×

Twisted particle filters. (English) Zbl 1302.60139

The authors introduce the so-called hidden Markov model (HMM) which operates as a Markov chain on a measurable space \(X\) and yields values on a suitable observation space \(Y\); for each \(X_n\), the value \(Y_n\) is conditionally independent on the rest of the process, given \(X_n\). The model is considered for the \(N\)-particle sytems. The simplest particle filter, known as the bootstrap algorithm, is a departure point for further introduction and discussion of more elaborate filters, with a focus on the \(N\rightarrow \infty \) regime. The main purpose of the present paper is to rigorously address comparative issues of how and why one algorithm may outperform another, and how it is possible to modify standard algorithms in order to improve performance. To this end, the twisted particle filters are introduced and validated by means of a traditional asymptotic analysis, in the regime where the number of particles tends to infinity.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
62M20 Inference from stochastic processes and prediction
60G35 Signal detection and filtering (aspects of stochastic processes)
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid

References:

[1] Andrieu, C., Doucet, A. and Holenstein, R. (2010). Particle Markov chain Monte Carlo methods. J. R. Stat. Soc. Ser. B Stat. Methodol. 72 269-342. · Zbl 1184.65001
[2] Andrieu, C. and Roberts, G. O. (2009). The pseudo-marginal approach for efficient Monte Carlo computations. Ann. Statist. 37 697-725. · Zbl 1185.60083
[3] Andrieu, C. and Vihola, M. (2012). Convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms. Available at [math.PR]. · Zbl 1326.65012
[4] Athreya, K. B. (2000). Change of measures for Markov chains and the \(L\log L\) theorem for branching processes. Bernoulli 6 323-338. · Zbl 0969.60076
[5] Bucklew, J. A., Ney, P. and Sadowsky, J. S. (1990). Monte Carlo simulation and large deviations theory for uniformly recurrent Markov chains. J. Appl. Probab. 27 44-59. · Zbl 0702.60028
[6] Cérou, F., Del Moral, P. and Guyader, A. (2011). A nonasymptotic theorem for unnormalized Feynman-Kac particle models. Ann. Inst. Henri Poincaré Probab. Stat. 47 629-649. · Zbl 1233.60047
[7] Chopin, N. (2004). Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference. Ann. Statist. 32 2385-2411. · Zbl 1079.65006
[8] Del Moral, P. (2004). Feynman-Kac Formulae : Genealogical and Interacting Particle Systems with Applications . Springer, New York. · Zbl 1130.60003
[9] Del Moral, P. and Guionnet, A. (1999). Central limit theorem for nonlinear filtering and interacting particle systems. Ann. Appl. Probab. 9 275-297. · Zbl 0938.60022
[10] Del Moral, P. and Guionnet, A. (2001). On the stability of interacting processes with applications to filtering and genetic algorithms. Ann. Inst. Henri Poincaré Probab. Stat. 37 155-194. · Zbl 0990.60005
[11] Del Moral, P. and Jacod, J. (2001). Interacting particle filtering with discrete time observations: Asymptotic behaviour in the Gaussian case. In Stochastics in Finite and Infinite Dimensions : In Honor of Gopinath Kallianpur (T. Hida, R. L. Karandikar, H. Kunita, B. S. Rajput, S. Watanabe and J. Xiong, eds.). Birkhäuser, Basel. · Zbl 1056.93574
[12] Doob, J. L. (1994). Measure Theory. Graduate Texts in Mathematics 143 . Springer, New York. · Zbl 0791.28001
[13] Douc, R. and Moulines, E. (2008). Limit theorems for weighted samples with applications to sequential Monte Carlo methods. Ann. Statist. 36 2344-2376. · Zbl 1155.62056
[14] Douc, R. and Moulines, E. (2011). Asymptotic properties of the maximum likelihood estimation in misspecified Hidden Markov models. Available at . · Zbl 1373.62436
[15] Douc, R., Moulines, É. and Olsson, J. (2009). Optimality of the auxiliary particle filter. Probab. Math. Statist. 29 1-28. · Zbl 1176.62092
[16] Douc, R., Moulines, E. and Olsson, J. (2012). Long-term stability of sequential Monte Carlo methods under verifiable conditions. Available at . · Zbl 1429.62364
[17] Doucet, A., Briers, M. and Sénécal, S. (2006). Efficient block sampling strategies for sequential Monte Carlo methods. J. Comput. Graph. Statist. 15 693-711.
[18] Favetto, B. (2012). On the asymptotic variance in the central limit theorem for particle filters. ESAIM Probab. Stat. 16 151-164. · Zbl 1273.60046
[19] Gordon, N. J., Salmond, D. J. and Smith, A. F. M. (1993). Novel approach to nonlinear/non-Gaussian Bayesian state estimation. Radar and Signal Processing , IEE Proceedings F 140 107-113.
[20] Johansen, A. M. and Doucet, A. (2008). A note on auxiliary particle filters. Statist. Probab. Lett. 78 1498-1504. · Zbl 1152.62066
[21] Kifer, Y. (1996). Perron-Frobenius theorem, large deviations, and random perturbations in random environments. Math. Z. 222 677-698. · Zbl 0863.60062
[22] Künsch, H. R. (2005). Recursive Monte Carlo filters: Algorithms and theoretical analysis. Ann. Statist. 33 1983-2021. · Zbl 1086.62106
[23] Le Gland, F. and Oudjane, N. (2004). Stability and uniform approximation of nonlinear filters using the Hilbert metric and application to particle filters. Ann. Appl. Probab. 14 144-187. · Zbl 1060.93094
[24] LeGland, F. and Oudjane, N. (2003). A robustification approach to stability and to uniform particle approximation of nonlinear filters: The example of pseudo-mixing signals. Stochastic Process. Appl. 106 279-316. · Zbl 1075.93541
[25] Leroux, B. G. (1992). Maximum-likelihood estimation for hidden Markov models. Stochastic Process. Appl. 40 127-143. · Zbl 0738.62081
[26] Oudjane, N. and Rubenthaler, S. (2005). Stability and uniform particle approximation of nonlinear filters in case of non ergodic signals. Stoch. Anal. Appl. 23 421-448. · Zbl 1140.93485
[27] Pitt, M. K. and Shephard, N. (1999). Filtering via simulation: Auxiliary particle filters. J. Amer. Statist. Assoc. 94 590-599. · Zbl 1072.62639
[28] van Handel, R. (2009). Uniform time average consistency of Monte Carlo particle filters. Stochastic Process. Appl. 119 3835-3861. · Zbl 1176.93076
[29] Whiteley, N. (2013). Stability properties of some particle filters. Ann. Appl. Probab. 23 2500-2537. · Zbl 1296.60098
[30] Whiteley, N. and Lee, A. (2014). Supplement to “Twisted particle filters.” . · Zbl 1302.60139
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.