Part of Advances in Neural Information Processing Systems 18 (NIPS 2005)

*Malte Kuss, Carl Rasmussen*

Gaussian processes are attractive models for probabilistic classification but unfortunately exact inference is analytically intractable. We compare Laplace's method and Expectation Propagation (EP) focusing on marginal likelihood estimates and predictive performance. We explain theoretically and corroborate empirically that EP is superior to Laplace. We also compare to a sophisticated MCMC scheme and show that EP is surprisingly accurate. In recent years models based on Gaussian process (GP) priors have attracted much attention in the machine learning community. Whereas inference in the GP regression model with Gaussian noise can be done analytically, probabilistic classification using GPs is analytically intractable. Several approaches to approximate Bayesian inference have been suggested, including Laplace's approximation, Expectation Propagation (EP), variational approximations and Markov chain Monte Carlo (MCMC) sampling, some of these in conjunction with generalisation bounds, online learning schemes and sparse approximations. Despite the abundance of recent work on probabilistic GP classifiers, most experimental studies provide only anecdotal evidence, and no clear picture has yet emerged, as to when and why which algorithm should be preferred. Thus, from a practitioners point of view probabilistic GP classification remains a jungle. In this paper, we set out to understand and compare two of the most wide-spread approximations: Laplace's method and Expectation Propagation (EP). We also compare to a sophisticated, but computationally demanding MCMC scheme to examine how close the approximations are to ground truth. We examine two aspects of the approximation schemes: Firstly the accuracy of approximations to the marginal likelihood which is of central importance for model selection and model comparison. In any practical application of GPs in classification (usually multiple) parameters of the covariance function (hyperparameters) have to be handled. Bayesian model selection provides a consistent framework for setting such parameters. Therefore, it is essential to evaluate the accuracy of the marginal likelihood approximations as a function of the hyperparameters, in order to assess the practical usefulness of the approach Secondly, we need to assess the quality of the approximate probabilistic predictions. In the past, the probabilistic nature of the GP predictions have not received much attention, the focus being mostly on classification error rates. This unfortunate state of affairs is caused primarily by typical benchmarking problems being considered outside of a realistic context. The ability of a classifier to produce class probabilities or confidences, have obvious

relevance in most areas of application, eg. medical diagnosis. We evaluate the predictive distributions of the approximate methods, and compare to the MCMC gold standard.

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The Gaussian Process Model for Binary Classification

Let y {-1, 1} denote the class label of an input x. Gaussian process classification (GPC) is discriminative in modelling p(y |x) for given x by a Bernoulli distribution. The probability of success p(y = 1|x) is related to an unconstrained latent function f (x) which is mapped to the unit interval by a sigmoid transformation, eg. the logit or the probit. For reasons of analytic convenience we exclusively use the probit model p(y = 1|x) = (f (x)), where denotes the cumulative density function of the standard Normal distribution. In the GPC model Bayesian inference is performed about the latent function f in the light of observed data D = {(yi , xi )|i = 1, . . . , m}. Let fi = f (xi ) and f = [f1 , . . . , fm ] e shorthand for the values of the latent function and y = [y1 , . . . , ym ] and X = [x1 , . . . , xm ] collect the class labels and inputs respectively. Given the latent function the class labels are independent Bernoulli variables, so the joint likelihood factories: b

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