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Particle-based likelihood inference in partially observed diffusion processes using generalised Poisson estimators. (English) Zbl 1274.62564

Summary: This paper concerns the use of the expectation-maximisation (EM) algorithm for inference in partially observed diffusion processes. In this context, a well known problem is that all except a few diffusion processes lack closed-form expressions of the transition densities. Thus, in order to estimate efficiently the EM intermediate quantity we construct, using novel techniques for unbiased estimation of diffusion transition densities, a random weight fixed-lag auxiliary particle smoother, which avoids the well known problem of particle trajectory degeneracy in the smoothing mode. The estimator is justified theoretically and demonstrated on a simulated example.

MSC:

62M09 Non-Markovian processes: estimation
65C05 Monte Carlo methods
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References:

[1] Aït-Sahalia, Y. (2008). Closed-form likelihood expansions for multivariate diffusions., Ann. Statist. , 36(2):906-937. · Zbl 1246.62180 · doi:10.1214/009053607000000622
[2] Ball, F. G. and Rice, J. H. (1992). Stochastic models for ion channels: Introduction and bibliography., Math. Biosci. , 112:189-206. · Zbl 0767.92007 · doi:10.1016/0025-5564(92)90023-P
[3] Beskos, A., Papaspiliopoulos, O., and Roberts, G. (2008). A factorisation of diffusion measure and finite sample path constructions., Methodology and Computing in Applied Probability , 10(1):85-104. · Zbl 1152.65013 · doi:10.1007/s11009-007-9060-4
[4] Beskos, A., Papaspiliopoulos, O., Roberts, G., and Fearnhead, P. (2006). Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes., J. Roy. Statist. Soc. Ser. B , 68(3):333-382. With discussion. · Zbl 1100.62079 · doi:10.1111/j.1467-9868.2006.00552.x
[5] Cappé, O., Moulines, E., and Rydén, T. (2005)., Inference in Hidden Markov Models . Springer. · Zbl 1080.62065
[6] Churchill, G. (1992). Hidden Markov chains and the analysis of genome structure., Computers & Chemistry , 16(2):107-115. · Zbl 0752.92015
[7] Del Moral, P., Jacod, J., and Protter, P. (2001). The Monte-Carlo method for filtering with discrete-time observations., Probability Theory and Related Fields , 120:346-368. · Zbl 0979.62072 · doi:10.1007/s004400100128
[8] Dempster, A., Laird, N., and Rubin, D. (1977). Maximum likelihood from incomplete data via the EM algorithm., J. Roy. Statist. Soc. Ser. B , 39:1-38. · Zbl 0364.62022
[9] Douc, R., Fort, G., Moulines, E., and Priouret, P. (2009). Forgetting the initial distribution for hidden markov models., Stoch. Process. Appl. , 119(4):1235-1256. · Zbl 1159.93357 · doi:10.1016/j.spa.2008.05.007
[10] Douc, R. and Moulines, E. (2008). Limit theorems for weighted samples with applications to sequential Monte Carlo methods., Ann. Statist. , 36(5):2344-2376. · Zbl 1155.62056 · doi:10.1214/07-AOS514
[11] Douc, R., Moulines, É., and Olsson, J. (2008). Optimality of the auxiliary particle filter., Probab. Math. Statist. , 29(1):1-28. · Zbl 1176.62092
[12] Fearnhead, P., Papaspiliopoulos, O., and Roberts, G. (2008). Particle filters for partially observed diffusions., J. Roy. Statist. Soc. Ser. B , 70(4):755-777. · Zbl 05563368 · doi:10.1111/j.1467-9868.2008.00661.x
[13] Fort, G. and Moulines, E. (2003). Convergence of the Monte Carlo expectation maximization for curved exponential families., Ann. Statist. , 31(4):1220-1259. · Zbl 1043.62015 · doi:10.1214/aos/1059655912
[14] Gordon, N., Salmond, D., and Smith, A. (1993). Novel approach to nonlinear/non-gaussian bayesian state estimation., IEE Proc. F, Radar signal Process. , 140:107-113.
[15] Handschin, J. and Mayne, D. (1969). Monte carlo techniques to estimate the conditional expectation in multi-stage non-linear filtering., Int. J. Control , 9:547-559. · Zbl 0174.51201 · doi:10.1080/00207176908905777
[16] Hürzeler, M. and Künsch, H. R. (2001). Approximating and maximising the likelihood for a general state-space model. In Doucet, A., de Freitas, N., and Gordon, N., editors, Sequential Monte Carlo Methods in Practice , pages 159-175. Springer. · Zbl 1056.93580
[17] Ionides, E. L., Bhadra, A., Atchadé, Y., and King, A. A. (2011). Iterated filtering., Ann. Statist. , 39(3):1776-1802. · Zbl 1220.62103 · doi:10.1214/11-AOS886
[18] Kitigawa, G. (1998). A self-organizing state-space-model., J. Am. Statist. Assoc. , 93(443):1203-1215.
[19] Kloeden, P. E. and Platen, E. (1992)., Numerical Solution of Stochastic Differential Equations . Springer. · Zbl 0752.60043
[20] Liu, J. and Chen, R. (1995). Blind deconvolution via sequential imputations., J. Am. Statist. Assoc. , 90(420):567-576. · Zbl 0826.62062 · doi:10.2307/2291068
[21] Olsson, J., Cappé, O., Douc, R., and Moulines, E. (2008). Sequential Monte Carlo smoothing with application to parameter estimation in nonlinear state space models., Bernoulli , 14(1):155-179. · Zbl 1155.62055 · doi:10.3150/07-BEJ6150
[22] Olsson, J. and Rydén, T. (2008). Asymptotic properties of the bootstrap particle filter maximum likelihood estimator for state space models., Stoch. Process. Appl. , 118:649-680. · Zbl 1132.62064 · doi:10.1016/j.spa.2007.05.007
[23] Olsson, J. and Ströjby, J. (2010). Convergence of random weight particle filters. Appears in J. Ströjby’s PhD thesis, On Inference in Partially Observed Markov Models using Sequential Monte Carlo Methods , Centre for Mathematical Sciences, Lund University, 2010.
[24] Pedersen, A. R. (1995). Consistency and Asymptotic Normality of an Approximative Maximum Likelihood Estimator for Discretely Observed Diffusion Processes., Bernoulli , 1(3):257-279. · Zbl 0839.62079 · doi:10.2307/3318480
[25] Pitt, M. and Shephard, N. (1999). Filtering via simulation: Auxiliary particle filters., J. Am. Statist. Assoc. , 87:493-499. · Zbl 1072.62639 · doi:10.1080/01621459.1999.10474153
[26] Rabiner, L. R. and Juang, B.-H. (1993)., Fundamentals of Speech Recognition . Prentice-Hall.
[27] Ristic, B., Arulampalam, M., and Gordon, A. (2004)., Beyond Kalman Filters: Particle Filters for Target Tracking . Artech House. · Zbl 1092.93041
[28] Wu, C. (1983). On the convergence properties of the EM algorithm., Ann. Statist. , 11:95-103. · Zbl 0517.62035 · doi:10.1214/aos/1176346060
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