The full Bayesian method for applying neural networks to a prediction problem is to set up the prior/hyperprior structure for the net and then perform the necessary integrals. However, these integrals are not tractable analytically, and Markov Chain Monte Carlo (MCMC) methods are slow, especially if the parameter space is high-dimensional. Using Gaussian processes we can approximate the weight space integral analytically, so that only a small number of hyperparameters need be integrated over by MCMC methods. We have applied this idea to classification problems, obtaining excellent results on the real-world problems investigated so far. 1 INTRODUCTION To make predictions based on a set of training data, fundamentally we need to combine our prior beliefs about possible predictive functions with the data at hand. In the Bayesian approach to neural networks a prior on the weights in the net induces a prior distribution over functions.

A classifier is called consistent with respect to a given set of classlabeled points if it correctly classifies the set. We consider classifiers defined by unions of local separators and propose algorithms for consistent classifier reduction. The expected complexities of the proposed algorithms are derived along with the expected classifier sizes. In particular, the proposed approach yields a consistent reduction of the nearest neighbor classifier, which performs "firm" classification, assigning each new object to a class, regardless of the data structure. The proposed reduction method suggests a notion of "soft" classification, allowing for indecision with respect to objects which are insufficiently or ambiguously supported by the data. The performances of the proposed classifiers in predicting stock behavior are compared to that achieved by the nearest neighbor method.

We consider the problem of prediction of stationary time series, using the architecture known as mixtures of experts (MEM). Here we suggest a mixture which blends several autoregressive models. This study focuses on some theoretical foundations of the prediction problem in this context. More precisely, it is demonstrated that this model is a universal approximator, with respect to learning the unknown prediction function. This statement is strengthened as upper bounds on the mean squared error are established. Based on these results it is possible to compare the MEM to other families of models (e.g., neural networks and state dependent models). It is shown that a degenerate version of the MEM is in fact equivalent to a neural network, and the number of experts in the architecture plays a similar role to the number of hidden units in the latter model.

The Support Vector (SV) method was recently proposed for estimating regressions, constructing multidimensional splines, and solving linear operator equations [Vapnik, 1995]. In this presentation we report results of applying the SV method to these problems.

We follow a similar approach to (Zhu & Rohwer, to appear 1996) in using a Gaussian process to define a prior over the space of functions, so that the expected generalisation cost under the posterior can be determined. The optimal model is defined in terms of the restriction of this posterior to the subspace defined by the model. The optimum is easily determined for linear models over a set of basis functions. We go on to compute the generalisation cost (with an error bar) for all models of this class, which we demonstrate to include the RAMnets.

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.

A new regression technique based on Vapnik's concept of support vectors is introduced. We compare support vector regression (SVR) with a committee regression technique (bagging) based on regression trees and ridge regression done in feature space. On the basis of these experiments, it is expected that SVR will have advantages in high dimensionality space because SVR optimization does not depend on the dimensionality of the input space.

Baum and Haussler [4] used these results to give sample size bounds for multi-layer threshold networks Generalization and the Size of the Weights in Neural Networks 135 that grow at least as quickly as the number of weights (see also [7]). However, for pattern classification applications the VC-bounds seem loose; neural networks often perform successfully with training sets that are considerably smaller than the number of weights. This paper shows that for classification problems on which neural networks perform well, if the weights are not too big, the size of the weights determines the generalization performance. In contrast with the function classes and algorithms considered in the VC-theory, neural networks used for binary classification problems have real-valued outputs, and learning algorithms typically attempt to minimize the squared error of the network output over a training set. As well as encouraging the correct classification, this tends to push the output away from zero and towards the target values of { -1, I}.

The parameter space of neural networks has a Riemannian metric structure. The natural Riemannian gradient should be used instead of the conventional gradient, since the former denotes the true steepest descent direction of a loss function in the Riemannian space. The behavior of the stochastic gradient learning algorithm is much more effective if the natural gradient is used. The present paper studies the information-geometrical structure of perceptrons and other networks, and prove that the online learning method based on the natural gradient is asymptotically as efficient as the optimal batch algorithm. Adaptive modification of the learning constant is proposed and analyzed in terms of the Riemannian measure and is shown to be efficient.

The weakly electric fish, Eigenmannia, generates a quasi sinusoidal, dipole-like electric field at individually fixed frequencies (250 - 600 Hz) by discharging an electric organ located in its tail (see Bullock and Heilgenberg, 1986 for reviews). The fish sense local changes in the electric field by means of two types of tuberous electroreceptors located on the body surface.