The relationship between the Bayesian approach and the minimum description length approach is established. We sharpen and clarify the general modeling principles MDL and MML, abstracted as the ideal MDL principle and defined from Bayes's rule by means of Kolmogorov complexity. The basic condition under which the ideal principle should be applied is encapsulated as the Fundamental Inequality, which in broad terms states that the principle is valid when the data are random, relative to every contemplated hypothesis and also these hypotheses are random relative to the (universal) prior. Basically, the ideal principle states that the prior probability associated with the hypothesis should be given by the algorithmic universal probability, and the sum of the log universal probability of the model plus the log of the probability of the data given the model should be minimized. If we restrict the model class to the finite sets then application of the ideal principle turns into Kolmogorov's minimal sufficient statistic. In general we show that data compression is almost always the best strategy, both in hypothesis identification and prediction.

Applications of Gaussian mixture models occur frequently in the fields of statistics and artificial neural networks.

In contrast to the maximum likelihood approach which finds only a single estimate for the regression parameters, the Bayesian approach yields a distribution of weight parameters, p(wID), conditional on the training data D, and predictions are ex- ·Present address: SNN, University of Nijmegen, Geert Grooteplein 21, Nijmegen, The Netherlands.

In this paper, we discuss regularisation in online/sequential learning algorithms. In environments where data arrives sequentially, techniques such as cross-validation to achieve regularisation or model selection are not possible. Further, bootstrapping to determine a confidence level is not practical. To surmount these problems, a minimum variance estimation approach that makes use of the extended Kalman algorithm for training multi-layer perceptrons is employed. The novel contribution of this paper is to show the theoretical links between extended Kalman filtering, Sutton's variable learning rate algorithms and Mackay's Bayesian estimation framework. In doing so, we propose algorithms to overcome the need for heuristic choices of the initial conditions and noise covariance matrices in the Kalman approach.

This paper reports about an application of Bayes' inferred neural network classifiers in the field of automatic sleep staging. The reason for using Bayesian learning for this task is twofold. First, Bayesian inference is known to embody regularization automatically. Second, a side effect of Bayesian learning leads to larger variance of network outputs in regions without training data. This results in well known moderation effects, which can be used to detect outliers. In a 5 fold cross-validation experiment the full Bayesian solution found with R. Neals hybrid Monte Carlo algorithm, was not better than a single maximum a-posteriori (MAP) solution found with D.J. MacKay's evidence approximation. In a second experiment we studied the properties of both solutions in rejecting classification of movement artefacts.

We address the problem oflearning structure in nonlinear Markov networks with continuous variables. This can be viewed as non-Gaussian multidimensional density estimation exploiting certain conditional independencies in the variables. Markov networks are a graphical way of describing conditional independencies well suited to model relationships which do not exhibit a natural causal ordering. We use neural network structures to model the quantitative relationships between variables. The main focus in this paper will be on learning the structure for the purpose of gaining insight into the underlying process. Using two data sets we show that interesting structures can be found using our approach. Inference will be briefly addressed.

Bayesian methods have been successfully applied to regression and classification problems in multi-layer perceptrons. We present a novel application of Bayesian techniques to Radial Basis Function networks by developing a Gaussian approximation to the posterior distribution which, for fixed basis function widths, is analytic in the parameters. The setting of regularization constants by crossvalidation is wasteful as only a single optimal parameter estimate is retained. We treat this issue by assigning prior distributions to these constants, which are then adapted in light of the data under a simple re-estimation formula. 1 Introduction Radial Basis Function networks are popular regression and classification tools[lO]. For fixed basis function centers, RBFs are linear in their parameters and can therefore be trained with simple one shot linear algebra techniques[lO]. The use of unsupervised techniques to fix the basis function centers is, however, not generally optimal since setting the basis function centers using density estimation on the input data alone takes no account of the target values associated with that data. Ideally, therefore, we should include the target values in the training procedure[7, 3, 9]. Unfortunately, allowing centers to adapt to the training targets leads to the RBF being a nonlinear function of its parameters, and training becomes more problematic. Most methods that perform supervised training of RBF parameters minimize the ·Present address: SNN, University of Nijmegen, Geert Grooteplein 21, Nijmegen, The Netherlands.

Prioritized sweeping is a model-based reinforcement learning method that attempts to focus an agent's limited computational resources to achieve a good estimate of the value of environment states. To choose effectively where to spend a costly planning step, classic prioritized sweeping uses a simple heuristic to focus computation on the states that are likely to have the largest errors. In this paper, we introduce generalized prioritized sweeping, a principled method for generating such estimates in a representation-specific manner. This allows us to extend prioritized sweeping beyond an explicit, state-based representation to deal with compact representations that are necessary for dealing with large state spaces. We apply this method for generalized model approximators (such as Bayesian networks), and describe preliminary experiments that compare our approach with classical prioritized sweeping.

This paper presents a new approach to the problem of modelling daily rainfall using neural networks. We first model the conditional distributions of rainfall amounts, in such a way that the model itself determines the order of the process, and the time-dependent shape and scale of the conditional distributions. After integrating over particular weather patterns, we are able to extract seasonal variations and long-term trends. 1 Introduction Analysis of rainfall data is important for many agricultural, ecological and engineering activities. Design of irrigation and drainage systems, for instance, needs to take account not only of mean expected rainfall, but also of rainfall volatility. Estimates of crop yields also depend on the distribution of rainfall during the growing season, as well as on the overall amount.

This paper reports about an application of Bayes' inferred neural network classifiers in the field of automatic sleep staging. The reason for using Bayesian learning for this task is twofold. First, Bayesian inference is known to embody regularization automatically. Second, a side effect of Bayesian learning leads to larger variance of network outputs in regions without training data. This results in well known moderation effects, which can be used to detect outliers. In a 5 fold cross-validation experiment the full Bayesian solution found with R. Neals hybrid Monte Carlo algorithm, was not better than a single maximum a-posteriori (MAP) solution found with D.J. MacKay's evidence approximation. In a second experiment we studied the properties of both solutions in rejecting classification of movement artefacts.