The essential property of BBNs is summarized by the Markov condition, which asserts that each variable is independent of its non-descendants given its parents.

When combining a set of learned models to form an improved estimator, the issue of redundancy or multicollinearity in the set of models must be addressed. A progression of existing approaches and their limitations with respect to the redundancy is discussed. A new approach, PCR *, based on principal components regression is proposed to address these limitations. An evaluation of the new approach on a collection of domains reveals that: 1) PCR* was the most robust combination method as the redundancy of the learned models increased, 2) redundancy could be handled without eliminating any of the learned models, and 3) the principal components of the learned models provided a continuum of "regularized" weights from which PCR * could choose.

To obtain classification systems with both good generalization performance and efficiency in space and time, we propose a learning method based on combinations of weak classifiers, where weak classifiers are linear classifiers (perceptrons) which can do a little better than making random guesses. A randomized algorithm is proposed to find the weak classifiers. They· are then combined through a majority vote. As demonstrated through systematic experiments, the method developed is able to obtain combinations of weak classifiers with good generalization performance and a fast training time on a variety of test problems and real applications.

This work investigates the representational and inductive capabilities of time-delay neural networks (TDNNs) in general, and of two subclasses of TDNN, those with delays only on the inputs (IDNN), and those which include delays on hidden units (HDNN). Both architectures are capable of representing the same class of languages, the definite memory machine (DMM) languages, but the delays on the hidden units in the HDNN helps it outperform the IDNN on problems composed of repeated features over short time windows.

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!

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.

The genetic algorithm (GA) is a heuristic search procedure based on mechanisms abstracted from population genetics. In a previous paper [Baluja & Caruana, 1995], we showed that much simpler algorithms, such as hillcIimbing and Population Based Incremental Learning (PBIL), perform comparably to GAs on an optimization problem custom designed to benefit from the GA's operators. This paper extends these results in two directions. First, in a large-scale empirical comparison of problems that have been reported in GA literature, we show that on many problems, simpler algorithms can perform significantly better than GAs. Second, we describe when crossover is useful, and show how it can be incorporated into PBIL. 1 IMPLICIT VS.

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.

It is shown that conventional computers can be exponentiallx faster than planar Hopfield networks: although there are planar Hopfield networks that take exponential time to converge, a stable state of an arbitrary planar Hopfield network can be found by a conventional computer in polynomial time.

This article presents a new result about the size of a multilayer neural network computing real outputs for exact learning of a finite set of real samples. The architecture of the network is feedforward, with one hidden layer and several outputs. Starting from a fixed training set, we consider the network as a function of its weights. We derive, for a wide family of transfer functions, a lower and an upper bound on the number of hidden units for exact learning, given the size of the dataset and the dimensions of the input and output spaces. The context of our work is rather similar to the well-known results of Baum et al. [1, 2,3,5, 10], but we consider both real inputs and outputs, instead ofthe dichotomies usually addressed.