We present a tree-structured architecture for supervised learning. The statistical model underlying the architecture is a hi-erarchical mixture model in which both the mixture coefficients and the mixture components are generalized linear models (GLIM’s). Learning is treated as a max-imum likelihood problem; in particular, we present an Expectation-Maximization (EM) algorithm for adjusting the parame-ters of the architecture. We also develop an on-line learning algorithm in which the pa-rameters are updated incrementally. Com-parative simulation results are presented in the robot dynamics domain. 1

We present an unsupervised technique for visual learning which is based on density estimation in high-dimensional spaces using an eigenspace decomposition. Two types of density estimates are derived for modeling the training data: a multivariate Gaussian (for unimodal distributions) and a Mixture-of-Gaussians model (for multimodal distributions). These probability densities are then used to formulate a maximum-likelihood estimation framework for visual search and target detection for automatic object recognition and coding. Our learning technique is applied to the probabilistic visual modeling, detection, recognition, and coding of human faces and non-rigid objects such as hands.

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.

In this paper a framework for maximum a posteriori (MAP) estimation of hidden Markov models (HMM) is presented. Three key issues of MAP estimation, namely the choice of prior distribution family, the specification of the parameters of prior densities and the evaluation of the MAP estimates, are addressed. Using HMMs with Gaussian mixture state observation densities as an example, it is assumed that the prior densities for the HMM parameters can be adequately represented as a product of Dirichlet and normal-Wishart densities. The classical maximum likelihood estimation algorithms, namely the forward-backward algorithm and the segmental k-means algorithm, are expanded and MAP estimation formulas are developed. Prior density estimation issues are discussed for two classes of applications: parameter smoothing and model adaptation, and some experimental results are given illustrating the practical interest of this approach. Because of its adaptive nature, Bayesian learning is shown to serve as a unified approach for a wide range of speech recognition applications

We describe the maximum-likelihood parameter estimation problem and how the Expectation-form of the EM algorithm as it is often given in the literature. We then develop the EM parameter estimation procedure for two applications: 1) finding the parameters of a mixture of Gaussian densities, and 2) finding the parameters of a hidden Markov model (HMM) (i.e., the Baum-Welch algorithm) for both discrete and Gaussian mixture observation models. We derive the update equations in fairly explicit detail but we do not prove any convergence properties. We try to emphasize intuition rather than mathematical rigor. ii 1 Maximum-likelihood Recall the definition of the maximum-likelihood estimation problem. We have a density function ¢¡¤£¦ ¥ §© ¨ that is governed by the set of parameters § (e.g.,   might be a set of Gaussians and § could be the means and covariances). We also have a data set of size � , supposedly drawn from this distribution, i.e., ���� � £�������������£��© �. That is, we assume that these data vectors are independent and

Wavelet-based statistical signal processing techniques such as denoising and detection typically model the wavelet coefficients as independent or jointly Gaussian. These models are unrealistic for many real-world signals. In this paper, we develop a new framework based on wavelet-domain hidden Markov models (HMMs). The framework enables us to concisely model the statistical dependencies and nonGaussian statistics often encountered in practice. Wavelet-domain HMMs are designed with the intrinsic properties of the wavelet transform in mind and provide powerful yet tractable probabilistic signal models. Efficient Expectation Maximization algorithms are developed for fitting the HMMs to observational signal data. The new framework is suitable for a wide range of applications, including signal estimation, detection, classification, prediction, and even synthesis. To demonstrate the utility of wavelet-domain HMMs, we develop novel algorithms for signal denoising, classification, and detectio...

Many approaches to Bayesian image segmentation have used maximum a posteriori (MAP) estimation in conjunction with Markov random fields (MRF). While this approach performs well, it has a number of disadvantages. In particular, exact MAP estimates cannot be computed, approximate MAP estimates are computationally expensive to compute, and unsupervised parameter estimation of the MRF is difficult. In this paper, we propose a new approach to Bayesian image segmentation which directly addresses these problems. The new method replaces the MRF model with a novel multiscale random field (MSRF), and replaces the MAP estimator with a sequential MAP (SMAP) estimator derived from a novel estimation criteria. Together, the proposed estimator and model result in a segmentation algorithm which is not iterative and can be computed in time proportional to MN where M is the number of classes and N is the number of pixels. We also develop a computationally effcient method for unsupervised estimation of m...

We present an unsupervised technique for visual learning which is based on density estimation in high-dimensional spaces using an eigenspace decomposition. Two types of density estimates are derived for modeling the training data: a multivariate Gaussian (for a unimodal distribution) and a multivariate Mixture-of-Gaussians model (for multimodal distributions). These probability densities are then used to formulate a maximum-likelihood estimation framework for visual search and target detection for automatic object recognition. This learning technique is tested in experiments with modeling and subsequent detection of human faces and non-rigid objects such as hands.

A wide variety of distortion functions are used for clustering, e.g., squared Euclidean distance, Mahalanobis distance and relative entropy. In this paper, we propose and analyze parametric hard and soft clustering algorithms based on a large class of distortion functions known as Bregman divergences. The proposed algorithms unify centroid-based parametric clustering approaches, such as classical kmeans and information-theoretic clustering, which arise by special choices of the Bregman divergence. The algorithms maintain the simplicity and scalability of the classical kmeans algorithm, while generalizing the basic idea to a very large class of clustering loss functions. There are two main contributions in this paper. First, we pose the hard clustering problem in terms of minimizing the loss in Bregman information, a quantity motivated by rate-distortion theory, and present an algorithm to minimize this loss. Secondly, we show an explicit bijection between Bregman divergences and exponential families. The bijection enables the development of an alternative interpretation of an ecient EM scheme for learning models involving mixtures of exponential distributions. This leads to a simple soft clustering algorithm for all Bregman divergences.

In many applications, it is necessary to determine the similarity of two strings. A widely-used notion of string similarity is the edit distance: the minimum number of insertions, deletions, and substitutions required to transform one string into the other. In this report, we provide a stochastic model for string edit distance. Our stochastic model allows us to learn a string edit distance function from a corpus of examples. We illustrate the utility of our approach by applying it to the difficult problem of learning the pronunciation of words in conversational speech. In this application, we learn a string edit distance with nearly one fifth the error rate of the untrained Levenshtein distance. Our approach is applicable to any string classification problem that may be solved using a similarity function against a database of labeled prototypes.