Bayesian methods for neural networks

Bayesian techniques have been developed over many years in a range of different fields, but have only recently been applied to the problem of learning in neural networks. As well as providing a consistent framework for statistical pattern recognition, the Bayesian approach offers a number of practical advantages including a potential solution to the problem of over-fitting. This chapter aims to provide an introductory overview of the application of Bayesian methods to neural networks. It assumes the reader is familiar with standard feed-forward network models and how to train them using conventional techniques.

Learning with regularizers in multilayer neural networks

We study the effect of regularization in an on-line gradient-descent learning scenario for a general two-layer student network with an arbitrary number of hidden units. Training examples are randomly drawn input vectors labelled by a two-layer teacher network with an arbitrary number of hidden units which may be corrupted by Gaussian output noise. We examine the effect of weight decay regularization on the dynamical evolution of the order parameters and generalization error in various phases of the learning process, in both noiseless and noisy scenarios.

Estimating conditional volatility with neural networks

It is well known that one of the obstacles to effective forecasting of exchange rates is heteroscedasticity (non-stationary conditional variance). The autoregressive conditional heteroscedastic (ARCH) model and its variants have been used to estimate a time dependent variance for many financial time series. However, such models are essentially linear in form and we can ask whether a non-linear model for variance can improve results just as non-linear models (such as neural networks) for the mean have done. In this paper we consider two neural network models for variance estimation. Mixture Density Networks (Bishop 1994, Nix and Weigend 1994) combine a Multi-Layer Perceptron (MLP) and a mixture model to estimate the conditional data density. They are trained using a maximum likelihood approach. However, it is known that maximum likelihood estimates are biased and lead to a systematic under-estimate of variance. More recently, a Bayesian approach to parameter estimation has been developed (Bishop and Qazaz 1996) that shows promise in removing the maximum likelihood bias. However, up to now, this model has not been used for time series prediction. Here we compare these algorithms with two other models to provide benchmark results: a linear model (from the ARIMA family), and a conventional neural network trained with a sum-of-squares error function (which estimates the conditional mean of the time series with a constant variance noise model). This comparison is carried out on daily exchange rate data for five currencies.

Using Bayesian neural networks to classify segmented images

We present results that compare the performance of neural networks trained with two Bayesian methods, (i) the Evidence Framework of MacKay (1992) and (ii) a Markov Chain Monte Carlo method due to Neal (1996) on a task of classifying segmented outdoor images. We also investigate the use of the Automatic Relevance Determination method for input feature selection.

Globally optimal parameters for on-line learning in multilayer neural networks

We present a framework for calculating globally optimal parameters, within a given time frame, for on-line learning in multilayer neural networks. We demonstrate the capability of this method by computing optimal learning rates in typical learning scenarios. A similar treatment allows one to determine the relevance of related training algorithms based on modifications to the basic gradient descent rule as well as to compare different training methods.

Globally optimal on-line learning rules for multi-layer neural networks

We present a method for determining the globally optimal on-line learning rule for a soft committee machine under a statistical mechanics framework. This rule maximizes the total reduction in generalization error over the whole learning process. A simple example demonstrates that the locally optimal rule, which maximizes the rate of decrease in generalization error, may perform poorly in comparison.

Globally optimal learning rates in multilayer neural networks

A method for calculating the globally optimal learning rate in on-line gradient-descent training of multilayer neural networks is presented. The method is based on a variational approach which maximizes the decrease in generalization error over a given time frame. We demonstrate the method by computing optimal learning rates in typical learning scenarios. The method can also be employed when different learning rates are allowed for different parameter vectors as well as to determine the relevance of related training algorithms based on modifications to the basic gradient descent rule.

Computation with infinite neural networks

For neural networks with a wide class of weight priors, it can be shown that in the limit of an infinite number of hidden units, the prior over functions tends to a gaussian process. In this article, analytic forms are derived for the covariance function of the gaussian processes corresponding to networks with sigmoidal and gaussian hidden units. This allows predictions to be made efficiently using networks with an infinite number of hidden units and shows, somewhat paradoxically, that it may be easier to carry out Bayesian prediction with infinite networks rather than finite ones.

Combining spatially distributed predictions from neural networks

In this report we discuss the problem of combining spatially-distributed predictions from neural networks. An example of this problem is the prediction of a wind vector-field from remote-sensing data by combining bottom-up predictions (wind vector predictions on a pixel-by-pixel basis) with prior knowledge about wind-field configurations. This task can be achieved using the scaled-likelihood method, which has been used by Morgan and Bourlard (1995) and Smyth (1994), in the context of Hidden Markov modelling

Point-wise confidence interval estimation by neural networks: A comparative study based on automotive engine calibration.

In developing neural network techniques for real world applications it is still very rare to see estimates of confidence placed on the neural network predictions. This is a major deficiency, especially in safety-critical systems. In this paper we explore three distinct methods of producing point-wise confidence intervals using neural networks. We compare and contrast Bayesian, Gaussian Process and Predictive error bars evaluated on real data. The problem domain is concerned with the calibration of a real automotive engine management system for both air-fuel ratio determination and on-line ignition timing. This problem requires real-time control and is a good candidate for exploring the use of confidence predictions due to its safety-critical nature.

