The perturbation method which we have presented overcomes the limitations of standard approaches, which are only appropriate for models with a single layer of adjustable weights, albeit at considerable computational expense. It has the added bonus of automatically taking into account the effect of regularisation techniques such as weight decay. The experimental results illustrate the application of the technique to two simple problems. As expected the number of degrees of freedom in the models is found to be related to the amount of weight decay used during training. The equivalent kernels are found to vary significantly in different regions of input space and the functions reconstructed from the estimated smoother matrices closely match the origna!

In the Bayes approach to statistical inference [Berger, 1985] one assumes that the prior uncertainty about parameters of an unknown data generating mechanism can be encoded in a probability distribution, the so called prior. Using the prior and the likelihood of the data given the parameters, the posterior distribution of the parameters can be derived from Bayes rule. From this posterior, various estimates for functions ofthe parameter, like predictions about unseen data, can be calculated. However, in general, those predictions cannot be realised by specific parameter values, but only by an ensemble average over parameters according to the posterior probability. Hence, exact implementations of Bayes method for neural networks require averages over network parameters which in general can be performed by time consuming 226 M. Opper and O. Winther Monte Carlo procedures.

In this paper we apply the method of complexity regularization to derive estimation bounds for nonlinear function estimation using a single hidden layer radial basis function network.

We propose a new method to compute prediction intervals. Especially for small data sets the width of a prediction interval does not only depend on the variance of the target distribution, but also on the accuracy of our estimator of the mean of the target, i.e., on the width of the confidence interval. The confidence interval follows from the variation in an ensemble of neural networks, each of them trained and stopped on bootstrap replicates of the original data set. A second improvement is the use of the residuals on validation patterns instead of on training patterns for estimation of the variance of the target distribution. As illustrated on a synthetic example, our method is better than existing methods with regard to extrapolation and interpolation in data regimes with a limited amount of data, and yields prediction intervals which actual confidence levels are closer to the desired confidence levels. 1 STATISTICAL INTERVALS In this paper we will consider feedforward neural networks for regression tasks: estimating an underlying mathematical function between input and output variables based on a finite number of data points possibly corrupted by noise.

Time series prediction is one of the major applications of neural networks. After a short introduction into the basic theoretical foundations we argue that the iterated prediction of a dynamical system may be interpreted as a model of the system dynamics. By means of RBF neural networks we describe a modeling approach and extend it to be able to model instationary systems. As a practical test for the capabilities of the method we investigate the modeling of musical and speech signals and demonstrate that the model may be used for synthesis of musical and speech signals.

Recently, there has been a vigorous debate concerning the nature of neural coding (Rieke et al. 1996; Stevens and Zador 1995; Shadlen and Newsome 1994). The prevailing view has been that the mean firing rate conveys all information about the sensory stimulus in a spike train and the precise timing of the individual spikes is noise. This belief is, in part, based on a lack of correlation between the precise timing of the spikes and the sensory qualities of the stimulus under study, particularly, on a lack of spike timing repeatability when identical stimulation is delivered. This view has been challenged by a number of recent studies, in which highly repeatable temporal patterns of spikes can be observed both in vivo (Bair and Koch 1996; Abeles et al. 1993) and in vitro (Mainen and Sejnowski 1994). Furthermore, application of information theory to the coding problem in the frog and house fly (Bialek et al. 1991; Bialek and Rieke 1992) suggested that additional information could be extracted from spike timing. In the absence of direct evidence for a timing code in the cerebral cortex, the role of spike timing in neural coding remains controversial.

We compare different methods to combine predictions from neural networks trained on different bootstrap samples of a regression problem. One of these methods, introduced in [6] and which we here call balancing, is based on the analysis of the ensemble generalization error into an ambiguity term and a term incorporating generalization performances of individual networks. We show how to estimate these individual errors from the residuals on validation patterns. Weighting factors for the different networks follow from a quadratic programming problem. On a real-world problem concerning the prediction of sales figures and on the well-known Boston housing data set, balancing clearly outperforms other recently proposed alternatives as bagging [1] and bumping [8]. 1 EARLY STOPPING AND BOOTSTRAPPING Stopped training is a popular strategy to prevent overfitting in neural networks.

To reduce the computational complexity of classification systems using tangent distance, Hastie et al. (HSS) developed an algorithm to devise rich models for representing large subsets of the data which computes automatically the "best" associated tangent subspace. Schwenk & Milgram proposed a discriminant modular classification system (Diabolo) based on several autoassociative multilayer perceptrons which use tangent distance as error reconstruction measure. We propose a gradient based constructive learning algorithm for building a tangent subspace model with discriminant capabilities which combines several of the the advantages of both HSS and Diabolo: devised tangent models hold discriminant capabilities, space requirements are improved with respect to HSS since our algorithm is discriminant and thus it needs fewer prototype models, dimension of the tangent subspace is determined automatically by the constructive algorithm, and our algorithm is able to learn new transformations.

Sometimes local features group into regions, as in texture segmentation; at other times they group into contours which may represent object boundaries. Although much is known about the processing steps that extract local features such as oriented input edges, it is still unclear how local features are grouped into global ones more meaningful for objects.

There is now considerable interest in using ensembles or committees of learning machines to improve the performance of the system over that of a single learning machine. In most neural network ensembles, the ensemble members are trained on either the same data (Hansen & Salamon 1990) or different subsets of the data (Perrone & Cooper 1993). The ensemble members typically have different initial conditions and/or different architectures. The subsets of the data may be chosen at random, with prior knowledge or by some principled approach e.g.