Vanhatalo, Jarno, Riihimäki, Jaakko, Hartikainen, Jouni, Jylänki, Pasi, Tolvanen, Ville, Vehtari, Aki

Gaussian processes (GP) are powerful tools for probabilistic modeling purposes. They can be used to define prior distributions over latent functions in hierarchical Bayesian models. The prior over functions is defined implicitly by the mean and covariance function, which determine the smoothness and variability of the function. The inference can then be conducted directly in the function space by evaluating or approximating the posterior process. Despite their attractive theoretical properties GPs provide practical challenges in their implementation. GPstuff is a versatile collection of computational tools for GP models compatible with Linux and Windows MATLAB and Octave. It includes, among others, various inference methods, sparse approximations and tools for model assessment. In this work, we review these tools and demonstrate the use of GPstuff in several models.

Vehtari, Aki, Mononen, Tommi, Tolvanen, Ville, Sivula, Tuomas, Winther, Ole

The future predictive performance of a Bayesian model can be estimated using Bayesian cross-validation. In this article, we consider Gaussian latent variable models where the integration over the latent values is approximated using the Laplace method or expectation propagation (EP). We study the properties of several Bayesian leave-one-out (LOO) cross-validation approximations that in most cases can be computed with a small additional cost after forming the posterior approximation given the full data. Our main objective is to assess the accuracy of the approximative LOO cross-validation estimators. That is, for each method (Laplace and EP) we compare the approximate fast computation with the exact brute force LOO computation. Secondarily, we evaluate the accuracy of the Laplace and EP approximations themselves against a ground truth established through extensive Markov chain Monte Carlo simulation. Our empirical results show that the approach based upon a Gaussian approximation to the LOO marginal distribution (the so-called cavity distribution) gives the most accurate and reliable results among the fast methods.

Vanhatalo, Jarno, Jylänki, Pasi, Vehtari, Aki

In the Gaussian process regression the observation model is commonly assumed to be Gaussian, which is convenient in computational perspective. However, the drawback is that the predictive accuracy of the model can be significantly compromised if the observations are contaminated by outliers. A robust observation model, such as the Student-t distribution, reduces the influence of outlying observations and improves the predictions. The problem, however, is the analytically intractable inference. In this work, we discuss the properties of a Gaussian process regression model with the Student-t likelihood and utilize the Laplace approximation for approximate inference. We compare our approach to a variational approximation and a Markov chain Monte Carlo scheme, which utilize the commonly used scale mixture representation of the Student-t distribution.

Riihimäki, Jaakko, Jylänki, Pasi, Vehtari, Aki

We consider probabilistic multinomial probit classification using Gaussian process (GP) priors. The challenges with the multiclass GP classification are the integration over the non-Gaussian posterior distribution, and the increase of the number of unknown latent variables as the number of target classes grows. Expectation propagation (EP) has proven to be a very accurate method for approximate inference but the existing EP approaches for the multinomial probit GP classification rely on numerical quadratures or independence assumptions between the latent values from different classes to facilitate the computations. In this paper, we propose a novel nested EP approach which does not require numerical quadratures, and approximates accurately all between-class posterior dependencies of the latent values, but still scales linearly in the number of classes. The predictive accuracy of the nested EP approach is compared to Laplace, variational Bayes, and Markov chain Monte Carlo (MCMC) approximations with various benchmark data sets. In the experiments nested EP was the most consistent method with respect to MCMC sampling, but the differences between the compared methods were small if only the classification accuracy is concerned.

Ghahramani, Zoubin, Rasmussen, Carl E.

We investigate Bayesian alternatives to classical Monte Carlo methods for evaluating integrals. Bayesian Monte Carlo (BMC) allows the incorporation ofprior knowledge, such as smoothness of the integrand, into the estimation. In a simple problem we show that this outperforms any classical importance sampling method. We also attempt more challenging multidimensionalintegrals involved in computing marginal likelihoods ofstatistical models (a.k.a.