First year qualifying report: neural networks for extracting wind vectors from satellite scatterometer data

The ERS-1 Satellite was launched in July 1991 by the European Space Agency into a polar orbit at about km800, carrying a C-band scatterometer. A scatterometer measures the amount of radar back scatter generated by small ripples on the ocean surface induced by instantaneous local winds. Operational methods that extract wind vectors from satellite scatterometer data are based on the local inversion of a forward model, mapping scatterometer observations to wind vectors, by the minimisation of a cost function in the scatterometer measurement space.par This report uses mixture density networks, a principled method for modelling conditional probability density functions, to model the joint probability distribution of the wind vectors given the satellite scatterometer measurements in a single cell (the `inverse' problem). The complexity of the mapping and the structure of the conditional probability density function are investigated by varying the number of units in the hidden layer of the multi-layer perceptron and the number of kernels in the Gaussian mixture model of the mixture density network respectively. The optimal model for networks trained per trace has twenty hidden units and four kernels. Further investigation shows that models trained with incidence angle as an input have results comparable to those models trained by trace. A hybrid mixture density network that incorporates geophysical knowledge of the problem confirms other results that the conditional probability distribution is dominantly bimodal.par The wind retrieval results improve on previous work at Aston, but do not match other neural network techniques that use spatial information in the inputs, which is to be expected given the ambiguity of the inverse problem. Current work uses the local inverse model for autonomous ambiguity removal in a principled Bayesian framework. Future directions in which these models may be improved are given.

Neural networks: A principled perspective

Introductory accounts of artificial neural networks often rely for motivation on analogies with models of information processing in biological networks. One limitation of such an approach is that it offers little guidance on how to find optimal algorithms, or how to verify the correct performance of neural network systems. A central goal of this paper is to draw attention to a quite different viewpoint in which neural networks are seen as algorithms for statistical pattern recognition based on a principled, i.e. theoretically well-founded, framework. We illustrate the concept of a principled viewpoint by considering a specific issue concerned with the interpretation of the outputs of a trained network. Finally, we discuss the relevance of such an approach to the issue of the validation and verification of neural network systems.

The interplay between microscopic and mesoscopic structures in complex networks

Understanding a complex network's structure holds the key to understanding its function. The physics community has contributed a multitude of methods and analyses to this cross-disciplinary endeavor. Structural features exist on both the microscopic level, resulting from differences between single node properties, and the mesoscopic level resulting from properties shared by groups of nodes. Disentangling the determinants of network structure on these different scales has remained a major, and so far unsolved, challenge. Here we show how multiscale generative probabilistic exponential random graph models combined with efficient, distributive message-passing inference techniques can be used to achieve this separation of scales, leading to improved detection accuracy of latent classes as demonstrated on benchmark problems. It sheds new light on the statistical significance of motif-distributions in neural networks and improves the link-prediction accuracy as exemplified for gene-disease associations in the highly consequential Online Mendelian Inheritance in Man database. © 2011 Reichardt et al.

Exploratory database visualisation: the application and assessment of data and dimensionality reduction

This thesis describes the development of a complete data visualisation system for large tabular databases, such as those commonly found in a business environment. A state-of-the-art 'cyberspace cell' data visualisation technique was investigated and a powerful visualisation system using it was implemented. Although allowing databases to be explored and conclusions drawn, it had several drawbacks, the majority of which were due to the three-dimensional nature of the visualisation. A novel two-dimensional generic visualisation system, known as MADEN, was then developed and implemented, based upon a 2-D matrix of 'density plots'. MADEN allows an entire high-dimensional database to be visualised in one window, while permitting close analysis in 'enlargement' windows. Selections of records can be made and examined, and dependencies between fields can be investigated in detail. MADEN was used as a tool for investigating and assessing many data processing algorithms, firstly data-reducing (clustering) methods, then dimensionality-reducing techniques. These included a new 'directed' form of principal components analysis, several novel applications of artificial neural networks, and discriminant analysis techniques which illustrated how groups within a database can be separated. To illustrate the power of the system, MADEN was used to explore customer databases from two financial institutions, resulting in a number of discoveries which would be of interest to a marketing manager. Finally, the database of results from the 1992 UK Research Assessment Exercise was analysed. Using MADEN allowed both universities and disciplines to be graphically compared, and supplied some startling revelations, including empirical evidence of the 'Oxbridge factor'.

Neural networks for perpetual grouping

A number of researchers have investigated the application of neural networks to visual recognition, with much of the emphasis placed on exploiting the network's ability to generalise. However, despite the benefits of such an approach it is not at all obvious how networks can be developed which are capable of recognising objects subject to changes in rotation, translation and viewpoint. In this study, we suggest that a possible solution to this problem can be found by studying aspects of visual psychology and in particular, perceptual organisation. For example, it appears that grouping together lines based upon perceptually significant features can facilitate viewpoint independent recognition. The work presented here identifies simple grouping measures based on parallelism and connectivity and shows how it is possible to train multi-layer perceptrons (MLPs) to detect and determine the perceptual significance of any group presented. In this way, it is shown how MLPs which are trained via backpropagation to perform individual grouping tasks, can be brought together into a novel, large scale network capable of determining the perceptual significance of the whole input pattern. Finally the applicability of such significance values for recognition is investigated and results indicate that both the NILP and the Kohonen Feature Map can be trained to recognise simple shapes described in terms of perceptual significances. This study has also provided an opportunity to investigate aspects of the backpropagation algorithm, particularly the ability to generalise. In this study we report the results of various generalisation tests. In applying the backpropagation algorithm to certain problems, we found that there was a deficiency in performance with the standard learning algorithm. An improvement in performance could however, be obtained when suitable modifications were made to the algorithm. The modifications and consequent results are reported here.