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Scaling limits for the transient phase of local Metropolis–Hastings algorithms

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  • Ole F. Christensen
  • Gareth O. Roberts
  • Jeffrey S. Rosenthal

Abstract

Summary. The paper considers high dimensional Metropolis and Langevin algorithms in their initial transient phase. In stationarity, these algorithms are well understood and it is now well known how to scale their proposal distribution variances. For the random‐walk Metropolis algorithm, convergence during the transient phase is extremely regular—to the extent that the algo‐rithm's sample path actually resembles a deterministic trajectory. In contrast, the Langevin algorithm with variance scaled to be optimal for stationarity performs rather erratically. We give weak convergence results which explain both of these types of behaviour and practical guidance on implementation based on our theory.

Suggested Citation

  • 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.
    • Handle: RePEc:bla:jorssb:v:67:y:2005:i:2:p:253-268
    DOI: 10.1111/j.1467-9868.2005.00500.x
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    Cited by:

    1. Agudze, Komla M. & Billio, Monica & Casarin, Roberto & Ravazzolo, Francesco, 2022. "Markov switching panel with endogenous synchronization effects," Journal of Econometrics, Elsevier, vol. 230(2), pages 281-298.
    2. Håvard Rue & Sara Martino & Nicolas Chopin, 2009. "Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(2), pages 319-392, April.
    3. Filippo Pagani & Martin Wiegand & Saralees Nadarajah, 2022. "An n‐dimensional Rosenbrock distribution for Markov chain Monte Carlo testing," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(2), pages 657-680, June.
    4. 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.
    5. Tore Selland Kleppe, 2016. "Adaptive Step Size Selection for Hessian-Based Manifold Langevin Samplers," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(3), pages 788-805, September.
    6. Jeremy Heng & Arnaud Doucet & Yvo Pokern, 2021. "Gibbs flow for approximate transport with applications to Bayesian computation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(1), pages 156-187, February.
    7. 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.

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