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.

Modelling sequential data is important in many areas of science and engineering. Hidden Markov models (HMMs) and Kalman filter models (KFMs) are popular for this because they are simple and flexible. For example, HMMs have been used for speech recognition and bio-sequence analysis, and KFMs have been used for problems ranging from tracking planes and missiles to predicting the economy. However, HMMs and KFMs are limited in their “expressive power”. Dynamic Bayesian Networks (DBNs) generalize HMMs by allowing the state space to be represented in factored form, instead of as a single discrete random variable. DBNs generalize KFMs by allowing arbitrary probability distributions, not just (unimodal) linear-Gaussian. In this thesis, I will discuss how to represent many different kinds of models as DBNs, how to perform exact and approximate inference in DBNs, and how to learn DBN models from sequential data. In particular, the main novel technical contributions of this thesis are as follows: a way of representing Hierarchical HMMs as DBNs, which enables inference to be done in O(T) time instead of O(T 3), where T is the length of the sequence; an exact smoothing algorithm that takes O(log T) space instead of O(T); a simple way of using the junction tree algorithm for online inference in DBNs; new complexity bounds on exact online inference in DBNs; a new deterministic approximate inference algorithm called factored frontier; an analysis of the relationship between the BK algorithm and loopy belief propagation; a way of applying Rao-Blackwellised particle filtering to DBNs in general, and the SLAM (simultaneous localization and mapping) problem in particular; a way of extending the structural EM algorithm to DBNs; and a variety of different applications of DBNs. However, perhaps the main value of the thesis is its catholic presentation of the field of sequential data modelling.

This paper analyses the recently suggested particle approach to filtering time series. We suggest that the algorithm is not robust to outliers for two reasons: the design of the simulators and the use of the discrete support to represent the sequentially updating prior distribution. Both problems are tackled in this paper. We believe we have largely solved the first problem and have reduced the order of magnitude of the second. In addition we introduce the idea of stratification into the particle filter which allows us to perform on-line Bayesian calculations about the parameters which index the models and maximum likelihood estimation. The new methods are illustrated by using a stochastic volatility model and a time series model of angles. Some key words: Filtering, Markov chain Monte Carlo, Particle filter, Simulation, SIR, State space. 1 1

A general framework for using Monte Carlo methods in dynamic systems is provided and its wide applications indicated. Under this framework, several currently available techniques are studied and generalized to accommodate more complex features. All of these methods are partial combinations of three ingredients: importance sampling and resampling, rejection sampling, and Markov chain iterations. We deliver a guideline on how they should be used and under what circumstance each method is most suitable. Through the analysis of differences and connections, we consolidate these methods into a generic algorithm by combining desirable features. In addition, we propose a general use of Rao-Blackwellization to improve performances. Examples from econometrics and engineering are presented to demonstrate the importance of Rao-Blackwellization and to compare different Monte Carlo procedures. Keywords: Blind deconvolution; Bootstrap filter; Gibbs sampling; Hidden Markov model; Kalman filter; Markov...

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In this paper, we propose a new particle filter based on sequential importance sampling. The algorithm uses a bank of unscented filters to obtain the importance proposal distribution. This proposal has two very "nice" properties. Firstly, it makes efficient use of the latest available information and, secondly, it can have heavy tails. As a result, we find that the algorithm outperforms standard particle filtering and other nonlinear filtering methods very substantially. This experimental finding is in agreement with the theoretical convergence proof for the algorithm. The algorithm also includes resampling and (possibly) Markov chain Monte Carlo (MCMC) steps.

In treating dynamic systems, sequential Monte Carlo methods use discrete samples to represent a complicated probability distribution and use rejection sampling, importance sampling, and weighted resampling to complete the on-line "filtering" task. In this article we propose a special sequential Monte Carlo method, the mixture Kalman filter, which uses random mixture of normal distributions to represent a target distribution. It is designed for on-line estimation and prediction of conditional and partial conditional dynamic linear models, which are themselves a class of widely used nonlinear system and also serve to approximate many other nonlinear systems. Compared with a few available filtering methods including Monte Carlo ones, the efficiency gain provided by the mixture Kalman filter can be very substantial. Another contribution of this article is the formulation of many nonlinear systems into conditional or partial conditional linear form, to which the mixture Kalman filter can be...

Jump Markov linear systems (JMLS) are linear systems whose parameters evolve with time according to a finite state Markov chain. In this paper, our aim is to recursively compute optimal state estimates for this class of systems. We present efficient simulation-based algorithms called particle filters to solve the optimal filtering problem as well as the optimal fixed-lag smoothing problem. Our algorithms combine sequential importance sampling, a selection scheme, and Markov chain Monte Carlo methods. They use several variance reduction methods to make the most of the statistical structure of JMLS.

We develop methods for performing smoothing computations in general state-space models. The methods rely on a particle representation of the filtering distributions, and their evolution through time using sequential importance sampling and resampling ideas. In particular, novel techniques are presented for generation of sample realizations of historical state sequences. This is carried out in a forwardfiltering backward-smoothing procedure which can be viewed as the non-linear, non-Gaussian counterpart of standard Kalman filter-based simulation smoothers in the linear Gaussian case. Convergence in the mean-squared error sense of the smoothed trajectories is proved, showing the validity of our proposed method. The methods are tested in a substantial application for the processing of speech signals represented by a time-varying autoregression and parameterised in terms of timevarying partial correlation coe#cients, comparing the results of our algorithm with those from a simple smoother based upon the filtered trajectories.

In dynamic statistical modeling situations, observations arise sequentially, causing the model to expand by progressive incorporation of new data items and new unknown parameters. For example, in clinical monitoring, new patient-specific parameters are introduced with each new patient. Markov chain Monte Carlo (MCMC) might be used for posterior inference, but would need to be redone at each expansion stage. Thus such methods are often too slow for real-time implementation. By combining MCMC with importance-resampling, we show how real-time posterior updating can be effected. The proposed dynamic sampling algorithms utilize posterior samples from previous expansion stages, and exploit conditional independence between groups of parameters to allow samples of parameters no longer of interest to be discarded, such as when a patient dies or is discharged. We apply the methods to monitoring of heart transplant recipients during infection from cytomegalovirus. KEY WORDS : Bayesian Inference, ...