Open Access
November 2017 Convergence of sequential quasi-Monte Carlo smoothing algorithms
Mathieu Gerber, Nicolas Chopin
Bernoulli 23(4B): 2951-2987 (November 2017). DOI: 10.3150/16-BEJ834


Gerber and Chopin [J. R. Stat. Soc. Ser. B. Stat. Methodol. 77 (2015) 509–579] recently introduced Sequential quasi-Monte Carlo (SQMC) algorithms as an efficient way to perform filtering in state–space models. The basic idea is to replace random variables with low-discrepancy point sets, so as to obtain faster convergence than with standard particle filtering. Gerber and Chopin (2015) describe briefly several ways to extend SQMC to smoothing, but do not provide supporting theory for this extension. We discuss more thoroughly how smoothing may be performed within SQMC, and derive convergence results for the so-obtained smoothing algorithms. We consider in particular SQMC equivalents of forward smoothing and forward filtering backward sampling, which are the most well-known smoothing techniques. As a preliminary step, we provide a generalization of the classical result of Hlawka and Mück [Computing (Arch. Elektron. Rechnen) 9 (1972) 127–138] on the transformation of QMC point sets into low discrepancy point sets with respect to non uniform distributions. As a corollary of the latter, we note that we can slightly weaken the assumptions to prove the consistency of SQMC.


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Mathieu Gerber. Nicolas Chopin. "Convergence of sequential quasi-Monte Carlo smoothing algorithms." Bernoulli 23 (4B) 2951 - 2987, November 2017.


Received: 1 June 2015; Revised: 1 February 2016; Published: November 2017
First available in Project Euclid: 23 May 2017

zbMATH: 1382.65010
MathSciNet: MR3654796
Digital Object Identifier: 10.3150/16-BEJ834

Keywords: Hidden Markov models , low discrepancy , particle filtering , quasi-Monte Carlo , sequential quasi-Monte Carlo , smoothing , state–space models

Rights: Copyright © 2017 Bernoulli Society for Mathematical Statistics and Probability

Vol.23 • No. 4B • November 2017
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