# Student-t Processes as Alternatives to Gaussian Processes

arXiv: https://arxiv.org/abs/1402.4306 · HTML: https://mlsanity.com/paper/1402.4306/

## Key facts

- Cited by 45 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Feb 2014.

## Built on by (most cited first, 30 of 45)

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- [Applications of Gaussian Processes at Extreme Lengthscales: From Molecules to Black Holes](https://mlsanity.com/paper/2303.14291/) (2023)
- [Online Student-t Processes with an Overall-local Scale Structure for Modelling Non-stationary Data](https://mlsanity.com/paper/2311.00564/) (2023)

## Abstract (arXiv)

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.