## On Sequential Monte Carlo Sampling Methods for Bayesian Filtering (2000)

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Venue: | STATISTICS AND COMPUTING |

Citations: | 735 - 66 self |

### BibTeX

@ARTICLE{Doucet00onsequential,

author = {Arnaud Doucet and Simon Godsill and Christophe Andrieu},

title = {On Sequential Monte Carlo Sampling Methods for Bayesian Filtering},

journal = {STATISTICS AND COMPUTING},

year = {2000},

volume = {10},

number = {3},

pages = {197--208}

}

### Years of Citing Articles

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### Abstract

In this article, we present an overview of methods for sequential simulation from posterior distributions. These methods are of particular interest in Bayesian filtering for discrete time dynamic models that are typically nonlinear and non-Gaussian. A general importance sampling framework is developed that unifies many of the methods which have been proposed over the last few decades in several different scientific disciplines. Novel extensions to the existing methods are also proposed. We show in particular how to incorporate local linearisation methods similar to those which have previously been employed in the determin-istic filtering literature; these lead to very effective importance distributions. Furthermore we describe a method which uses Rao-Blackwellisation in order to take advantage of the analytic structure present in some important classes of state-space models. In a final section we develop algorithms for prediction, smoothing and evaluation of the likelihood in dynamic models.

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(Show Context)
Citation Context ...ods have been proposed to approximate the importance function and the associated importance weight based on importance sampling (Doucet 1997, 1998) and Markov chain Monte Carlo (Berzuini et al. 1998, =-=Liu and Chen 1998-=-). These iterative algorithms are computationally intensive and there is a lack of theoretical convergence results. However, they may be useful when non-iterative schemes fail. In fact, the general fr... |

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Citation Context ...s below a fixed threshold Nthres, the SIR resampling procedure is used (Rubin 1988). Note that it is possible to implement the SIR procedure exactly in O(N) operations by using a classical algorithm (=-=Ripley 1987-=- p. 96) and Carpenter, Clifford and Fearnhead (1997), Doucet (1997, 1998) and Pitt and Shepherd (1999). Other resampling procedures which reduce the MC variation, such as stratified sampling (Carpente... |

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Citation Context ...normalised importance function π(x0:n | y0:n) which has a support including that of the state posterior. Then an estimate �IN ( fn) of the posterior expectation I ( fn) is obtained using Bayesian I=-=S (Geweke 1989): �IN ( fn) = N�-=-�� i=1 � (i) fn x 0:n � (i) ˜w n , ˜w(i) n = w ∗(i) n �N j) j=1 w∗( n where w ∗(i) n = p(y0:n | x0:n)p(x0:n)/π(x0:n | y0:n) is the unnormalised importance weight. Under weak assumptions... |

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Citation Context ... One cannot evaluate Neff exactly but, an estimate �Neff of Neff is given by: �Neff = � N i=1 1 � ˜w (i) k � 2 (35) When �Neff is below a fixed threshold Nthres, the SIR resampling proced=-=ure is used (Rubin 1988-=-). Note that it is possible to implement the SIR procedure exactly in O(N) operations by using a classical algorithm (Ripley 1987 p. 96) and Carpenter, Clifford and Fearnhead (1997), Doucet (1997, 199... |

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Citation Context ...arlo algorithms have been largely neglected until recent years. In the late 80’s, massive increases in computational power allowed the rebirth of numerical integration methods for Bayesian filtering=-= (Kitagawa 1987-=-). Current research has now focused on MC integration methods, which have the great advantage of not being subject to the assumption of linearity or Gaussianity in the model, and relevants198 Doucet, ... |

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Citation Context ...ssary to build these on a case by case basis, dependent on the model studied. To this end, it is possible to base these developments on previous work in suboptimal filtering (Anderson and Moore 1979, =-=West and Harrison 1997), -=-and this is considered in the next subsection. 2. Importance distribution obtained by local linearisation A simple choice selects as the importance function π(xk | xk−1, yk) a parametric distributi... |

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Citation Context ...and Fearnhead (1997), Doucet (1997, 1998) and Pitt and Shepherd (1999). Other resampling procedures which reduce the MC variation, such as stratified sampling (Carpenter, Clifford and Fearnhead 1997, =-=Kitagawa and Gersch 1996-=-) and residual resampling (Higuchi 1997, Liu and Chen 1998), may be applied as an alternative to SIR.s202 Doucet, Godsill and Andrieu An appropriate algorithm based on the SIR scheme proceeds as follo... |

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Citation Context ...blem or the optimal filtering problem. Practical applications include target tracking (Gordon, Salmond and Smith 1993), blind deconvolution of digital communications channels (Clapp and Godsill 1999, =-=Liu and Chen 1995),-=- estimation of stochastic volatility (Pitt and Shephard 0960-3174 C○ 2000 Kluwer Academic Publishers 1999) and digital enhancement of speech and audio signals (Godsill and Rayner 1998). Except in a ... |

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Citation Context ... approximating these filtering distributions, see for example Jazwinski (1970). The most popular algorithms, the extended Kalman filter and the Gaussian sum filter, rely on analytical approximations (=-=Anderson and Moore 1979). I-=-nteresting work in the automatic control field was carried out during the 1960’s and 70’s using sequential Monte Carlo (MC) integration methods, see Akashi and Kumamoto (1975), Handschin and Mayne... |

