A unified framework for closed-form nonparametric regression, classification, preference and mixed problems with Skew Gaussian Processes
This work provides a more flexible and unified modeling approach for researchers and practitioners in machine learning and data science who deal with asymmetric data distributions and require closed-form solutions for various learning tasks.
This paper introduces a unified framework using Skew Gaussian Processes (SkewGPs) that can handle various nonparametric problems including regression, classification, and preference learning. By proving the conjugacy of SkewGP with normal and affine probit likelihoods, the authors derive closed-form posterior distributions for these problems, leading to improved performance over Gaussian processes in active learning and Bayesian optimization.
Skew-Gaussian processes (SkewGPs) extend the multivariate Unified Skew-Normal distributions over finite dimensional vectors to distribution over functions. SkewGPs are more general and flexible than Gaussian processes, as SkewGPs may also represent asymmetric distributions. In a recent contribution we showed that SkewGP and probit likelihood are conjugate, which allows us to compute the exact posterior for non-parametric binary classification and preference learning. In this paper, we generalize previous results and we prove that SkewGP is conjugate with both the normal and affine probit likelihood, and more in general, with their product. This allows us to (i) handle classification, preference, numeric and ordinal regression, and mixed problems in a unified framework; (ii) derive closed-form expression for the corresponding posterior distributions. We show empirically that the proposed framework based on SkewGP provides better performance than Gaussian processes in active learning and Bayesian (constrained) optimization. These two tasks are fundamental for design of experiments and in Data Science.