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Skew-Gaussian processes (SkewGPs) extend the multivariate Unified Skew-Normaldistributions over finite dimensional vectors to distribution over functions.SkewGPs are more general and flexible than Gaussian processes, as SkewGPs mayalso represent asymmetric distributions. In a recent contribution we showedthat SkewGP and probit likelihood are conjugate, which allows us to compute theexact posterior for non-parametric binary classification and preferencelearning. In this paper, we generalize previous results and we prove thatSkewGP is conjugate with both the normal and affine probit likelihood, and morein general, with their product. This allows us to (i) handle classification,preference, numeric and ordinal regression, and mixed problems in a unifiedframework; (ii) derive closed-form expression for the corresponding posteriordistributions. We show empirically that the proposed framework based on SkewGPprovides better performance than Gaussian processes in active learning andBayesian (constrained) optimization. These two tasks are fundamental for designof experiments and in Data Science.

A unified framework for closed-form nonparametric regression, classification, preference and mixed problems with Skew Gaussian Processes