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.

Markov networks are widely used in a wide variety of applications, in problems ranging from computer vision, to natural language, to computational biology. In most current applications, even those that rely heavily on learned models, the structure of the Markov network is constructed by hand, due to the lack of effective algorithms for learning Markov network structure from data. In this paper, we provide a computationally effective method for learning Markov network structure from data. Our method is based on the use of L1 regularization on the weights of the log-linear model, which has the effect of biasing the model towards solutions where many of the parameters are zero. This formulation converts the Markov network learning problem into a convex optimization problem in a continuous space, which can be solved using efficient gradient methods. A key issue in this setting is the (unavoidable) use of approximate inference, which can lead to errors in the gradient computation when the network structure is dense. Thus, we explore the use of different feature introduction schemes and compare their performance. We provide results for our method on synthetic data, and on two real world data sets: modeling the joint distribution of pixel values in the MNIST data, and modeling the joint distribution of genetic sequence variations in the human HapMap data. We show that our L1-based method achieves considerably higher generalization performance than the more standard L2-based method (a Gaussian parameter prior) or pure maximum-likelihood learning. We also show that we can learn MRF network structure at a computational cost that is not much greater than learning parameters alone, demonstrating the existence of a feasible method for this important problem. 1

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We study the computational and sample complexity of parameter and structure learning in graphical models. Our main result shows that the class of factor graphs with bounded degree can be learned in polynomial time and from a polynomial number of training examples, assuming that the data is generated by a network in this class. This result covers both parameter estimation for a known network structure and structure learning. It implies as a corollary that we can learn factor graphs for both Bayesian networks and Markov networks of bounded degree, in polynomial time and sample complexity. Importantly, unlike standard maximum likelihood estimation algorithms, our method does not require inference in the underlying network, and so applies to networks where inference is intractable. We also show that the error of our learned model degrades gracefully when the generating distribution is not a member of the target class of networks. In addition to our main result, we show that the sample complexity of parameter learning in graphical models has an O(1) dependence on the number of variables in the model when using the KL-divergence normalized by the number of variables as the performance criterion.

We describe a serial algorithm called feature-inclusion stochastic search, or FINCS, that uses online estimates of edge-inclusion probabilities to inform the process of Bayesian model determination in Gaussian graphical models. FINCS is compared to Metropolis-based search methods and found to be superior along a variety of dimensions, leading to more accurate and less volatile estimates of edge-inclusion probabilities and greater speed in finding good models. Though FINCS is conceived as a method for characterizing model uncertainty in moderate-dimensional problems, we also find that it performs well as a stochastic hill-climber in bigger problems. We illustrate its use on an example involving mutual-fund data, where we compare the model-averaged predictive performance of models discovered with FINCS to those discovered with the Metropolis algorithm.

Most RDBMSs maintain a set of histograms for estimating the selectivities of given queries.

We describe two classes of software systems that release tabular summaries of an underlying database. Table servers respond to user queries for (marginal) sub-tables of the "full" table summarizing the entire database, and are characterized by dynamic assessment of disclosure risk, in light of previously answered queries. Optimal tabular releases are static releases of sets of sub-tables that are characterized by maximizing the amount of information released, as given by a measure of data utility, subject to a constraint on disclosure risk. Underlying abstractions --- primarily associated with the query space, as well as released and unreleasable sub-tables and frontiers, computational algorithms and issues, especially scalability, and prototype software implementations are discussed.

For a chordal graph G = (V, E), we study the problem of whether a new vertex u � ∈ V and a given set of edges between u and vertices in V can be added to G so that the resulting graph remains chordal. We show how to resolve this efficiently, and at the same time, if the answer is no, specify a maximal subset of the proposed edges that can be added along with u, or conversely, a minimal set of extra edges that can be added in addition to the given set, so that the resulting graph is chordal. In order to do this, we give a new characterization of chordal graphs and, for each potential new edge uv, a characterization of the set of edges incident to u that also must be added to G along with uv. We propose a data structure that can compute and add each such set in O(n) time. Based on these results, we present an algorithm that computes both a minimal triangulation and a maximal chordal subgraph of an arbitrary input graph in O(nm) time, using a totally new vertex incremental approach. In contrast to previous algorithms, our process is on-line in that each new vertex is added without reconsidering any choice made at previous steps, and without requiring any knowledge of the vertices that might be added subsequently. 1

We propose a new approach for learning a sparse graphical model approximation to a specified multivariate probability distribution (such as the empirical distribution of sample data). The selection of sparse graph structure arises naturally in our approach through solution of a convex optimization problem, which differentiates our method from standard combinatorial approaches. We seek the maximum entropy relaxation (MER) within an exponential family, which maximizes entropy subject to constraints that marginal distributions on small subsets of variables are close to the prescribed marginals in relative entropy. To solve MER, we present a modified primal-dual interior point method that exploits sparsity of the Fisher information matrix in models defined on chordal graphs. This leads to a tractable, scalable approach provided the level of relaxation in MER is sufficient to obtain a thin graph. The merits of our approach are investigated by recovering the structure of some simple graphical models from sample data. 1

my family-- especially my father, Donald. iv Abstract Many important data analysis tasks can be addressed by formulating them as probability estimation problems. For example, a popular general approach to automatic classification problems is to learn a probabilistic model of each class from data in which the classes are known, and then use Bayes's rule with these models to predict the correct classes of other data for which they are not known. Anomaly detection and scientific discovery tasks can often be addressed by learning probability models over possible events and then looking for events to which these models assign low probabilities. Many data compression algorithms such as Huffman coding and arithmetic coding rely on probabilistic models of the data stream in order achieve high compression rates.