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.

The Bayes Net Toolbox (BNT) is an open-source Matlab package for directed graphical models. BNT supports many kinds of nodes (probability distributions), exact and approximate inference, parameter and structure learning, and static and dynamic models. BNT is widely used in teaching and research: the web page has received over 28,000 hits since May 2000. In this paper, we discuss a broad spectrum of issues related to graphical models (directed and undirected), and describe, at a high-level, how BNT was designed to cope with them all. We also compare BNT to other software packages for graphical models, and to the nascent OpenBayes effort.

Simultaneous Localization and Mapping (SLAM) is a fundamental problem in mobile robotics: while a robot navigates in an unknown environment, it must incrementally build a map of its surroundings and localize itself within that map. Traditional approaches to the problem are based upon Kalman filters, but suffer from complexity issues: the size of the belief state and the time complexity of the filtering operation grow quadratically in the size of the map. This paper presents a filtering technique that maintains a tractable approximation of the filtered belief state as a thin junction tree. The junction tree grows under measurement and motion updates and is periodically "thinned" to remain tractable via efficient maximum likelihood projections. When applied to the SLAM problem, these thin junction tree filters have a linear-space belief state representation, and use a linear-time filtering operation. Further approximation can yield a constant-time filtering operation, at the expense of delaying the incorporation of observations into the majority of the map. Experiments on a suite of SLAM problems validate the approach.

We describe expectation propagation for approximate inference in dynamic Bayesian networks as a natural extension of Pearl's exact belief propagation. Expectation propagation is a greedy algorithm, converges in many practical cases, but not always. We derive a double-loop algorithm, guaranteed to converge to a local minimum of a Bethe free energy. Furthermore, we show that stable fixed points of (damped) expectation propagation correspond to local minima of this free energy, but that the converse need not be the case. We illustrate the algorithms by applying them to switching linear dynamical systems and discuss implications for approximate inference in general Bayesian networks.

We present a technique that, given a sequence of musical note onset times, performs simultaneous identification of the norated rhythm and the variable tempo associated with the times. Our formulation is probabilistic: We develop a stochastic model for the interconnected evolution of a rhythm process, a tempo process, and an observable process. This model allows the globally optimal identification of the most likely rhythm and tempo sequence, given the observed onset times. We demonstrate applications to a sequence of times derived from a sampled audio file and to MIDI data.

A method is presented for the rhythmic parsing problem: Given a sequence of observed musical note onset times, we simultaneously estimate the corresponding norated rhythm and tempo process. A graphical model is developed that represents the evolution of tempo and rhythm and relates these hidden quantities to an observable performance. The rhythm variables are discrete and the tempo and observation variables are continuous. We show how to compute the globally most likely configuration of the tempo and rhythm variables given an observation of note onset times. Experiments are presented on both MIDI data and a data set derived from an audio signal. A generalization to computing MAP estimates for arbitrary conditional Gaussian distributions is outlined.

Mixtures of truncated exponentials (MTE) potentials are an alternative to discretization and Monte Carlo methods for solving hybrid Bayesian networks. Any probability density function (PDF) can be approximated by an MTE potential, which can always be marginalized in closed form. This allows propagation to be done exactly using the Shenoy-Shafer architecture for computing marginals, with no restrictions on the construction of a join tree. This paper presents MTE potentials that approximate standard PDF’s and applications of these potentials for solving inference problems in hybrid Bayesian networks. These approximations will extend the types of inference problems that can be modeled with Bayesian networks, as demonstrated using three examples.

Mixtures of truncated exponentials (MTE) potentials are an alternative to discretization for solving hybrid Bayesian networks. Any probability density function can be approximated with an MTE potential, which can always by marginalized in closed form. This allows propagation to be done exactly using the Shenoy-Shafer architecture for computing marginals, with no restrictions on the construction of a join tree.

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.

We describe a computer system that provides a real-time musical accompaniment for a live soloist in a piece of non-improvised music for soloist and accompaniment. A Bayesian network is developed that represents the joint distribution on the times at which the solo and accompaniment notes are played, relating the two parts through a layer of hidden variables. The network is first constructed using the rhythmic information contained in the musical score. The network is then trained to capture the musical interpretations of the soloist and accompanist in an off-line rehearsal phase.