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Sequential quasi Monte Carlo

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  • Mathieu Gerber
  • Nicolas Chopin

Abstract

type="main" xml:id="rssb12104-abs-0001"> We derive and study sequential quasi Monte Carlo (SQMC), a class of algorithms obtained by introducing QMC point sets in particle filtering. SQMC is related to, and may be seen as an extension of, the array-RQMC algorithm of L'Ecuyer and his colleagues. The complexity of SQMC is O { N log ( N ) } , where N is the number of simulations at each iteration, and its error rate is smaller than the Monte Carlo rate O P ( N − 1 / 2 ) . The only requirement to implement SQMC algorithms is the ability to write the simulation of particle x t n given x t − 1 n as a deterministic function of x t − 1 n and a fixed number of uniform variates. We show that SQMC is amenable to the same extensions as standard SMC, such as forward smoothing, backward smoothing and unbiased likelihood evaluation. In particular, SQMC may replace SMC within a particle Markov chain Monte Carlo algorithm. We establish several convergence results. We provide numerical evidence that SQMC may significantly outperform SMC in practical scenarios.

Suggested Citation

  • Mathieu Gerber & Nicolas Chopin, 2015. "Sequential quasi Monte Carlo," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 77(3), pages 509-579, June.
    • Handle: RePEc:bla:jorssb:v:77:y:2015:i:3:p:509-579
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    File URL: http://hdl.handle.net/10.1111/rssb.2015.77.issue-3
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    Citations

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    Cited by:

    1. Dick, Josef & Rudolf, Daniel & Zhu, Houying, 2019. "A weighted discrepancy bound of quasi-Monte Carlo importance sampling," Statistics & Probability Letters, Elsevier, vol. 149(C), pages 100-106.
    2. Z. I. Botev, 2017. "The normal law under linear restrictions: simulation and estimation via minimax tilting," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(1), pages 125-148, January.
    3. Mathieu GERBER & Nicolas CHOPIN & Nick WHITELEY, 2017. "Negative association, ordering and convergence of resampling methods," Working Papers 2017-36, Center for Research in Economics and Statistics.
    4. Chopin, Nicolas & Gadat, Sébastien & Guedj, Benjamin & Guyader, Arnaud & Vernet, Elodie, 2015. "On some recent advances in high dimensional Bayesian Statistics," TSE Working Papers 15-557, Toulouse School of Economics (TSE).
    5. Duan, Jin-Chuan & Fulop, Andras & Hsieh, Yu-Wei, 2020. "Data-cloning SMC2: A global optimizer for maximum likelihood estimation of latent variable models," Computational Statistics & Data Analysis, Elsevier, vol. 143(C).
    6. Karol Gellert & Erik Schlögl, 2021. "Parameter Learning and Change Detection Using a Particle Filter with Accelerated Adaptation," Risks, MDPI, vol. 9(12), pages 1-18, December.
    7. 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.
    8. Jean-Jacques Forneron, 2019. "A Scrambled Method of Moments," Papers 1911.09128, arXiv.org.
    9. Crucinio, Francesca R. & Johansen, Adam M., 2023. "Properties of marginal sequential Monte Carlo methods," Statistics & Probability Letters, Elsevier, vol. 203(C).
    10. Nicolas Chopin & Mathieu Gerber, 2017. "Sequential quasi-Monte Carlo: Introduction for Non-Experts, Dimension Reduction, Application to Partly Observed Diffusion Processes," Working Papers 2017-35, Center for Research in Economics and Statistics.
    11. 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.
    12. Zhijian He & Art B. Owen, 2016. "Extensible grids: uniform sampling on a space filling curve," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(4), pages 917-931, September.
    13. Fabian Goessling, 2018. "Human Capital, Growth, and Asset Prices," CQE Working Papers 6918, Center for Quantitative Economics (CQE), University of Muenster.
    14. Fabian Goessling, 2018. "Randomized Quasi Sequential Markov Chain Monte Carlo²," CQE Working Papers 7018, Center for Quantitative Economics (CQE), University of Muenster.
    15. Burkhart, Michael C., 2019. "A Discriminative Approach to Bayesian Filtering with Applications to Human Neural Decoding," Thesis Commons 4j3fu, Center for Open Science.
    16. Karamé, Frédéric, 2018. "A new particle filtering approach to estimate stochastic volatility models with Markov-switching," Econometrics and Statistics, Elsevier, vol. 8(C), pages 204-230.
    17. Gael M. Martin & David T. Frazier & Christian P. Robert, 2020. "Computing Bayes: Bayesian Computation from 1763 to the 21st Century," Monash Econometrics and Business Statistics Working Papers 14/20, Monash University, Department of Econometrics and Business Statistics.
    18. Gael M. Martin & David T. Frazier & Christian P. Robert, 2021. "Approximating Bayes in the 21st Century," Monash Econometrics and Business Statistics Working Papers 24/21, Monash University, Department of Econometrics and Business Statistics.
    19. L’Ecuyer, Pierre & Munger, David & Lécot, Christian & Tuffin, Bruno, 2018. "Sorting methods and convergence rates for Array-RQMC: Some empirical comparisons," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 143(C), pages 191-201.
    20. Bhattacharya, Arnab & Wilson, Simon P., 2018. "Sequential Bayesian inference for static parameters in dynamic state space models," Computational Statistics & Data Analysis, Elsevier, vol. 127(C), pages 187-203.
    21. Golightly, Andrew & Bradley, Emma & Lowe, Tom & Gillespie, Colin S., 2019. "Correlated pseudo-marginal schemes for time-discretised stochastic kinetic models," Computational Statistics & Data Analysis, Elsevier, vol. 136(C), pages 92-107.
    22. Dellaportas, Petros & Titsias, Michalis K. & Petrova, Katerina & Plataniotis, Anastasios, 2023. "Scalable inference for a full multivariate stochastic volatility model," Journal of Econometrics, Elsevier, vol. 232(2), pages 501-520.
    23. Fulop, Andras & Li, Junye, 2019. "Bayesian estimation of dynamic asset pricing models with informative observations," Journal of Econometrics, Elsevier, vol. 209(1), pages 114-138.

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