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.

Abstract not found

Two or more Bayesian network structures are Markov equivalent when the corresponding acyclic digraphs encode the same set of conditional independencies. Therefore, the search space of Bayesian network structures may be organized in equivalence classes, where each of them represents a different set of conditional independencies. The collection of sets of conditional independencies obeys a partial order, the so-called “inclusion order.” This paper discusses in depth the role that the inclusion order plays in learning the structure of Bayesian networks. In particular, this role involves the way a learning algorithm traverses the search space. We introduce a condition for traversal operators, the inclusion boundary condition, which, when it is satisfied, guarantees that the search strategy can avoid local maxima. This is proved under the assumptions that the data is sampled from a probability distribution which is faithful to an acyclic digraph, and the length of the sample is unbounded. The previous discussion leads to the design of a new traversal operator and two new learning algorithms in the context of heuristic search and the Markov Chain Monte Carlo method. We carry out a set of experiments with synthetic and real-world data that show empirically the benefit of striving for the inclusion order when learning Bayesian networks from data.

We propose a decision theoretic approach for deciding which interventions to perform so as to learn the causal structure of a model as quickly as possible. Without such interventions, it is impossible to distinguish between Markov equivalent models, even given infinite data. We perform online MCMC to estimate the posterior over graph structures, and use importance sampling to find the best action to perform at each step. We assume the data is discrete-valued and fully observed.

Traditional approaches to Bayes net structure learning typically assume little regularity in graph structure other than sparseness. However, in many cases, we expect more systematicity: variables in real-world systems often group into classes that predict the kinds of probabilistic dependencies they participate in. Here we capture this form of prior knowledge in a hierarchical Bayesian framework, and exploit it to enable structure learning and type discovery from small datasets. Specifically, we present a nonparametric generative model for directed acyclic graphs as a prior for Bayes net structure learning. Our model assumes that variables come in one or more classes and that the prior probability of an edge existing between two variables is a function only of their classes. We derive an MCMC algorithm for simultaneous inference of the number of classes, the class assignments of variables, and the Bayes net structure over variables. For several realistic, sparse datasets, we show that the bias towards systematicity of connections provided by our model can yield more accurate learned networks than the traditional approach of using a uniform prior, and that the classes found by our model are appropriate. 1

Motivation: We have hypothesized that the construction of transcriptional regulatory networks using a method that optimizes connectivity would lead to regulation consistent with biological expectations. A key expectation is that the hypothetical networks should produce a few, very strong attractors, highly similar to the original observations, mimicking biological state stability and determinism. Another central expectation is that, since it is expected that the biological control is distributed and mutually reinforcing, interpretation of the observations should lead to a very small number of connection schemes.

There are essentially two kinds of approaches for learning the structure of Bayesian Networks (BNs) from data. The first approach tries to find a graph which satis es all the constraints implied by the empirical conditional independencies measured in the data [PV91, SGS00a, Shi00]. The second approach searches through the space of models (either DAGs or PDAGs), and uses some scoring metric (typically Bayesian or some approximation, such as BIC/MDL) to evaluate the models [CH92, Hec95, Hec98, Kra98], typically returning the highest scoring model found. Our main interest is in learning BN structure from gene expression data [FLNP00, HGJY01, MM99, SGS00b]. In domains such as this, where the ratio of the number of observations to the number of variables is low (i.e., when we have sparse data), selecting a threshold for the conditional independence (CI) tests can be tricky, and repeated use of such tests can lead to inconsistencies [DD99]. Bayesian s...

We present the Mambo algorithm for the discovery of association rules. Mambo is driven by conditional independence relations between the variables instead of the minimum support restrictions of algorithms like Apriori. We argue that making use of conditional independencies is an intuitively appealing way to restrict the set of association rules considered. Since we only have a finite sample from the probability distribution of interest, we have to deal with uncertainty concerning the conditional independencies present. Bayesian methods are used to quantify this uncertainty, and the posterior probabilities of conditional independence relations are estimated with the Markov Chain Monte Carlo technique. We analyse an insurance data set with Mambo and illustrate the differences in results compared to Apriori.

We show how to significantly speed up MCMC sampling of DAG structures by using a powerful non-local proposal based on Koivisto’s dynamic programming (DP) algorithm (11; 10), which computes the exact marginal posterior edge probabilities by analytically summing over orders. Furthermore, we show how sampling in DAG space can avoid subtle biases that are introduced by approaches that work only with orders, such as Koivisto’s DP algorithm and MCMC order samplers (6; 5). 1

Two or more Bayesian Networks are Markov equivalent when their corresponding acyclic digraphs encode the same set of conditional independence (# CI) restrictions. Therefore, the search space of Bayesian Networks may be organized in classes of equivalence, where each of them consists of a particular set of CI restrictions. The collection of sets of CI restrictions obeys a partial order, the graphical Markov model inclusion partial order, or inclusion order for short.