The prior can be obtained by placing prior distributions on the weights in a neural 494 P. W Goldberg, C. K. L Williams and C. M. Bishop network, although we would argue that it is perhaps more natural to place priors directly overfunctions. One tractable way of doing this is to create a Gaussian process prior. This has the advantage that predictions can be made from the posterior using only matrix multiplication for fixed hyperparameters and a global noise level. In contrast, for neural networks (with fixed hyperparameters and a global noise level) it is necessary to use approximations or Markov chain Monte Carlo (MCMC) methods. Rasmussen(1996) has demonstrated that predictions obtained with Gaussian processes are as good as or better than other state-of-the art predictors. In much of the work on regression problems in the statistical and neural networks literatures, it is assumed that there is a global noise level, independent of the input vector x. The book by Bishop (1995) and the papers by Bishop (1994), MacKay (1995) and Bishop and Qazaz (1997) have examined the case of input-dependent noise for parametric models such as neural networks.

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 iswasteful 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 betrained 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, allowingcenters 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.

We address the problem oflearning structure in nonlinear Markov networks with continuous variables. This can be viewed as non-Gaussian multidimensional densityestimation exploiting certain conditional independencies in the variables. Markov networks are a graphical way of describing conditional independencieswell suited to model relationships which do not exhibit a natural causal ordering. We use neural network structures to model the quantitative relationships between variables.

A simple linear averaging of the outputs of several networks as e.g. in bagging [3], seems to follow naturally from a bias/variance decomposition of the sum-squared error. The sum-squared error of the average model is a quadratic function of the weighting factors assigned to the networks in the ensemble [7], suggesting a quadratic programming algorithm for finding the "optimal" weighting factors. If we interpret the output of a network as a probability statement, the sum-squared error corresponds to minus the loglikelihood or the Kullback-Leibler divergence, and linear averaging of the outputs tologarithmic averaging of the probability statements: the logarithmic opinion pool. The crux of this paper is that this whole story about model averaging, bias/variancedecompositions, and quadratic programming to find the optimal weighting factors, is not specific for the sumsquared error,but applies to the combination of probability statements of any kind in a logarithmic opinion pool, as long as the Kullback-Leibler divergence plays the role of the error measure. As examples we treat model averaging for classification models under a cross-entropy error measure and models for estimating variances.

Volker Tresp and Thomas Briegel * Siemens AG Corporate Technology Otto-Hahn-Ring 6 81730 Miinchen, Germany Abstract We consider neural network models for stochastic nonlinear dynamical systems where measurements of the variable of interest are only available atirregular intervals i.e. most realizations are missing. Difficulties arise since the solutions for prediction and maximum likelihood learning withmissing data lead to complex integrals, which even for simple cases cannot be solved analytically. In this paper we propose a specific combinationof a nonlinear recurrent neural predictive model and a linear error model which leads to tractable prediction and maximum likelihood adaptation rules. In particular, the recurrent neural network can be trained using the real-time recurrent learning rule and the linear error model can be trained by an EM adaptation rule, implemented using forward-backwardKalman filter equations. The model is applied to predict the glucose/insulin metabolism of a diabetic patient where blood glucose measurements are only available a few times a day at irregular intervals.

We present a new approximate learning algorithm for Boltzmann Machines, using a systematic expansion of the Gibbs free energy to second order in the weights. The linear response correction to the correlations is given by the Hessian of the Gibbs free energy. The computational complexity of the algorithm is cubic in the number of neurons. We compare the performance of the exact BM learning algorithm with first order (Weiss) mean field theory and second order (TAP) mean field theory. The learning task consists of a fully connected Ising spin glass model on 10 neurons. We conclude that 1) the method works well for paramagnetic problems 2) the TAP correction gives a significant improvement over the Weiss mean field theory, both for paramagnetic and spin glass problems and 3) that the inclusion of diagonal weights improves the Weiss approximation for paramagnetic problems, but not for spin glass problems.

A neural network approach to stereovision is presented based on aliasing effects of simple disparity estimators and a fast coherencedetection scheme.Within a single network structure, a dense disparity map with an associated validation map and, additionally, the fused cyclopean view of the scene are available. The network operations are based on simple, biological plausible circuitry; the algorithm is fully parallel and non-iterative. 1 Introduction Humans experience the three-dimensional world not as it is seen by either their left or right eye, but from a position of a virtual cyclopean eye, located in the middle between the two real eye positions. The different perspectives between the left and right eyes cause slight relative displacements of objects in the two retinal images (disparities), which make a simple superposition of both images without diplopia impossible. Proper fusion of the retinal images into the cyclopean view requires the registration of both images to a common coordinate system, which in turn requires calculation of disparities for all image areas which are to be fused.

We apply information maximization / maximum likelihood blind source separation [2, 6) to complex valued signals mixed with complex valuednonstationary matrices. This case arises in radio communications withbaseband signals. We incorporate known source signal distributions in the adaptation, thus making the algorithms less "blind". This results in drastic reduction of the amount of data needed for successful convergence. Adaptation to rapidly changing signal mixing conditions, such as to fading in mobile communications, becomesnow feasible as demonstrated by simulations. 1 Introduction In SDMA (spatial division multiple access) the purpose is to separate radio signals of interfering users (either intentional or accidental) from each others on the basis of the spatial characteristics of the signals using smart antennas, array processing, and beamforming [5, 8).

We present a method for the analysis of nonstationary time series withmultiple operating modes. In particular, it is possible to detect and to model both a switching of the dynamics and a less abrupt, time consuming drift from one mode to another. This is achieved in two steps. First, an unsupervised training method provides predictionexperts for the inherent dynamical modes. Then, the trained experts are used in a hidden Markov model that allows to model drifts. An application to physiological wake/sleep data demonstrates that analysis and modeling of real-world time series can be improved when the drift paradigm is taken into account.

The S-Map is a network with a simple learning algorithm that combines theself-organization capability of the Self-Organizing Map (SOM) and the probabilistic interpretability of the Generative Topographic Mapping(GTM). The simulations suggest that the S Map algorithm has a stronger tendency to self-organize from random initialconfiguration than the GTM. The S-Map algorithm can be further simplified to employ pure Hebbian learning, without changingthe qualitative behaviour of the network. 1 Introduction The self-organizing map (SOM; for a review, see [1]) forms a topographic mapping from the data space onto a (usually two-dimensional) output space. The SOM has been succesfully used in a large number of applications [2]; nevertheless, there are some open theoretical questions, as discussed in [1, 3]. Most of these questions arise because of the following two facts: the SOM is not a generative model, i.e. it does not generate a density in the data space, and it does not have a well-defined objective function that the training process would strictly minimize.