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Quasi‐stationary Monte Carlo and the ScaLE algorithm

Author

Listed:
  • Murray Pollock
  • Paul Fearnhead
  • Adam M. Johansen
  • Gareth O. Roberts
Abstract
This paper introduces a class of Monte Carlo algorithms which are based on the simulation of a Markov process whose quasi‐stationary distribution coincides with a distribution of interest. This differs fundamentally from, say, current Markov chain Monte Carlo methods which simulate a Markov chain whose stationary distribution is the target. We show how to approximate distributions of interest by carefully combining sequential Monte Carlo methods with methodology for the exact simulation of diffusions. The methodology introduced here is particularly promising in that it is applicable to the same class of problems as gradient‐based Markov chain Monte Carlo algorithms but entirely circumvents the need to conduct Metropolis–Hastings type accept–reject steps while retaining exactness: the paper gives theoretical guarantees ensuring that the algorithm has the correct limiting target distribution. Furthermore, this methodology is highly amenable to ‘big data’ problems. By employing a modification to existing naive subsampling and control variate techniques it is possible to obtain an algorithm which is still exact but has sublinear iterative cost as a function of data size.

Suggested Citation

  • Murray Pollock & Paul Fearnhead & Adam M. Johansen & Gareth O. Roberts, 2020. "Quasi‐stationary Monte Carlo and the ScaLE algorithm," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(5), pages 1167-1221, December.
    • Handle: RePEc:bla:jorssb:v:82:y:2020:i:5:p:1167-1221
    DOI: 10.1111/rssb.12365
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    References listed on IDEAS

    as
    1. Murray Pollock, 2015. "On the Exact Simulation of (Jump) Diffusion Bridges," Papers 1505.03030, arXiv.org.
    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. Devroye, Luc, 2009. "On exact simulation algorithms for some distributions related to Jacobi theta functions," Statistics & Probability Letters, Elsevier, vol. 79(21), pages 2251-2259, November.
    4. Alexandre Bouchard-Côté & Sebastian J. Vollmer & Arnaud Doucet, 2018. "The Bouncy Particle Sampler: A Nonreversible Rejection-Free Markov Chain Monte Carlo Method," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(522), pages 855-867, April.
    5. Alexandros Beskos & Omiros Papaspiliopoulos & Gareth O. Roberts, 2008. "A Factorisation of Diffusion Measure and Finite Sample Path Constructions," Methodology and Computing in Applied Probability, Springer, vol. 10(1), pages 85-104, March.
    6. Burq, Zaeem A. & Jones, Owen D., 2008. "Simulation of Brownian motion at first-passage times," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 77(1), pages 64-71.
    7. Bierkens, Joris & Bouchard-Côté, Alexandre & Doucet, Arnaud & Duncan, Andrew B. & Fearnhead, Paul & Lienart, Thibaut & Roberts, Gareth & Vollmer, Sebastian J., 2018. "Piecewise deterministic Markov processes for scalable Monte Carlo on restricted domains," Statistics & Probability Letters, Elsevier, vol. 136(C), pages 148-154.
    8. Alexandros Beskos & Omiros Papaspiliopoulos & Gareth O. Roberts & Paul Fearnhead, 2006. "Exact and computationally efficient likelihood‐based estimation for discretely observed diffusion processes (with discussion)," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(3), pages 333-382, June.
    9. Medina-Aguayo, Felipe & Rudolf, Daniel & Schweizer, Nikolaus, 2020. "Perturbation bounds for Monte Carlo within Metropolis via restricted approximations," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 2200-2227.
    10. Johansen, Adam M. & Doucet, Arnaud, 2008. "A note on auxiliary particle filters," Statistics & Probability Letters, Elsevier, vol. 78(12), pages 1498-1504, September.
    11. Cheng Li & Sanvesh Srivastava & David B. Dunson, 2017. "Simple, scalable and accurate posterior interval estimation," Biometrika, Biometrika Trust, vol. 104(3), pages 665-680.
    12. Martin, Andrew D. & Quinn, Kevin M. & Park, Jong Hee, 2011. "MCMCpack: Markov Chain Monte Carlo in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 42(i09).
    13. Pierre E. Jacob & John O’Leary & Yves F. Atchadé, 2020. "Unbiased Markov chain Monte Carlo methods with couplings," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(3), pages 543-600, July.
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    Cited by:

    1. Gael M. Martin & David T. Frazier & Christian P. Robert, 2022. "Computing Bayes: From Then `Til Now," Monash Econometrics and Business Statistics Working Papers 14/22, Monash University, Department of Econometrics and Business Statistics.

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