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133
Gaussian processes for machine learning
 in: Adaptive Computation and Machine Learning
, 2006
"... 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 hyperpar ..."
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Cited by 289 (2 self)
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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
Correcting sample selection bias by unlabeled data
"... 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 prese ..."
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Cited by 129 (9 self)
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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.
Sparse online gaussian processes
 Neural Computation
"... Minor corrections included a a The authors acknowledge reader feedbacks We develop an approach for sparse representations of Gaussian Process (GP) models (which are Bayesian types of kernel machines) in order to overcome their limitations for large data sets. The method is based on a combination of ..."
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Cited by 128 (6 self)
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Minor corrections included a a The authors acknowledge reader feedbacks We develop an approach for sparse representations of Gaussian Process (GP) models (which are Bayesian types of kernel machines) in order to overcome their limitations for large data sets. The method is based on a combination of a Bayesian online algorithm together with a sequential construction of a relevant subsample of the data which fully specifies the prediction of the GP model. By using an appealing parametrisation and projection techniques that use the RKHS norm, recursions for the effective parameters and a sparse Gaussian approximation of the posterior process are obtained. This allows both for a propagation of predictions as well as of Bayesian error measures. The significance and robustness of our approach is demonstrated on a variety of experiments. Sparse Online Gaussian Processes 2
Implementing approximate Bayesian inference for latent Gaussian models using integrated nested Laplace approximations: A manual for the inlaprogram
, 2008
"... 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, smoothingspline models, statespace models, semiparametric regression, spatial and spatiotemp ..."
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Cited by 79 (16 self)
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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, smoothingspline models, statespace models, semiparametric regression, spatial and spatiotemporal models, logGaussian Coxprocesses, 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 nonGaussian response variables. The posterior marginals are not available in closed form due to the nonGaussian 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
Gaussian processes for ordinal regression
 Journal of Machine Learning Research
, 2004
"... 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 ..."
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Cited by 71 (1 self)
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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 realworld 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 for machine learning
 International Journal of Neural Systems
, 2004
"... Gaussian processes (GPs) are natural generalisations of multivariate Gaussian random variables to infinite (countably or continuous) index sets. GPs have been applied in a large number of fields to a diverse range of ends, and very many deep theoretical analyses of various properties are available. ..."
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Cited by 67 (15 self)
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Gaussian processes (GPs) are natural generalisations of multivariate Gaussian random variables to infinite (countably or continuous) index sets. GPs have been applied in a large number of fields 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. Gaussian process models are routinely used to solve hard machine learning problems. They are attractive because of their flexible nonparametric nature and computational simplicity. Treated within a Bayesian framework, very powerful statistical methods can be implemented which offer valid estimates of uncertainties in our predictions and generic model selection procedures cast as nonlinear optimization problems. Their main drawback of heavy computational scaling has recently been alleviated by the introduction of generic sparse approximations [13, 78, 31]. The mathematical literature on GPs is large and often uses deep
Selective Sampling For Nearest Neighbor Classifiers
 MACHINE LEARNING
, 2004
"... 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, ..."
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Cited by 60 (3 self)
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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 LSSa 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.
Maximum likelihood estimation of a stochastic integrateandfire neural model
 NIPS
, 2003
"... We examine a cascade encoding model for neural response in which a linear filtering stage is followed by a noisy, leaky, integrateandfire spike generation mechanism. This model provides a biophysically more realistic alternative to models based on Poisson (memoryless) spike generation, and can eff ..."
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Cited by 60 (20 self)
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We examine a cascade encoding model for neural response in which a linear filtering stage is followed by a noisy, leaky, integrateandfire spike generation mechanism. This model provides a biophysically more realistic alternative to models based on Poisson (memoryless) spike generation, and can effectively reproduce a variety of spiking behaviors seen in vivo. We describe the maximum likelihood estimator for the model parameters, given only extracellular spike train responses (not intracellular voltage data). Specifically, we prove that the log likelihood function is concave and thus has an essentially unique global maximum that can be found using gradient ascent techniques. We develop an efficient algorithm for computing the maximum likelihood solution, demonstrate the effectiveness of the resulting estimator with numerical simulations, and discuss a method of testing the model’s validity using timerescaling and density evolution techniques. Paninski et al., November 30, 2004 2 1
Bayesian Gaussian Process Models: PACBayesian Generalisation Error Bounds and Sparse Approximations
, 2003
"... ii Nonparametric 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 ..."
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Cited by 53 (14 self)
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ii Nonparametric 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 distributionfree finite sample bound on the difference between generalisation and empirical (training) error for GP classification methods. While the general theorem (the PACBayesian 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
Bayesian model selection for Support Vector machines, Gaussian processes and other kernel classifiers
"... 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 ca ..."
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Cited by 46 (6 self)
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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.