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.

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.

We present a simple sparse greedy technique to approximate the maximum a posteriori estimate of Gaussian Processes with much improved scaling behaviour in the sample size m.

We propose a method of approximate dynamic programming for Markov decision processes (MDPs) using algebraic decision diagrams (ADDs). We produce near-optimal value functions and policies with much lower time and space requirements than exact dynamic programming. Our method reduces the sizes of the intermediate value functions generated during value iteration by replacing the values at the terminals of the ADD with ranges of values. Our method is demonstrated on a class of large MDPs (with up to 34 billion states), and we compare the results with the optimal value functions.

In this paper we will discuss a variational mean-field theory and its application to BNs, sigmoidal BNs in particular. We present a variational derivation of the mean-field theory, proposed by Plefka[2].

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).

Antarctica contains the most fertile meteorite hunting grounds on Earth. The pristine, dry and cold environment ensures that meteorites deposited there are preserved for long periods. Subsequent glacial flow of the ice sheets where they land concentrates them in particular areas. To date, most meteorites recovered throughout history have been done so in Antarctica in the last 20 years. Furthermore, they are less likely to be contaminated by terrestrial compounds.

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.

When trying to recover 3D structure from a set of images, the most difficult problem is establishing the correspondence between the measurements. Most existing approaches assume that features can be tracked across frames, whereas methods that exploit rigidity constraints to facilitate matching do so only under restricted camera motion. In this paper we propose a Bayesian approach that avoids the brittleness associated with singling out one "best" correspondence, and instead consider the distribution over all possible correspondences. We treat both a fully Bayesian approach that yields a posterior distribution, and a MAP approach that makes use of EM to maximize this posterior. We show how Markov chain Monte Carlo methods can be used to implement these techniques in practice, and present experimental results on real data.