Abstract. We give a basic introduction to Gaussian Process regression models. We focus on understanding the role of the stochastic process and how it is used to define a distribution over functions. We present the simple equations for incorporating training data and examine how to learn the hyperparameters using the marginal likelihood. We explain the practical advantages of Gaussian Process and end with conclusions and a look at the current trends in GP work. Supervised learning in the form of regression (for continuous outputs) and classification (for discrete outputs) is an important constituent of statistics and machine learning, either for analysis of data sets, or as a subgoal of a more complex problem. Traditionally parametric 1 models have been used for this purpose. These have a possible advantage in ease of interpretability, but for complex data sets, simple parametric models may lack expressive power, and their more complex counterparts (such as feed forward neural networks) may not be easy to work with in practice. The advent of kernel machines, such as Support Vector Machines and Gaussian Processes has opened the possibility of flexible models which are practical to work with. In this short tutorial we present the basic idea on how Gaussian Process models can be used to formulate a Bayesian framework for regression. We will focus on understanding the stochastic process and how it is used in supervised learning. Secondly, we will discuss practical matters regarding the role of hyperparameters in the covariance function, the marginal likelihood and the automatic Occam’s razor. For broader introductions to Gaussian processes, consult [1], [2]. 1 Gaussian Processes In this section we define Gaussian Processes and show how they can very naturally be used to define distributions over functions. In the following section we continue to show how this distribution is updated in the light of training examples. 1 By a parametric model, we here mean a model which during training “absorbs ” the information from the training data into the parameters; after training the data can be discarded.

We consider the scenario where training and test data are drawn from different distributions, commonly referred to as sample selection bias. Most algorithms for this setting try to first recover sampling distributions and then make appropriate corrections based on the distribution estimate. We present a nonparametric method which directly produces resampling weights without distribution estimation. Our method works by matching distributions between training and testing sets in feature space. Experimental results demonstrate that our method works well in practice.

We present a probabilistic kernel approach to ordinal regression based on Gaussian processes. A threshold model that generalizes the probit function is used as the likelihood function for ordinal variables. Two inference techniques, based on the Laplace approximation and the expectation propagation algorithm respectively, are derived for hyperparameter learning and model selection. We compare these two Gaussian process approaches with a previous ordinal regression method based on support vector machines on some benchmark and real-world data sets, including applications of ordinal regression to collaborative filtering and gene expression analysis. Experimental results on these data sets verify the usefulness of our approach.

Gaussian processes (GPs) are natural generalisations of multivariate Gaussian random variables to in nite (countably or continuous) index sets. GPs have been applied in a large number of elds to a diverse range of ends, and very many deep theoretical analyses of various properties are available. This paper gives an introduction to Gaussian processes on a fairly elementary level with special emphasis on characteristics relevant in machine learning. It draws explicit connections to branches such as spline smoothing models and support vector machines in which similar ideas have been investigated.

Most existing inductive learning algorithms work under the assumption that their training examples are already tagged. There are domains, however, where the tagging procedure requires significant computation resources or manual labor. In such cases, it may be beneficial for the learner to be active, intelligently selecting the examples for labeling with the goal of reducing the labeling cost. In this paper we present LSS---a lookahead algorithm for selective sampling of examples for nearest neighbor classifiers. The algorithm is looking for the example with the highest utility, taking its effect on the resulting classifier into account. Computing the expected utility of an example requires estimating the probability of its possible labels. We propose to use the random field model for this estimation. The LSS algorithm was evaluated empirically on seven real and artificial data sets, and its performance was compared to other selective sampling algorithms. The experiments show that the proposed algorithm outperforms other methods in terms of average error rate and stability.

Structured additive regression models are perhaps the most commonly used class of models in statistical applications. It includes, among others, (generalised) linear models, (generalised) additive models, smoothing-spline models, state-space models, semiparametric regression, spatial and spatio-temporal models, log-Gaussian Cox-processes, geostatistical and geoadditive models. In this paper we consider approximate Bayesian inference in a popular subset of structured additive regression models, latent Gaussian models, where the latent field is Gaussian, controlled by a few hyperparameters and with non-Gaussian response variables. The posterior marginals are not available in closed form due to the non-Gaussian response variables. For such models, Markov chain Monte Carlo methods can be implemented, but they are not without problems, both in terms of convergence and computational time. In some practical applications, the extent of these problems is such that Markov chain Monte Carlo is simply not an appropriate tool for routine analysis. We show that, by using an integrated nested Laplace approximation and its simplified version, we can directly compute very accurate approximations to the posterior marginals. The main benefit of these approximations

ii Non-parametric models and techniques enjoy a growing popularity in the field of machine learning, and among these Bayesian inference for Gaussian process (GP) models has recently received significant attention. We feel that GP priors should be part of the standard toolbox for constructing models relevant to machine learning in the same way as parametric linear models are, and the results in this thesis help to remove some obstacles on the way towards this goal. In the first main chapter, we provide a distribution-free finite sample bound on the difference between generalisation and empirical (training) error for GP classification methods. While the general theorem (the PAC-Bayesian bound) is not new, we give a much simplified and somewhat generalised derivation and point out the underlying core technique (convex duality) explicitly. Furthermore, the application to GP models is novel (to our knowledge). A central feature of this bound is that its quality depends crucially on task knowledge being encoded

We present a variational Bayesian method for model selection over families of kernels classifiers like Support Vector machines or Gaussian processes. The algorithm needs no user interaction and is able to adapt a large number of kernel parameters to given data without having to sacrifice training cases for validation. This opens the possibility to use sophisticated families of kernels in situations where the small "standard kernel" classes are clearly inappropriate. We relate the method to other work done on Gaussian processes and clarify the relation between Support Vector machines and certain Gaussian process models.

The computation required for kernel machines with N training samples is O(N ). Such computational complexity is significant even for moderate size problems and is prohibitive for large datasets. We present an approximation technique based on the improved fast Gauss transform to reduce the computation to O(N). We also give an error bound for the approximation, and provide experimental results on the UCI datasets.

Many real-world classification tasks involve the prediction of multiple, inter-dependent class labels. A prototypical case of this sort deals with prediction of a sequence of labels for a sequence of observations. Such problems arise naturally in the context of annotating and segmenting observation sequences.