The Elliptical Processes: a Family of Fat-tailed Stochastic Processes

TL;DR

This paper introduces elliptical processes, a flexible family of non-parametric probabilistic models that generalize Gaussian and Student-t processes, capturing fat-tailed behaviors while remaining computationally feasible.

Contribution

It proposes elliptical processes based on elliptical distributions as mixtures of Gaussians, providing closed-form marginals and conditionals, and demonstrates their advantages in robust regression tasks.

Findings

Elliptical processes include Gaussian and Student-t processes as special cases.

Numerical experiments show elliptical processes outperform Gaussian processes in robust regression.

The models effectively capture fat-tailed behaviors and are computationally tractable.

Abstract

We present the elliptical processes -- a family of non-parametric probabilistic models that subsumes the Gaussian process and the Student-t process. This generalization includes a range of new fat-tailed behaviors yet retains computational tractability. We base the elliptical processes on a representation of elliptical distributions as a continuous mixture of Gaussian distributions and derive closed-form expressions for the marginal and conditional distributions. We perform numerical experiments on robust regression using an elliptical process defined by a piecewise constant mixing distribution, and show advantages compared with a Gaussian process. The elliptical processes may become a replacement for Gaussian processes in several settings, including when the likelihood is not Gaussian or when accurate tail modeling is critical.