevier, vol. 130(1), pages 297-327.
    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.
    3. 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.

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