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.

The Factored Frontier (FF) algorithm is a simple approximate inference algorithm for Dynamic Bayesian Networks (DBNs). It is very similar to the fully factorized version of the Boyen-Koller (BK) algorithm, but instead of doing an exact update at every step followed by marginalisation (projection), it always works with factored distributions. Hence it can be applied to models for which the exact update step is intractable. We show that FF is equivalent to (one iteration of) loopy belief propagation (LBP) on the original DBN, and that BK is equivalent (to one iteration of) LBP on a DBN where we cluster some of the nodes. We then show empirically that by iterating more than once, LBP can improve on the accuracy of both FF and BK. We compare these algorithms on two real-world DBNs: the first is a model of a water treatment plant, and the second is a coupled HMM, used to model freeway trac.

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We survey the literature on methods for inference and learning in Bayesian Networks composed of discrete and continuous nodes, in which the continuous nodes have a multivariate Gaussian distribution, whose mean and variance depends on the values of the discrete nodes. We also briefly consider hybrid Dynamic Bayesian Networks, an extension of switching Kalman filters. This report is meant to summarize what is known at a sufficient level of detail to enable someone to implement the algorithms, but without dwelling on formalities.

this paper, we derive Pearl's algorithm [Pea88] for belief propogation in polytrees, using what we consider to be a somewhat clearer notation. (A polytree is a directed graph, whose undirected version is a tree, i.e., has no loops. In a polytree, a node may have multiple parents. Pearl's algorithm is essentially the wellknown forwards-backwards algorithm for chains [Rab89] generalized to polytrees.) In addition, we derive the update equations for a new class of deterministic nodes called multiplexer nodes, in a way which avoids summing over all possible parent values. This allows us to handle nodes with very high fan-in. 1 Pearl's algorithm