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.

斜高斯过程 (SkewGP) 将有限维向量上的多元统一斜正态分布扩展到函数上的分布。SkewGPs 比高斯过程更通用和灵活，因为 SkewGPs 也可能代表非对称分布。在最近的一个贡献中，我们表明 SkewGP 和概率似然是共轭的，这使我们能够计算非参数二元分类和偏好学习的确切后验。在本文中，我们概括了之前的结果，并证明 SkewGP 与正态和仿射概率似然共轭，更一般地说，与它们的乘积共轭。这使我们能够 (i) 在统一框架中处理分类、偏好、数字和序数回归以及混合问题；(ii) 推导出相应后验分布的闭式表达式。我们凭经验表明，所提出的基于 SkewGP 的框架在主动学习和贝叶斯（约束）优化方面提供了比高斯过程更好的性能。这两项任务是实验设计和数据科学的基础。