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.

Jensen's inequality is a powerful mathematical tool and one of the workhorses in statistical learning. Its applications therein include the EM algorithm, Bayesian estimation and Bayesian inference. Jensen computes simplelower bounds on otherwise intractable quantities such as products of sums and latent log-likelihoods. This simplification then permits operationslike integration and maximization. Quite often (i.e. in discriminative learning) upper bounds are needed as well. We derive and prove an efficient analytic inequality that provides such variational upper bounds. This inequality holds for latent variable mixtures of exponential family distributions and thus spans a wide range of contemporary statistical models.We also discuss applications of the upper bounds including maximum conditional likelihood, large margin discriminative models and conditional Bayesian inference. Convergence, efficiency and prediction results are shown.

Our focus, however, is on the discovery of scene statistics which are useful for solving visual inference problems. For example, in related work [5] we have analyzed the statistics of filter responses on and off edges and hence derived effective edge detectors.

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 beused for quantifying the dependencies in graphical models. 1 Introduction Gaussian processes are typically used for regression where it is assumed that the underlying functionis generated by one infinite-dimensional Gaussian distribution (i.e.

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. The practical relevance of the result lies in the fact that the well-known support vector machine is optimal w.r.t. the new bound only if the feature vectors are all of the same length.

Learning of a smooth but nonparametric probability density can be regularized usingmethods of Quantum Field Theory. We implement a field theoretic prior numerically, test its efficacy, and show that the free parameter ofthe theory (,smoothness scale') can be determined self consistently bythe 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].

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 amongthe 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, weaddress the related problem of detecting hidden variables that interact with the observed variables. This problem is of interest both for improving our understanding of the domain and as a preliminary step that guides the learning procedure towards promising models.

A Bayes network based classifier for distinguishing terrestrial rocks from meteorites is implemented onboard the Nomad robot. Equipped with a camera, spectrometer and eddy current sensor, this robot searched the ice sheets of Antarctica and autonomously made the first robotic identification of a meteorite, in January 2000 at the Elephant Moraine. This paper discusses rock classification from a robotic platform, and describes the system onboard Nomad. 1 Introduction Figure 1: Human meteorite search with snowmobiles on the Antarctic ice sheets, and on foot in the moraines. 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.

We use graphical models to explore the question of how people learn simple causalrelationships from data. The two leading psychological theories canboth be seen as estimating the parameters of a fixed graph. We argue that a complete account of causal induction should also consider how people learn the underlying causal graph structure, and we propose to model this inductive process as a Bayesian inference. Our argument is supported through the discussion of three data sets. 1 Introduction Causality plays a central role in human mental life. Our behavior depends upon our understanding ofthe causal structure of our environment, and we are remarkably good at inferring causation from mere observation. Constructing formal models of causal induction is currently a major focus of attention in computer science [7], psychology [3,6], and philosophy [5].This paper attempts to connect these literatures, by framing the debate between two major psychological theories in the computational language of graphical models. We show that existing theories equate human causal induction with maximum likelihood parameter estimationon a fixed graphical structure, and we argue that to fully account for human behavioral data, we must also postulate that people make Bayesian inferences about the underlying causal graph structure itself.