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

• Main Authors: Bånkestad, Maria, Sjölund, Jens, Taghia, Jalil, Schön, Thomas
• Format: Journal Article
• Language: English
• Published: 13-03-2020
• Subjects: Computer Science - Learning; Statistics - Machine Learning; Statistics - Methodology
• DOI: 10.48550/arxiv.2003.07201

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.