Both Reviewers #3 and #4 ask for concrete examples to motivate the use of elliptical distributions.

We present elliptical processes--a family of non-parametric probabilistic models that subsumes Gaussian processes and Student's t processes. This generalization includes a range of new heavy-tailed behaviors while retaining computational tractability. Elliptical processes are based on a representation of elliptical distributions as a continuous mixture of Gaussian distributions. We parameterize this mixture distribution as a spline normalizing flow, which we train using variational inference. The proposed form of the variational posterior enables a sparse variational elliptical process applicable to large-scale problems. We highlight advantages compared to Gaussian processes through regression and classification experiments. Elliptical processes can supersede Gaussian processes in several settings, including cases where the likelihood is non-Gaussian or when accurate tail modeling is essential.

We present the elliptical processes-a new family of stochastic processes that subsumes the Gaussian process and the Student-t process. This generalization retains computational tractability while substantially increasing the range of tail behaviors that can be modeled. We base the elliptical processes on a representation of elliptical distributions as mixtures of Gaussian distributions and derive closed-form expressions for the marginal and conditional distributions. We perform an in-depth study of a particular elliptical process, where the mixture distribution is piecewise constant, and show some of its advantages over the Gaussian process through a number of experiments on robust regression. Looking forward, we believe there are several settings, e.g. when the likelihood is not Gaussian or when accurate tail modeling is critical, where the elliptical processes could become the stochastic processes of choice.

We investigate the Student-t process as an alternative to the Gaussian process as a nonparametric prior over functions. We derive closed form expressions for the marginal likelihood and predictive distribution of a Student-t process, by integrating away an inverse Wishart process prior over the covariance kernel of a Gaussian process model. We show surprising equivalences between different hierarchical Gaussian process models leading to Student-t processes, and derive a new sampling scheme for the inverse Wishart process, which helps elucidate these equivalences. Overall, we show that a Student-t process can retain the attractive properties of a Gaussian process -- a nonparametric representation, analytic marginal and predictive distributions, and easy model selection through covariance kernels -- but has enhanced flexibility, and predictive covariances that, unlike a Gaussian process, explicitly depend on the values of training observations. We verify empirically that a Student-t process is especially useful in situations where there are changes in covariance structure, or in applications like Bayesian optimization, where accurate predictive covariances are critical for good performance. These advantages come at no additional computational cost over Gaussian processes.