# Aspects of reparametrization in Gaussian process regression with the Weibull model.

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http://urn.fi/URN:NBN:fi:hulib-202002251403
 Julkaisun nimi: Aspects of reparametrization in Gaussian process regression with the Weibull model. Tekijä: Tanskanen, Ville Muu tekijä: Helsingin yliopisto, Matemaattis-luonnontieteellinen tiedekunta Julkaisija: Helsingin yliopisto Päiväys: 2018 Kieli: eng URI: http://urn.fi/URN:NBN:fi:hulib-202002251403 http://hdl.handle.net/10138/312288 Opinnäytteen taso: pro gradu -tutkielmat Oppiaine: Soveltava matematiikka Tiivistelmä: Gaussian processes can be used through Bayesian models so that they are formed through a multidimensional Gaussian prior with a special covariance matrix structure and an arbitrary likelihood model. They often include a latent variable structure between the features and the response variable. Bayesian modeling's drawbacks are usually related to the normalizing constants that normalize the product of a prior probability density function and a likelihood function to a proper probability distribution. These integrals are hard or even impossible to calculate analytically and hence some approximations are required. One popular approximation is the Laplace approximation, which is a Gaussian approximation for the unnormalized log-posterior distribution. Reparametrization of the observation model can lead to changes in properties of the posterior distribution such as shape and convergence. The performance of approximations made for the posterior distribution also change along with the parametrization. The changes are often related to either computational complexity or the predictive performance of the approximation. This thesis presents the Gaussian processes starting from Bayes' formula and moves quickly towards key concepts in Bayesian modeling such as predictive distributions and hierarchy. An approximation of interest for the posterior distribution, the Laplace approximation, is derived. Traditional optimization algorithm for the Laplace approximation is the Newton method, which is replaced by an algorithm called natural gradient adaptation in this thesis. Then the focus is turned from general introduction of Gaussian processes to more specific treatment of them by choosing the Weibull distribution as an observation model. Two different parametrizations for the Weibull model are studied, one which acts as a baseline and can be thought as traditional parametrization for the model, and another one for which the parameters are orthogonal. The predictive performance of the Laplace approximation is then compared within the two parametrizations in two different kind of data sets. Finally the results show decrease in computation time required for the Laplace approximation but no improvement in the predictive performance for orthogonal parametrization.
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