Bayesian regression filter and the issue of priors

We propose a Bayesian framework for regression problems, which covers areas which are usually dealt with by function approximation. An online learning algorithm is derived which solves regression problems with a Kalman filter. Its solution always improves with increasing model complexity, without the risk of over-fitting. In the infinite dimension limit it approaches the true Bayesian posterior. The issues of prior selection and over-fitting are also discussed, showing that some of the commonly held beliefs are misleading. The practical implementation is summarised. Simulations using 13 popular publicly available data sets are used to demonstrate the method and highlight important issues concerning the choice of priors.

On the relationship between Bayesian error bars and the input data density

We investigate the dependence of Bayesian error bars on the distribution of data in input space. For generalized linear regression models we derive an upper bound on the error bars which shows that, in the neighbourhood of the data points, the error bars are substantially reduced from their prior values. For regions of high data density we also show that the contribution to the output variance due to the uncertainty in the weights can exhibit an approximate inverse proportionality to the probability density. Empirical results support these conclusions.

Classification of oximetry signals using Bayesian neural networks to assist in the detection of obstructive sleep apnoea syndrome

In the present study, multilayer perceptron (MLP) neural networks were applied to help in the diagnosis of obstructive sleep apnoea syndrome (OSAS). Oxygen saturation (SaO2) recordings from nocturnal pulse oximetry were used for this purpose. We performed time and spectral analysis of these signals to extract 14 features related to OSAS. The performance of two different MLP classifiers was compared: maximum likelihood (ML) and Bayesian (BY) MLP networks. A total of 187 subjects suspected of suffering from OSAS took part in the study. Their SaO2 signals were divided into a training set with 74 recordings and a test set with 113 recordings. BY-MLP networks achieved the best performance on the test set with 85.58% accuracy (87.76% sensitivity and 82.39% specificity). These results were substantially better than those provided by ML-MLP networks, which were affected by overfitting and achieved an accuracy of 76.81% (86.42% sensitivity and 62.83% specificity). Our results suggest that the Bayesian framework is preferred to implement our MLP classifiers. The proposed BY-MLP networks could be used for early OSAS detection. They could contribute to overcome the difficulties of nocturnal polysomnography (PSG) and thus reduce the demand for these studies.

Bayesian training of mixture density networks

Mixture Density Networks (MDNs) are a well-established method for modelling the conditional probability density which is useful for complex multi-valued functions where regression methods (such as MLPs) fail. In this paper we extend earlier research of a regularisation method for a special case of MDNs to the general case using evidence based regularisation and we show how the Hessian of the MDN error function can be evaluated using R-propagation. The method is tested on two data sets and compared with early stopping.

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.

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.

Regression with input-dependent noise: A Bayesian treatment

In most treatments of the regression problem it is assumed that the distribution of target data can be described by a deterministic function of the inputs, together with additive Gaussian noise having constant variance. The use of maximum likelihood to train such models then corresponds to the minimization of a sum-of-squares error function. In many applications a more realistic model would allow the noise variance itself to depend on the input variables. However, the use of maximum likelihood to train such models would give highly biased results. In this paper we show how a Bayesian treatment can allow for an input-dependent variance while overcoming the bias of maximum likelihood.

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.

Bayesian classification with Gaussian processes

We consider the problem of assigning an input vector bfx to one of m classes by predicting P(c|bfx) for c = 1, ldots, m. For a two-class problem, the probability of class 1 given bfx is estimated by s(y(bfx)), where s(y) = 1/(1 + e^-y). A Gaussian process prior is placed on y(bfx), and is combined with the training data to obtain predictions for new bfx points. We provide a Bayesian treatment, integrating over uncertainty in y and in the parameters that control the Gaussian process prior; the necessary integration over y is carried out using Laplace's approximation. The method is generalized to multi-class problems (m >2) using the softmax function. We demonstrate the effectiveness of the method on a number of datasets.

Training Bayesian networks for image segmentation

We are concerned with the problem of image segmentation in which each pixel is assigned to one of a predefined finite number of classes. In Bayesian image analysis, this requires fusing together local predictions for the class labels with a prior model of segmentations. Markov Random Fields (MRFs) have been used to incorporate some of this prior knowledge, but this not entirely satisfactory as inference in MRFs is NP-hard. The multiscale quadtree model of Bouman and Shapiro (1994) is an attractive alternative, as this is a tree-structured belief network in which inference can be carried out in linear time (Pearl 1988). It is an hierarchical model where the bottom-level nodes are pixels, and higher levels correspond to downsampled versions of the image. The conditional-probability tables (CPTs) in the belief network encode the knowledge of how the levels interact. In this paper we discuss two methods of learning the CPTs given training data, using (a) maximum likelihood and the EM algorithm and (b) emphconditional maximum likelihood (CML). Segmentations obtained using networks trained by CML show a statistically-significant improvement in performance on synthetic images. We also demonstrate the methods on a real-world outdoor-scene segmentation task.

Bayesian inference for wind field retrieval

In many problems in spatial statistics it is necessary to infer a global problem solution by combining local models. A principled approach to this problem is to develop a global probabilistic model for the relationships between local variables and to use this as the prior in a Bayesian inference procedure. We show how a Gaussian process with hyper-parameters estimated from Numerical Weather Prediction Models yields meteorologically convincing wind fields. We use neural networks to make local estimates of wind vector probabilities. The resulting inference problem cannot be solved analytically, but Markov Chain Monte Carlo methods allow us to retrieve accurate wind fields.

Efficient Bayesian inference for learning in the Ising linear perceptron and signal detection in CDMA

Efficient new Bayesian inference technique is employed for studying critical properties of the Ising linear perceptron and for signal detection in code division multiple access (CDMA). The approach is based on a recently introduced message passing technique for densely connected systems. Here we study both critical and non-critical regimes. Results obtained in the non-critical regime give rise to a highly efficient signal detection algorithm in the context of CDMA; while in the critical regime one observes a first-order transition line that ends in a continuous phase transition point. Finite size effects are also studied. © 2006 Elsevier B.V. All rights reserved.

Evaluating the effectiveness of Bayesian feature selection

A practical Bayesian approach for inference in neural network models has been available for ten years, and yet it is not used frequently in medical applications. In this chapter we show how both regularisation and feature selection can bring significant benefits in diagnostic tasks through two case studies: heart arrhythmia classification based on ECG data and the prognosis of lupus. In the first of these, the number of variables was reduced by two thirds without significantly affecting performance, while in the second, only the Bayesian models had an acceptable accuracy. In both tasks, neural networks outperformed other pattern recognition approaches.

Bayesian inference for wind field retrieval

In many problems in spatial statistics it is necessary to infer a global problem solution by combining local models. A principled approach to this problem is to develop a global probabilistic model for the relationships between local variables and to use this as the prior in a Bayesian inference procedure. We show how a Gaussian process with hyper-parameters estimated from Numerical Weather Prediction Models yields meteorologically convincing wind fields. We use neural networks to make local estimates of wind vector probabilities. The resulting inference problem cannot be solved analytically, but Markov Chain Monte Carlo methods allow us to retrieve accurate wind fields.