Regression with input-dependent noise: A Gaussian process treatment

Gaussian processes provide natural non-parametric prior distributions over regression functions. In this paper we consider regression problems where there is noise on the output, and the variance of the noise depends on the inputs. If we assume that the noise is a smooth function of the inputs, then it is natural to model the noise variance using a second Gaussian process, in addition to the Gaussian process governing the noise-free output value. We show that prior uncertainty about the parameters controlling both processes can be handled and that the posterior distribution of the noise rate can be sampled from using Markov chain Monte Carlo methods. Our results on a synthetic data set give a posterior noise variance that well-approximates the true variance.

Non-zero mean Gaussian process prior wind field models

This report outlines the derivation and application of a non-zero mean, polynomial-exponential covariance function based Gaussian process which forms the prior wind field model used in 'autonomous' disambiguation. It is principally used since the non-zero mean permits the computation of realistic local wind vector prior probabilities which are required when applying the scaled-likelihood trick, as the marginals of the full wind field prior. As the full prior is multi-variate normal, these marginals are very simple to compute.

Regression with Gaussian processes

The Bayesian analysis of neural networks is difficult because the prior over functions has a complex form, leading to implementations that either make approximations or use Monte Carlo integration techniques. In this paper I investigate the use of Gaussian process priors over functions, which permit the predictive Bayesian analysis to be carried out exactly using matrix operations. The method has been tested on two challenging problems and has produced excellent results.

Gaussian processes for regression

The Bayesian analysis of neural networks is difficult because a simple prior over weights implies a complex prior distribution over functions. In this paper we investigate the use of Gaussian process priors over functions, which permit the predictive Bayesian analysis for fixed values of hyperparameters to be carried out exactly using matrix operations. Two methods, using optimization and averaging (via Hybrid Monte Carlo) over hyperparameters have been tested on a number of challenging problems and have produced excellent results.

General gaussian priors for improved generalisation

We explore the dependence of performance measures, such as the generalization error and generalization consistency, on the structure and the parameterization of the prior on `rules', instanced here by the noisy linear perceptron. Using a statistical mechanics framework, we show how one may assign values to the parameters of a model for a `rule' on the basis of data instancing the rule. Information about the data, such as input distribution, noise distribution and other `rule' characteristics may be embedded in the form of general gaussian priors for improving net performance. We examine explicitly two types of general gaussian priors which are useful in some simple cases. We calculate the optimal values for the parameters of these priors and show their effect in modifying the most probable, MAP, values for the rules.

Gaussian regression and optimal finite dimensional linear models

The problem of regression under Gaussian assumptions is treated generally. The relationship between Bayesian prediction, regularization and smoothing is elucidated. The ideal regression is the posterior mean and its computation scales as O(n³), where n is the sample size. We show that the optimal m-dimensional linear model under a given prior is spanned by the first m eigenfunctions of a covariance operator, which is a trace-class operator. This is an infinite dimensional analogue of principal component analysis. The importance of Hilbert space methods to practical statistics is also discussed.

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.

Adding constrained discontinuities to Gaussian process models of wind fields

Gaussian Processes provide good prior models for spatial data, but can be too smooth. In many physical situations there are discontinuities along bounding surfaces, for example fronts in near-surface wind fields. We describe a modelling method for such a constrained discontinuity and demonstrate how to infer the model parameters in wind fields with MCMC sampling.

Bayesian classification with Gaussian processes

We consider the problem of assigning an input vector to one of m classes by predicting P(c|x) for c=1,...,m. For a two-class problem, the probability of class one given x is estimated by s(y(x)), where s(y)=1/(1+e-y). A Gaussian process prior is placed on y(x), and is combined with the training data to obtain predictions for new x 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 multiclass problems (m>2) using the softmax function. We demonstrate the effectiveness of the method on a number of datasets.

Adding constrained discontinuities to Gaussian process models of wind fields

Gaussian Processes provide good prior models for spatial data, but can be too smooth. In many physical situations there are discontinuities along bounding surfaces, for example fronts in near-surface wind fields. We describe a modelling method for such a constrained discontinuity and demonstrate how to infer the model parameters in wind fields with MCMC sampling.

Mean field methods for Gaussian process classifiers

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