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

TL;DR

This paper introduces a unified Skew-Gaussian process framework that extends Gaussian processes to model asymmetric distributions, enabling exact posteriors for various regression and classification tasks, and demonstrates improved performance in active learning and optimization.

Contribution

It generalizes SkewGP conjugacy to include normal and affine probit likelihoods, allowing a unified approach for diverse problems with closed-form posteriors.

Findings

SkewGP outperforms Gaussian processes in active learning.

The framework handles classification, preference, and regression in a unified way.

Empirical results show improved Bayesian optimization performance.

Abstract

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.