Key to the application of such models in safety-critical domains is the quantification of their model error. Gaussian processes provide such a measure anduniform error bounds havebeen derived,which allowsafe control based on thesemodels.

By now Bayesian methods are routinely used in practice for solving inverse problems.

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We propose two novel tricks to tackle this challenge.

We introduce a novel variational method that allows to approximately integrate out kernel hyperparameters, such as length-scales, in Gaussian process regression. This approach consists of a novel variant of the variational framework that has been recently developed for the Gaussian process latent variable model which additionally makes use of a standardised representation of the Gaussian process. We consider this technique for learning Mahalanobis distance metrics in a Gaussian process regression setting and provide experimental evaluations and comparisons with existing methods by considering datasets with high-dimensional inputs.