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65
Dynamic Bayesian Networks: Representation, Inference and Learning
, 2002
"... 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 biosequence analysis, and KFMs have bee ..."
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Cited by 598 (3 self)
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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 biosequence 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) linearGaussian. 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 RaoBlackwellised 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.
Filtering via simulation: auxiliary particle filter
 Journal of the American Statistical Association
, 1999
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Learning Dynamic Bayesian Networks
 In Adaptive Processing of Sequences and Data Structures, Lecture Notes in Artificial Intelligence
, 1998
"... Suppose we wish to build a model of data from a finite sequence of ordered observations, {Y1, Y2,..., Yt}. In most realistic scenarios, from modeling stock prices to physiological data, the observations are not related deterministically. Furthermore, there is added uncertainty resulting from the li ..."
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Cited by 142 (0 self)
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Suppose we wish to build a model of data from a finite sequence of ordered observations, {Y1, Y2,..., Yt}. In most realistic scenarios, from modeling stock prices to physiological data, the observations are not related deterministically. Furthermore, there is added uncertainty resulting from the limited size of our data set and any mismatch between our model and the true process. Probability theory provides a powerful tool for expressing both randomness and uncertainty in our model [23]. We can express the uncertainty in our prediction of the future outcome Yt+l via a probability density P(Yt+llY1,..., Yt). Such a probability density can then be used to make point predictions, define error bars, or make decisions that are expected to minimize some loss function. This chapter presents a probabilistic framework for learning models of temporal data. We express these models using the Bayesian network formalism (a.k.a. probabilistic graphical models or belief networks)a marriage of probability theory and graph theory in which dependencies between variables are expressed graphically. The graph not only allows the user to understand which variables
Switching Kalman Filters
, 1998
"... We show how many different variants of Switching Kalman Filter models can be represented in a unified way, leading to a single, generalpurpose inference algorithm. We then show how to find approximate Maximum Likelihood Estimates of the parameters using the EM algorithm, extending previous results ..."
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Cited by 61 (2 self)
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We show how many different variants of Switching Kalman Filter models can be represented in a unified way, leading to a single, generalpurpose inference algorithm. We then show how to find approximate Maximum Likelihood Estimates of the parameters using the EM algorithm, extending previous results on learning using EM in the nonswitching case [DRO93, GH96a] and in the switching, but fully observed, case [Ham90]. 1 Introduction Dynamical systems are often assumed to be linear and subject to Gaussian noise. This model, called the Linear Dynamical System (LDS) model, can be defined as x t = A t x t\Gamma1 + v t y t = C t x t +w t where x t is the hidden state variable at time t, y t is the observation at time t, and v t ¸ N(0; Q t ) and w t ¸ N(0; R t ) are independent Gaussian noise sources. Typically the parameters of the model \Theta = f(A t ; C t ; Q t ; R t )g are assumed to be timeinvariant, so that they can be estimated from data using e.g., EM [GH96a]. One of the main adva...
Monte Carlo Methods for Tempo Tracking and Rhythm Quantization
 JOURNAL OF ARTIFICIAL INTELLIGENCE RESEARCH
, 2003
"... We present a probabilistic generarive model for timing deviations in expressive music performance. The structure of the proposed model is equivalent to a switching state space model. The switch variables correspond to discrete note locations as in a musical score. The continuous hidden variables ..."
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Cited by 56 (9 self)
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We present a probabilistic generarive model for timing deviations in expressive music performance. The structure of the proposed model is equivalent to a switching state space model. The switch variables correspond to discrete note locations as in a musical score. The continuous hidden variables denote the tempo. We formulate two well known music recognition problems, namely tempo tracking and automatic transcription (rhythm quantization) as filtering and maximum a posteriori (MAP) state estimation tasks. Ex act computation of posterior features such as the MAP state is intractable in this model class, so we introduce Monte Carlo methods for integration and optimization. We compare Markov Chain Monte Carlo (MCMC) methods (such as Gibbs sampling, simulated annealing and iterative improvement) and sequential Monte Carlo methods (particle filters). Our simulation results suggest better results with sequential methods. The methods can be applied in both online and batch scenarios such as tempo tracking and transcription and are thus potentially useful in a number of music applications such as adaptive automatic accompaniment, score typesetting and music information retrieval.
Statistical Reconstruction And Analysis Of Autoregressive Signals In Impulsive Noise
, 1998
"... Modelling and reconstruction methods are presented for noise reduction of autocorrelated signals in nonGaussian, impulsive noise environments. A Bayesian probabilistic framework is adopted and Markov chain Monte Carlo methods are developed for detection and correction of impulses. Individual noise ..."
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Cited by 48 (16 self)
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Modelling and reconstruction methods are presented for noise reduction of autocorrelated signals in nonGaussian, impulsive noise environments. A Bayesian probabilistic framework is adopted and Markov chain Monte Carlo methods are developed for detection and correction of impulses. Individual noise sources are modelled as Gaussian with unknown scale (variance), allowing for robustness to `heavytailed' impulse distributions, while the underlying signal is modelled as autoregressive (AR). Results are presented for both artificial and real data from voice and music recordings and comparisons are made with existing techniques. The new techniques are found to give improved detection and elimination of impulses in adverse noise conditions at the expense of some extra computational complexity.
