Student-t Processes as Alternatives to Gaussian Processes

TL;DR

This paper introduces Student-t processes as flexible, nonparametric alternatives to Gaussian processes, with closed-form expressions and improved predictive covariances, useful in non-stationary settings and Bayesian optimization.

Contribution

It derives closed-form expressions for Student-t processes, reveals equivalences with hierarchical Gaussian process models, and introduces a new sampling scheme for the inverse Wishart process.

Findings

Student-t processes retain Gaussian process properties with added flexibility.

They have predictive covariances that depend on training data values.

Empirical results show advantages in non-stationary and Bayesian optimization tasks.

Abstract

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.