Statistics and Computing (2000) 10, 197–208
On sequential Monte Carlo sampling
methods for Bayesian filtering
ARNAUD DOUCET, SIMON GODSILL and CHRISTOPHE ANDRIEU
Signal Processing Group, Department of Engineering, University of Cambridge,
Trumpington Street, CB2 1PZ Cambridge, UK
ad2@eng.cam.ac.uk, sjg@eng.cam.ac.uk, ca226@eng.cam.ac.uk
Received July 1998 and accepted August 1999
In this article, we present an overview of methods for sequential simulation from posterior distribu-
tions. 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 devel-
oped 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 deterministic 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.
Keywords: Bayesian filtering, nonlinear non-Gaussian state space models, sequential Monte Carlo
methods, particle filtering, importance sampling, Rao-Blackwellised estimates
I. Introduction
Many problems in applied statistics, statistical signal process-
ing, time series analysis and econometrics can be stated in a
state space form as follows. A transition equation describes the
prior distribution of a hidden Markov process {x
k
; k ∈N}, the
so-called hidden state process, and an observation equation de-
scribes the likelihood of the observations {y
k
; k ∈N}, k being a
discrete time index. Within a Bayesian framework, all relevant
information about {x
0
, x
1
,...,x
k
}given observations up to and
including time k can be obtained from the posterior distribu-
tion p(x
0
, x
1
,...,x
k
|y
0
,y
1
,...,y
k
). In many applications we
are interested in estimating recursively in time this distribution,
and particularly one of its marginals, the so-called filtering dis-
tribution p(x
k
|y
0
, y
1
,...,y
k
). Given the filtering distribution
one can then routinely proceed to filtered point estimates such
as the posterior mode or mean of the state. This problem is
known as the Bayesian filtering problem or the optimal filtering
problem. Practical applications include target tracking (Gordon,
Salmond and Smith 1993), blind deconvolution of digital com-
munications channels (Clapp and Godsill 1999, Liu and Chen
1995), estimation of stochastic volatility (Pitt and Shephard
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
analytically. From the mid 1960’s, a great deal of attention has
been devoted to 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). Inter-
esting work in the automatic control field was carried out during
the 1960’s and 70’s using sequential Monte Carlo (MC) inte-
gration methods, see Akashi and Kumamoto (1975), Handschin
and Mayne (1969), Handschin (1970), and Zaritskii, Svetnik and
Shimelevich (1975). Possibly owing to the severe computational
limitations of the time, these Monte Carlo algorithms have been
largely neglected until recent years. In the late 80’s, massive
increases in computational power allowed the rebirth of numeri-
cal 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 as-
sumption of linearity or Gaussianity in the model, and relevant
0960-3174
C
°
2000 Kluwer Academic Publishers
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