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Central limit theorems for coupled particle filters

Published online by Cambridge University Press:  24 September 2020

Ajay Jasra
Affiliation:
King Abdullah University of Science & Technology
Fangyuan Yu
Affiliation:
King Abdullah University of Science & Technology

Abstract

In this article we prove new central limit theorems (CLTs) for several coupled particle filters (CPFs). CPFs are used for the sequential estimation of the difference of expectations with respect to filters which are in some sense close. Examples include the estimation of the filtering distribution associated to different parameters (finite difference estimation) and filters associated to partially observed discretized diffusion processes (PODDP) and the implementation of the multilevel Monte Carlo (MLMC) identity. We develop new theory for CPFs, and based upon several results, we propose a new CPF which approximates the maximal coupling (MCPF) of a pair of predictor distributions. In the context of ML estimation associated to PODDP with time-discretization $\Delta_l=2^{-l}$ , $l\in\{0,1,\dots\}$ , we show that the MCPF and the approach of Jasra, Ballesio, et al. (2018) have, under certain assumptions, an asymptotic variance that is bounded above by an expression that is of (almost) the order of $\Delta_l$ ( $\mathcal{O}(\Delta_l)$ ), uniformly in time. The $\mathcal{O}(\Delta_l)$ bound preserves the so-called forward rate of the diffusion in some scenarios, which is not the case for the CPF in Jasra et al. (2017).

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Original Article
Copyright
© Applied Probability Trust 2020

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References

Bally, V. and Talay, D. (1996). The law of the Euler scheme for stochastic differential equations II: Approximation of the density. Monte Carlo Meth. Appl. 2, 93128.CrossRefGoogle Scholar
Beskos, A. et al. (2017). Multilevel sequential Monte Carlo samplers. Stoch. Process. Appl. 127, 14171440.CrossRefGoogle Scholar
Billingsley, P. (1995). Probability and Measure. Wiley, New York.Google Scholar
Cappe, O., Moulines, E. and Ryden, T. (2005). Inference in Hidden Markov Models. Springer, New York.10.1007/0-387-28982-8CrossRefGoogle Scholar
Chopin, N. (2004). Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference. Ann. Statist. 32, 23852411.CrossRefGoogle Scholar
Chopin, N. and Singh, S. S. (2015). On particle Gibbs sampling. Bernoulli 21, 18551883.10.3150/14-BEJ629CrossRefGoogle Scholar
Crisan, D. and Bain, A. (2008). Fundamentals of Stochastic Filtering. Springer, New York.Google Scholar
Del Moral, P. (2004). Feynman–Kac Formulae: Genealogical and Interacting Particle Systems with Applications. Springer, New York.CrossRefGoogle Scholar
Del Moral, P. and Miclo, L. (2000). Branching and interacting particle systems approximation of Feynman–Kac formulae with applications to non-linear filtering. In Séminaire de Probabilités XXXIV (Lecture Notes in Mathematics 1729), eds. J. Azéma, M. Émery, M. Ledoux, and M. Yor, Springer, Berlin, pp. 1145.CrossRefGoogle Scholar
Del Moral, P., Doucet, A. and Jasra, A. (2012). On adaptive resampling procedures for sequential Monte Carlo methods. Bernoulli 18, 252272.CrossRefGoogle Scholar
Del Moral, P., Jacod, J. and Protter, P. (2001). The Monte Carlo method for filtering with discrete-time observations. Prob. Theory Relat. Fields 120, 346368.10.1007/PL00008786CrossRefGoogle Scholar
Del Moral, P., Jasra, A. and Law, K. J. H. (2017). Multilevel sequential Monte Carlo: Mean square error bounds under verifiable conditions. Stoch. Anal. 35, 478498.CrossRefGoogle Scholar
Giles, M. B. (2008). Multilevel Monte Carlo path simulation. Operat. Res. 56, 607617.CrossRefGoogle Scholar
Gregory, A., Cotter, C. and Reich, S. (2016). Multilevel ensemble transform particle filtering. SIAM J. Sci. Comput. 38, A1317A1338.CrossRefGoogle Scholar
Heinrich, S. (2001). Multilevel Monte Carlo methods. In Large-Scale Scientific Computing, eds. S. Margenov, J. Wasniewski, and P. Yalamov, Springer, Berlin, pp. 5867.CrossRefGoogle Scholar
Jacob, P., Lindsten, F. and Schön, T. (2016). Coupling of particle filters. Preprint. Available at https://arxiv.org/abs/1606.01156.Google Scholar
Jacob, P., Lindsten, F. and Schön, T. (2019). Smoothing with couplings of conditional particle filters. To appear in J. Amer. Statist. Assoc.Google Scholar
Jacob, P., O’Leary, J. and Atachde, Y. (2020). Unbiased Markov chain Monte Carlo with couplings. To appear in J. R. Statist. Soc. B.CrossRefGoogle Scholar
Jasra, A. (2015). On the behaviour of the backward interpretation of Feynman–Kac formulae under verifiable conditions. J. Appl. Prob. 52, 339359.CrossRefGoogle Scholar
Jasra, A., Ballesio, M., von Schwerin, E. and Tempone, R. (2020). A Wasserstein coupled particle filter for multilevel estimation. Preprint. Available at https://arxiv.org/abs/2004.0398.Google Scholar
Jasra, A., Kamatani, K., Osei, P. P. and Zhou, Y. (2018). Multilevel particle filters: Normalizing constant estimation. Statist. Comput. 28, 4760.CrossRefGoogle Scholar
Jasra, A., Kamatani, K., Law, K. J. H. and Zhou, Y. (2017). Multilevel particle filters. SIAM J. Numer. Anal. 55, 30683096.CrossRefGoogle Scholar
Lee, A., Singh, S. S. and Vihola, M. (2019). Coupled conditional backward sampling particle filter. To appear in Ann. Statist.Google Scholar
Lo, A. (2017). Functional generalizations of Hoeffding’s covariance lemma and a formula for Kendall’s tau. Statist. Prob. Lett. 122, 218226.CrossRefGoogle Scholar
Mao, X. (2007). Stochastic Differential Equations and Applications, 2nd edn. Woodhead, Cambridge.Google Scholar
McDiarmid, C. (1989). On the method of bounded differences. In Surveys in Combinatorics (London Math. Soc. Lecture Notes 141), ed. J. Siemons, Cambridge University Press, pp. 148188.Google Scholar
Rachev, T. and Rüschendorf, S. (1998). Mass Transportation Problems, Volume 1: Theory. Springer, New York.Google Scholar
Sen, D., Thiery, A. and Jasra, A. (2018). On coupling particle filters. Statist. Comput. 28, 461475.CrossRefGoogle Scholar
Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York.10.1007/978-1-4612-1236-2CrossRefGoogle Scholar
Whiteley, N. P. (2013). Stability properties of some particle filters. Ann. Appl. Prob. 23, 25002537.CrossRefGoogle Scholar
Whiteley, N. P., Kantas, N. and Jasra, A. (2012). Linear variance bounds for particle approximations of time homogenous Feynman–Kac formulae. Stoch. Process. Appl. 122, 18401865.CrossRefGoogle Scholar

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