One way to approximate inference in richly-connected graphical models is to apply the sum-product algorithm (a.k.a. The sum-product algorithm can be directly applied in Gaussian networks and in graphs for coding, but for many conditional probability functions - including the sigmoid function - direct application of the sum-product algorithm is not possible. We introduce "accumulator networks" that have low local complexity (but exponential global complexity) so the sum-product algorithm can be directly applied. In an accumulator network, the probability of a child given its parents is computed by accumulating the inputs from the parents in a Markov chain or more generally a tree. After giving expressions for inference and learning in accumulator networks, we give results on the "bars problem" and on the problem of extracting translated, overlapping faces from an image. 1 Introduction Graphical probability models with hidden variables are capable of representing complex dependencies between variables, filling in missing data and making Bayesoptimal decisions using probabilistic inferences (Hinton and Sejnowski 1986; Pearl 1988; Neal 1992).

How should we decide among competing explanations of a cognitive process given limited observations? The problem of model selection is at the heart of progress in cognitive science. In this paper, Minimum Description Length (MDL) is introduced as a method for selecting among computational models of cognition. We also show that differential geometry provides an intuitive understanding of what drives model selection in MDL. Finally, adequacy of MDL is demonstrated in two areas of cognitive modeling.

We introduce the mixture of Gaussian processes (MGP) model which is useful for applications in which the optimal bandwidth of a map is input dependent. The MGP is derived from the mixture of experts model and can also be used for modeling general conditional probability densities. We discuss how Gaussian processes -in particular in form of Gaussian process classification, the support vector machine and the MGP modelcan be used for quantifying the dependencies in graphical models. 1 Introduction Gaussian processes are typically used for regression where it is assumed that the underlying function is generated by one infinite-dimensional Gaussian distribution (i.e.

This paper presents a unified probabilistic framework for denoising and dereverberation of speech signals. The framework transforms the denoising and dereverberation problems into Bayes-optimal signal estimation. The key idea is to use a strong speech model that is pre-trained on a large data set of clean speech. Computational efficiency is achieved by using variational EM, working in the frequency domain, and employing conjugate priors. The framework covers both single and multiple microphones. We apply this approach to noisy reverberant speech signals and get results substantially better than standard methods.

Two well known classes of unsupervised procedures that can be cast in this manner are generative and recoding models. In a generative unsupervised framework, the environment generates training exampleswhich we will refer to as observations-by sampling from one distribution; the other distribution is embodied in the model. Examples of generative frameworks are mixtures of Gaussians (MoG) [2], factor analysis [4], and Boltzmann machines [8]. In the recoding unsupervised framework, the model transforms points from an obser- vation space to an output space, and the output distribution is compared either to a reference distribution or to a distribution derived from the output distribution. An example is independent component analysis (leA) [11], a method that discovers a representation of vector-valued observations in which the statistical dependence among the vector elements in the output space is minimized.

A serious problem in learning probabilistic models is the presence of hidden variables. These variables are not observed, yet interact with several of the observed variables. As such, they induce seemingly complex dependencies among the latter. In recent years, much attention has been devoted to the development of algorithms for learning parameters, and in some cases structure, in the presence of hidden variables. In this paper, we address the related problem of detecting hidden variables that interact with the observed variables.

We present a bound on the generalisation error of linear classifiers in terms of a refined margin quantity on the training set. The result is obtained in a PAC-Bayesian framework and is based on geometrical arguments in the space of linear classifiers. The new bound constitutes an exponential improvement of the so far tightest margin bound by Shawe-Taylor et al. [8] and scales logarithmically in the inverse margin. Even in the case of less training examples than input dimensions sufficiently large margins lead to nontrivial bound values and - for maximum margins - to a vanishing complexity term. Furthermore, the classical margin is too coarse a measure for the essential quantity that controls the generalisation error: the volume ratio between the whole hypothesis space and the subset of consistent hypotheses.

Learning of a smooth but nonparametric probability density can be regularized using methods of Quantum Field Theory. We implement a field theoretic prior numerically, test its efficacy, and show that the free parameter of the theory (,smoothness scale') can be determined self consistently by the data; this forms an infinite dimensional generalization of the MDL principle. Finally, we study the implications of one's choice of the prior and the parameterization and conclude that the smoothness scale determination makes density estimation very weakly sensitive to the choice of the prior, and that even wrong choices can be advantageous for small data sets. One of the central problems in learning is to balance'goodness of fit' criteria against the complexity of models. An important development in the Bayesian approach was thus the realization that there does not need to be any extra penalty for model complexity: if we compute the total probability that data are generated by a model, there is a factor from the volume in parameter space-the'Occam factor' -that discriminates against models with more parameters [1, 2].

Bayesian networks are graphical representations of probability distributions. In virtually all of the work on learning these networks, the assumption is that we are presented with a data set consisting of randomly generated instances from the underlying distribution. In many situations, however, we also have the option of active learning, where we have the possibility of guiding the sampling process by querying for certain types of samples. This paper addresses the problem of estimating the parameters of Bayesian networks in an active learning setting. We provide a theoretical framework for this problem, and an algorithm that chooses which active learning queries to generate based on the model learned so far. We present experimental results showing that our active learning algorithm can significantly reduce the need for training data in many situations.

Subsequently, SVMs have been modified to handle regression [12] and GPs have been adapted to the problem of classification [8]. Both schemes essentially work in the same function space that is characterised by kernels (SVM) and covariance functions (GP), respectively. While the formal similarity of the two methods is striking the underlying paradigms of inference are very different. The SVM was inspired by results from statistical/PAC learning theory while GPs are usually considered in a Bayesian framework. This ideological clash can be viewed as a continuation in machine learning of the by now classical disagreement between Bayesian and frequentistic statistics.