The Elliptical Processes: a Family of Fat-tailed Stochastic Processes
This work addresses the need for more flexible probabilistic models in machine learning, particularly for scenarios requiring accurate tail modeling or non-Gaussian likelihoods, representing a novel method for a known bottleneck.
The authors tackled the problem of modeling fat-tailed data by introducing elliptical processes, a family of non-parametric probabilistic models that generalize Gaussian and Student-t processes, and demonstrated advantages in robust regression experiments compared to Gaussian processes.
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.