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Citation Context ...on methods which are designed to make the most of any structure within the model studied. Numerous methods have been developed for reducing the variance of MC estimates including antithetic sampling (=-=Handschin and Mayne 1969-=-, Handschin 1970) and control variates (Akashi and Kumamoto 1975, Handschin 1970). We apply here the Rao-Blackwellisation method, see Casella and Robert (1996) for a general reference on the topic. In... |

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Citation Context ..., conditional upon x1 0:n , x20:n is a linear Gaussian state space model and the integrations required by the Rao-Blackwellisation method can be realized using the Kalman filter. Akashi and Kumamoto (=-=Akashi and Kumamoto 1977-=-, Tugnait 1982) introduced this algorithm under the name of RSA (Random Sampling Algorithm) in the particular case where x1 k is a homogeneous scalar finite state-space Markov chain. In this case, the... |

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Citation Context ...ned to make the most of any structure within the model studied. Numerous methods have been developed for reducing the variance of MC estimates including antithetic sampling (Handschin and Mayne 1969, =-=Handschin 1970-=-) and control variates (Akashi and Kumamoto 1975, Handschin 1970). We apply here the Rao-Blackwellisation method, see Casella and Robert (1996) for a general reference on the topic. In a sequential fr... |

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(Show Context)
Citation Context ..., x20:n is a linear Gaussian state space model and the integrations required by the Rao-Blackwellisation method can be realized using the Kalman filter. Akashi and Kumamoto (Akashi and Kumamoto 1977, =-=Tugnait 1982-=-) introduced this algorithm under the name of RSA (Random Sampling Algorithm) in the particular case where x1 k is a homogeneous scalar finite state-space Markov chain. In this case, they adopted the ... |

35 |
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Citation Context ...and Shepherd (1999). Other resampling procedures which reduce the MC variation, such as stratified sampling (Carpenter, Clifford and Fearnhead 1997, Kitagawa and Gersch 1996) and residual resampling (=-=Higuchi 1997-=-, Liu and Chen 1998), may be applied as an alternative to SIR.s202 Doucet, Godsill and Andrieu An appropriate algorithm based on the SIR scheme proceeds as follows at time k. SIS/Resampling Monte Carl... |

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20 |
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Citation Context ...e Bayesian filtering problem or the optimal filtering problem. Practical applications include target tracking (Gordon, Salmond and Smith 1993), blind deconvolution of digital communications channels (=-=Clapp and Godsill 1999, -=-Liu and Chen 1995), estimation of stochastic volatility (Pitt and Shephard 0960-3174 C○ 2000 Kluwer Academic Publishers 1999) and digital enhancement of speech and audio signals (Godsill and Rayner ... |

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Citation Context ... importance function which limits seriously the number of resampling steps. B. Nonlinear series We consider here the following nonlinear reference model (Gordon, Salmon and Smith 1993, Kitagawa 1987, =-=Tanizaki and Mariano 1998): xk = f -=-(xk−1) + vk = 1 2 xk−1 + 25 yk = g(xk) + wk = (xk) 2 + wk 20 xk−1 + 8 cos (1.2k) + vk 1 + (xk−1) 2 (73) (74) where x0 ∼ N (0, 5), vk and wk are mutually independent white Gaussian noise, vk ... |

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Citation Context ...roximation of the optimal importance function. Several Monte Carlo methods have been proposed to approximate the importance function and the associated importance weight based on importance sampling (=-=Doucet 1997-=-, 1998) and Markov chain Monte Carlo (Berzuini et al. 1998, Liu and Chen 1998). These iterative algorithms are computationally intensive and there is a lack of theoretical convergence results. However... |

11 | Monte Carlo integration in general dynamic models - Müller - 1991 |

9 | Filtering via simulation: Auxiliary particle filters - K, Shephard - 1999 |

7 |
Construction of Discrete-time Nonlinear Filter by Monte Carlo Methods with Variance-reducing Techniques
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(Show Context)
Citation Context ...within the model studied. Numerous methods have been developed for reducing the variance of MC estimates including antithetic sampling (Handschin and Mayne 1969, Handschin 1970) and control variates (=-=Akashi and Kumamoto 1975-=-, Handschin 1970). We apply here the Rao-Blackwellisation method, see Casella and Robert (1996) for a general reference on the topic. In a sequential framework, MacEachern, Clyde and Liu (1999) have a... |

6 |
Digital Audio Restoration—A Statistical ModelBased Approach
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(Show Context)
Citation Context ...pp and Godsill 1999, Liu and Chen 1995), estimation of stochastic volatility (Pitt and Shephard 0960-3174 C○ 2000 Kluwer Academic Publishers 1999) and digital enhancement of speech and audio signals=-= (Godsill and Rayner 1998-=-). Except in a few special cases, including linear Gaussian state space models (Kalman filter) and hidden finite-state space Markov chains, it is impossible to evaluate these distributions analyticall... |

4 | The use of Bayesian belief networks to fuse continuous and discrete information for target recognition, tracking and situation assessment - Stewart, McCarty - 1992 |

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1 | Kumamoto H.(1977) Random Sampling Approach to State Estimation in Switching Environments - Akashi |