Real-world learning tasks may involve high-dimensional data sets with arbitrary patterns of missing data. In this paper we present a framework based on maximum likelihood density estimation for learning from such data set.s. VVe use mixture models for the density estimatesand make two distinct appeals to the Expectation Maximization (EM) principle (Dempster et al., 1977) in deriving a learning algorithm-EM is used both for the estimation of mixture componentsand for coping wit.h missing dat.a. The resulting algorithm is applicable t.o a wide range of supervised as well as unsupervised learning problems.

Data clustering amounts to a combinatorial optimization problem to reduce thecomplexity of a data representation and to increase its precision. Central and pairwise data clustering are studied in the maximum entropy framework.For central clustering we derive a set of reestimation equations and a minimization procedure which yields an optimal number ofclusters, their centers and their cluster probabilities. A meanfield approximation for pairwise clustering is used to estimate assignment probabilities. A se1fconsistent solution to multidimensional scaling and pairwise clustering is derived which yields an optimal embedding and clustering of data points in a d-dimensional Euclidian space. 1 Introduction A central problem in information processing is the reduction of the data complexity with minimal loss in precision to discard noise and to reveal basic structure of data sets. Data clustering addresses this tradeoff by optimizing a cost function which preserves the original data as complete as possible and which simultaneously favors prototypes with minimal complexity (Linde et aI., 1980; Gray, 1984; Chou et aI., 1989; Rose et ai., 1990). We discuss anobjective function for the joint optimization of distortion errors and the complexity of a reduced data representation.

Such decisions include which function approximator to use, how to trade smoothness for goodness of fit and which features arerelevant.

Most connectionist research has focused on learning mappings from one space to another (eg.

In this procedure one transforms variables to a basis in which the covariance isdiagonal and then projects out the low variance directions. While application of PCA to remove input variables is useful in some cases (Leen et al., 1990), there is no guarantee that low variance variables have little effect on error. We propose a saliency measure, based on PCA, that identifies those variables that have the least effect on error. Our proposed Principal Components Pruning algorithm applies this measure to obtain a simple and cheap pruning technique in the context of supervised learning. Fast Pruning Using Principal Components 37 Special Case: PCP in Linear Regression In unbiased linear models, one can bound the bias introduced from pruning the principal degrees of freedom in the model.

This paper presents a formulation for unsupervised learning of clusters reflectingmultiple causal structure in binary data. Unlike the standard mixture model, a multiple cause model accounts for observed databy combining assertions from many hidden causes, each of which can pertain to varying degree to any subset of the observable dimensions.A crucial issue is the mixing-function for combining beliefs from different cluster-centers in order to generate data reconstructions whose errors are minimized both during recognition and learning. We demonstrate a weakness inherent to the popular weighted sum followed by sigmoid squashing, and offer an alternative formof the nonlinearity. Results are presented demonstrating the algorithm's ability successfully to discover coherent multiple causal representat.ions of noisy test data and in images of printed characters. 1 Introduction The objective of unsupervised learning is to identify patterns or features reflecting underlying regularities in data. Single-cause techniques, including the k-means algorithm andthe standard mixture-model (Duda and Hart, 1973), represent clusters of data points sharing similar patterns of Is and Os under the assumption that each data point belongs to, or was generated by, one and only one cluster-center; output activity is constrained to sum to 1. In contrast, a multiple-cause model permits more than one cluster-center to become fully active in accounting for an observed data vector. The advantage of a multiple cause model is that a relatively small number 27 28 Saund of hidden variables can be applied combinatorially to generate a large data set.

Researchers often try to understand-post hoc-representations that emerge in the hidden layers of a neural net following training. Interpretation is difficult because these representations are typically highly distributed and continuous. By "continuous," wemean that if one constructed a scatterplot over the hidden unit activity space of patterns obtained in response to various inputs, examination at any scale would reveal the patterns to be broadly distributed over the space.

This paper is a multidisciplinary review of empirical, statistical learning from a graphical model perspective. Well-known examples of graphical models include Bayesian networks, directed graphs representing a Markov chain, and undirected networks representing a Markov field. These graphical models are extended to model data analysis and empirical learning using the notation of plates. Graphical operations for simplifying and manipulating a problem are provided including decomposition, differentiation, andthe manipulation of probability models from the exponential family. Two standard algorithm schemas for learning are reviewed in a graphical framework: Gibbs sampling and the expectation maximizationalgorithm. Using these operations and schemas, some popular algorithms can be synthesized from their graphical specification. This includes versions of linear regression, techniques for feed-forward networks, and learning Gaussian and discrete Bayesian networks from data. The paper concludes by sketching some implications for data analysis and summarizing how some popular algorithms fall within the framework presented. The main original contributions here are the decompositiontechniques and the demonstration that graphical models provide a framework for understanding and developing complex learning algorithms.

Given a knowledge base KB containing first-order and statistical facts, we consider a principled method, called the random-worlds method, for computing a degree of belief that some formula Phi holds given KB. If we are reasoning about a world or system consisting of N individuals, then we can consider all possible worlds, or first-order models, withdomain {1,...,N} that satisfy KB, and compute thefraction of them in which Phi is true. We define the degree of belief to be the asymptotic value of this fraction as N grows large. We show that when the vocabulary underlying Phi andKB uses constants and unary predicates only, we can naturally associate an entropy with each world. As N grows larger,there are many more worlds with higher entropy. Therefore, we can usea maximum-entropy computation to compute the degree of belief. This result is in a similar spirit to previous work in physics and artificial intelligence, but is far more general. Of equal interest to the result itself are the limitations on its scope. Most importantly, the restriction to unary predicates seems necessary. Although the random-worlds method makes sense in general, the connection to maximum entropy seems to disappear in the non-unary case. These observations suggest unexpected limitations to the applicability of maximum-entropy methods.

This book (originally published in 1994 by Ellis Horwood) is now out of print. The copyright now resides with the editors who have decided to make the material freely available on the web.This book is based on the EC (ESPRIT) project StatLog which compare and evaluated a range of classification techniques, with an assessment of their merits, disadvantages and range of application. This integrated volume provides a concise introduction to each method, and reviews comparative trials in large-scale commercial and industrial problems. It makes accessible to a wide range of workers the complex issue of classification as approached through machine learning, statistics and neural networks, encouraging a cross-fertilization between these discplines.