A companion set of lecture slides is available at

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.

We discuss Bayesian methods for learning Bayesian networks when data sets are incomplete. In particular, we examine asymptotic approximations for the marginal likelihood of incomplete data given a Bayesian network. We consider the Laplace approximation and the less accurate but more efficient BIC/MDL approximation. We also consider approximations proposed by Draper (1993) and Cheeseman and Stutz (1995). These approximations are as efficient as BIC/MDL, but their accuracy has not been studied in any depth. We compare the accuracy of these approximations under the assumption that the Laplace approximation is the most accurate. In experiments using synthetic data generated from discrete naive-Bayes models having a hidden root node, we find that (1) the BIC/MDL measure is the least accurate, having a bias in favor of simple models, and (2) the Draper and CS measures are the most accurate. 1

We hope to be able to provide answers to the following questions: 1) Has there been a structural break in postwar U.S. real GDP growth toward more stabilization? 2) If so, when would it have been? 3) What's the nature of the structural break? For this purpose, we employ a Bayesian approach to dealing with structural break at an unknown changepoint in a Markov-switching model of business cycle. Empirical results suggest that there has been a structural break in U.S. real GDP growth toward more stabilization, with the posterior mode of the break date around 1984:1. Furthermore, we #nd a narrowing gap between growth rates during recessions and booms is at least as important as a decline in the volatility of shocks. Key Words: Bayes Factor, Gibbs sampling, Marginal Likelihood, Markov-Switching, Stabilization, Structural Break. JEL Classi#cations: C11, C12, C22, E32. 1. Introduction In the literature, the issue of postwar stabilization of the U.S. economy relative to the prewar period has...

This paper surveys the fundamental principles of subjective Bayesian inference in econometrics and the implementation of those principles using posterior simulation methods. The emphasis is on the combination of models and the development of predictive distributions. Moving beyond conditioning on a fixed number of completely specified models, the paper introduces subjective Bayesian tools for formal comparison of these models with as yet incompletely specified models. The paper then shows how posterior simulators can facilitate communication between investigators (for example, econometricians) on the one hand and remote clients (for example, decision makers) on the other, enabling clients to vary the prior distributions and functions of interest employed by investigators. A theme of the paper is the practicality of subjective Bayesian methods. To this end, the paper describes publicly available software for Bayesian inference, model development, and communication and provides illustrations using two simple econometric models. *This paper was originally prepared for the Australasian meetings of the Econometric Society in Melbourne, Australia,

Computing (ratios of) normalizing constants of probability models is a fundamental computational problem for many statistical and scientific studies. Monte Carlo simulation is an effective technique, especially with complex and high-dimensional models. This paper aims to bring to the attention of general statistical audiences of some effective methods originating from theoretical physics and at the same time to explore these methods from a more statistical perspective, through establishing theoretical connections and illustrating their uses with statistical problems. We show that the acceptance ratio method and thermodynamic integration are natural generalizations of importance sampling, which is most familiar to statistical audiences. The former generalizes importance sampling through the use of a single "bridge" density and is thus a case of bridge sampling in the sense of Meng and Wong (1996). Thermodynamic integration, which is also known in the numerical analysis literature as Oga...

This paper is concerned with the Bayesian estimation of non-linear stochastic differential equations when only discrete observations are available. The estimation is carried out using a tuned MCMC method, in particular a blocked Metropolis-Hastings algorithm, by introducing auxiliary points and using the Euler-Maruyama discretisation scheme. Techniques for computing the likelihood function, the marginal likelihood and diagnostic measures (all based on the MCMC output) are presented. Examples using simulated and real data are presented and discussed in detail.

This paper deals with both exploration and interpretation problems related to posterior distributions for mixture models. The specification of mixture posterior distributions means that the presence of k! modes is known immediately. Standard Markov chain Monte Carlo techniques usually have difficulties with well-separated modes such as occur here; the Markov chain Monte Carlo sampler stays within a neighbourhood of a local mode and fails to visit other equally important modes. We show that exploration of these modes can be imposed on the Markov chain Monte Carlo sampler using tempered transitions based on Langevin algorithms. However, as the prior distribution does not distinguish between the different components, the posterior mixture distribution is symmetric and thus standard estimators such as posterior means cannot be used. Since this is also true for most non-symmetric priors, we propose alternatives for Bayesian inference for permutation invariant posteriors, including a cluster...

Latent Dirichlet allocation (LDA) and other related topic models are increasingly popular tools for summarization and manifold discovery in discrete data. However, LDA does not capture correlations between topics. In this paper, we introduce the pachinko allocation model (PAM), which captures arbitrary, nested, and possibly sparse correlations between topics using a directed acyclic graph (DAG). The leaves of the DAG represent individual words in the vocabulary, while each interior node represents a correlation among its children, which may be words or other interior nodes (topics). PAM provides a flexible alternative to recent work by Blei and Lafferty (2006), which captures correlations only between pairs of topics. Using text data from newsgroups, historic NIPS proceedings and other research paper corpora, we show improved performance of PAM in document classification, likelihood of held-out data, the ability to support finer-grained topics, and

In a Bayesian analysis of finite mixture models, parameter estimation and clustering are sometimes less straightforward that might be expected. In particular, the common practice of estimating parameters by their posterior mean, and summarising joint posterior distributions by marginal distributions, often leads to nonsensical answers. This is due to the so-called “labelswitching” problem, which is caused by symmetry in the likelihood of the model parameters. A frequent response to this problem is to remove the symmetry using artificial identifiability constraints. We demonstrate that this fails in general to solve the problem, and describe an alternative class of approaches, relabelling algorithms, which arise from attempting to minimise the posterior expected loss under a class of loss functions. We describe in detail one particularly simple and general relabelling algorithm, and illustrate its success in dealing with the labelswitching problem on two examples.