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.

Prediction with Gaussian processes: from linear regression to linear prediction and beyond

The main aim of this paper is to provide a tutorial on regression with Gaussian processes. We start from Bayesian linear regression, and show how by a change of viewpoint one can see this method as a Gaussian process predictor based on priors over functions, rather than on priors over parameters. This leads in to a more general discussion of Gaussian processes in section 4. Section 5 deals with further issues, including hierarchical modelling and the setting of the parameters that control the Gaussian process, the covariance functions for neural network models and the use of Gaussian processes in classification problems.

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.

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.

Hierarchical GTM: Constructing localized non-linear projection manifolds in a principled way

It has been argued that a single two-dimensional visualization plot may not be sufficient to capture all of the interesting aspects of complex data sets, and therefore a hierarchical visualization system is desirable. In this paper we extend an existing locally linear hierarchical visualization system PhiVis ¸iteBishop98a in several directions: bf(1) We allow for em non-linear projection manifolds. The basic building block is the Generative Topographic Mapping. bf(2) We introduce a general formulation of hierarchical probabilistic models consisting of local probabilistic models organized in a hierarchical tree. General training equations are derived, regardless of the position of the model in the tree. bf(3) Using tools from differential geometry we derive expressions for local directional curvatures of the projection manifold. Like PhiVis, our system is statistically principled and is built interactively in a top-down fashion using the EM algorithm. It enables the user to interactively highlight those data in the parent visualization plot which are captured by a child model. We also incorporate into our system a hierarchical, locally selective representation of magnification factors and directional curvatures of the projection manifolds. Such information is important for further refinement of the hierarchical visualization plot, as well as for controlling the amount of regularization imposed on the local models. We demonstrate the principle of the approach on a toy data set and apply our system to two more complex 12- and 19-dimensional data sets.

Hierarchical GTM: constructing localized non-linear projection manifolds in a principled way

It has been argued that a single two-dimensional visualization plot may not be sufficient to capture all of the interesting aspects of complex data sets, and therefore a hierarchical visualization system is desirable. In this paper we extend an existing locally linear hierarchical visualization system PhiVis ¸iteBishop98a in several directions: bf(1) We allow for em non-linear projection manifolds. The basic building block is the Generative Topographic Mapping (GTM). bf(2) We introduce a general formulation of hierarchical probabilistic models consisting of local probabilistic models organized in a hierarchical tree. General training equations are derived, regardless of the position of the model in the tree. bf(3) Using tools from differential geometry we derive expressions for local directional curvatures of the projection manifold. Like PhiVis, our system is statistically principled and is built interactively in a top-down fashion using the EM algorithm. It enables the user to interactively highlight those data in the ancestor visualization plots which are captured by a child model. We also incorporate into our system a hierarchical, locally selective representation of magnification factors and directional curvatures of the projection manifolds. Such information is important for further refinement of the hierarchical visualization plot, as well as for controlling the amount of regularization imposed on the local models. We demonstrate the principle of the approach on a toy data set and apply our system to two more complex 12- and 18-dimensional data sets.

A principled approach to interactive hierarchical non-linear visualization of high-dimensional data

Hierarchical visualization systems are desirable because a single two-dimensional visualization plot may not be sufficient to capture all of the interesting aspects of complex high-dimensional data sets. We extend an existing locally linear hierarchical visualization system PhiVis [1] in several directions: bf(1) we allow for em non-linear projection manifolds (the basic building block is the Generative Topographic Mapping -- GTM), bf(2) we introduce a general formulation of hierarchical probabilistic models consisting of local probabilistic models organized in a hierarchical tree, bf(3) we describe folding patterns of low-dimensional projection manifold in high-dimensional data space by computing and visualizing the manifold's local directional curvatures. Quantities such as magnification factors [3] and directional curvatures are helpful for understanding the layout of the nonlinear projection manifold in the data space and for further refinement of the hierarchical visualization plot. Like PhiVis, our system is statistically principled and is built interactively in a top-down fashion using the EM algorithm. We demonstrate the visualization system principle of the approach on a complex 12-dimensional data set and mention possible applications in the pharmaceutical industry.

Semi-supervised learning of hierarchical latent trait models for data visualisation

An interactive hierarchical Generative Topographic Mapping (HGTM) ¸iteH_GTM has been developed to visualise complex data sets. In this paper, we build a more general visualisation system by extending the HGTM visualisation system in 3 directions: bf (1) We generalize HGTM to noise models from the exponential family of distributions. The basic building block is the Latent Trait Model (LTM) developed in ¸iteKaban_pami. bf (2) We give the user a choice of initializing the child plots of the current plot in either em interactive, or em automatic mode. In the interactive mode the user interactively selects ``regions of interest'' as in ¸iteH_GTM, whereas in the automatic mode an unsupervised minimum message length (MML)-driven construction of a mixture of LTMs is employed. bf (3) We derive general formulas for magnification factors in latent trait models. Magnification factors are a useful tool to improve our understanding of the visualisation plots, since they can highlight the boundaries between data clusters. The unsupervised construction is particularly useful when high-level plots are covered with dense clusters of highly overlapping data projections, making it difficult to use the interactive mode. Such a situation often arises when visualizing large data sets. We illustrate our approach on a toy example and apply our system to three more complex real data sets.

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.

A Bayesian approach to on-line learning

Online learning is discussed from the viewpoint of Bayesian statistical inference. By replacing the true posterior distribution with a simpler parametric distribution, one can define an online algorithm by a repetition of two steps: An update of the approximate posterior, when a new example arrives, and an optimal projection into the parametric family. Choosing this family to be Gaussian, we show that the algorithm achieves asymptotic efficiency. An application to learning in single layer neural networks is given.