On the Relationship Between Markov Chain Monte Carlo Methods for Model Uncertainty
 JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS
, 2001
"... This article considers Markov chain computational methods for incorporating uncertainty about the dimension of a parameter when performing inference within a Bayesian setting. A general class of methods is proposed for performing such computations, based upon a product space representation of the ..."
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Cited by 32 (3 self)
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This article considers Markov chain computational methods for incorporating uncertainty about the dimension of a parameter when performing inference within a Bayesian setting. A general class of methods is proposed for performing such computations, based upon a product space representation of the problem which is similar to that of Carlin and Chib. It is shown that all of the existing algorithms for incorporation of model uncertainty into Markov chain Monte Carlo (MCMC) can be derived as special cases of this general class of methods. In particular, we show that the popular reversible jump method is obtained when a special form of MetropolisHastings (MH) algorithm is applied to the product space. Furthermore, the Gibbs sampling method and the variable selection method are shown to derive straightforwardly from the general framework. We believe that these new relationships between methods, which were until now seen as diverse procedures, are an important aid to the understanding of MCMC model selection procedures and may assist in the future development of improved procedures. Our discussion also sheds some light upon the important issues of "pseudoprior" selection in the case of the Carlin and Chib sampler and choice of proposal distribution in the case of reversible jump. Finally, we propose efficient reversible jump proposal schemes that take advantage of any analytic structure that may be present in the model. These proposal schemes are compared with a standard reversible jump scheme for the problem of model order uncertainty in autoregressive time series, demonstrating the improvements which can be achieved through careful choice of proposals
Expectation correction for smoothed inference in switching linear dynamical systems
 Journal of Machine Learning Research
"... We introduce a method for approximate smoothed inference in a class of switching linear dynamical systems, based on a novel form of Gaussian Sum smoother. This class includes the switching Kalman ‘Filter ’ and the more general case of switch transitions dependent on the continuous latent state. The ..."
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Cited by 24 (7 self)
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We introduce a method for approximate smoothed inference in a class of switching linear dynamical systems, based on a novel form of Gaussian Sum smoother. This class includes the switching Kalman ‘Filter ’ and the more general case of switch transitions dependent on the continuous latent state. The method improves on the standard Kim smoothing approach by dispensing with one of the key approximations, thus making fuller use of the available future information. Whilst the central assumption required is projection to a mixture of Gaussians, we show that an additional conditional independence assumption results in a simpler but accurate alternative. Our method consists of a single Forward and Backward Pass and is reminiscent of the standard smoothing ‘correction’ recursions in the simpler linear dynamical system. The method is numerically stable and compares favourably against alternative approximations, both in cases where a single mixture component provides a good posterior approximation, and where a multimodal approximation is required.
MCMC Methods for Financial Econometrics
 Handbook of Financial Econometrics
, 2002
"... This chapter discusses Markov Chain Monte Carlo (MCMC) based methods for es timating continuoustime asset pricing models. We describe the Bayesian approach to empirical asset pricing, the mechanics of MCMC algorithms and the strong theoretical underpinnings of MCMC algorithms. We provide a tuto ..."
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Cited by 24 (4 self)
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This chapter discusses Markov Chain Monte Carlo (MCMC) based methods for es timating continuoustime asset pricing models. We describe the Bayesian approach to empirical asset pricing, the mechanics of MCMC algorithms and the strong theoretical underpinnings of MCMC algorithms. We provide a tutorial on building MCMC algo rithms and show how to estimate equity price models with factors such as stochastic expected returns, stochastic volatility and jumps, multifactor term structure models with stochastic volatility, timevarying central tenclancy or jumps and regime switching models.
Semiparametric Bayesian inference for time series with mixed spectra
 J. Royal Statist. Soc. Ser. B
, 1996
"... This paper provides a Bayesian analysis of such a model. The main contribution of our paper is that different features of the datasuch as the spectral density of the stationary term, the regression parameters, unknown frequencies and missing observationsare combined in a hierarchical Bayesian fr ..."
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Cited by 22 (1 self)
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This paper provides a Bayesian analysis of such a model. The main contribution of our paper is that different features of the datasuch as the spectral density of the stationary term, the regression parameters, unknown frequencies and missing observationsare combined in a hierarchical Bayesian framework and estimated simultaneously. A Bayesian test to detect the presence of deterministic components in the data is also constructed. Applications of our methods to simulated and real data suggest that they perform well. We place a smoothness prior, similar to that in Wahba (1980), on the logarithm of the spectral density. To make the estimation of the spectral density computationally tractable, Whittle's (1957) approximation to the Gaussian likelihood is used. This results in a nonparametric regression problem with the logarithm of the periodogram as the dependent variable, the logarithm of the spectral density as the unknown regression curve, and observation errors having log chisquared distributions. By approximating the logarithm of a chisquared distribution as a mixture of normals, the approximate log likelihood together with the prior for the spectral density can be expressed as a state space model with errors that are mixtures of normals. The computation is carried out efficiently by Markov chain Monte Carlo using the sampling approach in Carter and Kohn (1994). To make the paper easier to read the full model is introduced in a number of steps. Section 2 shows how to estimate the spectral density of a stationary process in the absence of deterministic components. Section 3 extends the estimation to the signal plus noise model with missing observations. Section 4 shows by example how the results in Sections 2 and 3 can be combined to analyze data and studies emp...