Chan, Hock Peng; Lai, Tze Leung A general theory of particle filters in hidden Markov models and some applications. (English) Zbl 1293.60071 Ann. Stat. 41, No. 6, 2877-2904 (2013). Let \({\text{X}} = \left\{ {{X_t}: t \geqslant 1} \right\}\) be a Markov chain and let \({\text{Y}} = \left\{ {{Y_t}: t \geqslant 1} \right\}\) be conditionally independent given X, such that \({X_t} \sim {p_t}\left( { \cdot \left| {{X_{t - 1}}} \right.} \right),{Y_t} \sim {g_t}\left( { \cdot \left| {{X_t}} \right.} \right)\), where \({p_t}\) and \({g_t}\) are density functions with respect to some measures \({\nu _X}\) and \({\nu _Y}\), and \({p_1}\left( { \cdot \left| {{X_0}} \right.} \right)\) denotes the initial density \({p_1}\left( \cdot \right)\) of \({X_1}\). Letting \({{\text{X}}_t} = ({X_1},\dots,{X_t})\), \({{\text{Y}}_t} = ({Y_1},\dots,{Y_t})\), and \(\psi \) be a measurable real-valued function of \({{\text{X}}_t}\), the paper considers estimation of \({\psi _T}: = \operatorname{E}[\psi ({{\text{X}}_T}){\left| {\text{Y}} \right._T}]\). The density function of \({{\text{X}}_T}\) conditional on \({{\text{Y}}_T}\) is often difficult to sample from and the normalizing constant is also to difficult to compute. Particle filters that use sequential Monte Carlo (SMC) methods involving importance sampling and resampling have been developed to circumvent this difficulty. The paper provides a comprehensive theory of the SMC estimate \({\hat \psi _T}\), which includes asymptotic normality and consistent standard error estimation. Reviewer: Oleg K. Zakusilo (Kyïv) Cited in 1 ReviewCited in 21 Documents MSC: 60J22 Computational methods in Markov chains 60F10 Large deviations 65C05 Monte Carlo methods 65C40 Numerical analysis or methods applied to Markov chains 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B43 Percolation 82C43 Time-dependent percolation in statistical mechanics Keywords:hidden Markov models; particle filter; importance sampling; resampling; standard error PDFBibTeX XMLCite \textit{H. P. Chan} and \textit{T. L. Lai}, Ann. Stat. 41, No. 6, 2877--2904 (2013; Zbl 1293.60071) 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] Baker, J. E. (1985). Adaptive selection methods for genetic algorithms. In Proc. International Conference on Genetic Algorithms and Their Applications (J. Grefenstette, ed.) 101-111. 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