Student-t Processes as Alternatives to Gaussian Processes (1402.4306v2)

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.

• The paper introduces Student-t processes, derived via an inverse Wishart process, as a robust alternative to traditional Gaussian processes.
• It provides closed-form expressions for marginal likelihood and predictive distributions, enabling seamless integration with existing GP frameworks.
• Empirical results demonstrate enhanced performance in capturing variable covariance and managing outliers, beneficial for Bayesian optimization applications.

Exploring Student-$t$ Processes as Alternatives to Gaussian Processes

The paper "Student-$t$ Processes as Alternatives to Gaussian Processes" by Amar Shah, Andrew Gordon Wilson, and Zoubin Ghahramani focuses on evaluating the Student-$t$ process (TP) as a robust alternative to the widely used Gaussian process (GP) for Bayesian nonparametric modeling. By proposing the Student-$t$ process as a prior over functions, the authors aim to harness the advantages of GPs while addressing their limitations, particularly in terms of modeling flexibility and predictive uncertainty.

Gaussian processes have been significant in nonparametric regression due to their flexibility, interpretability, and consistent performance across various applications. However, GPs assume a fixed form of covariance structure, which may not always adequately capture the variability in data. This paper investigates the potential of TPs, derived by placing an inverse Wishart process prior on the covariance of a Gaussian process, enabling more adaptive modeling of covariance structures.

Key Contributions and Methods

1. Inverse Wishart Process as a Covariance Prior: The authors introduce the inverse Wishart process as a viable nonparametric prior for covariance matrices of arbitrary size. This approach overcomes the limitations of the Wishart distribution, which necessitates an unbounded number of degrees of freedom to retain flexibility with large data sets.
2. Derivation and Properties of Student-$t$ Processes: The paper derives TPs from hierarchical GP models with an inverse Wishart process over kernels. TPs emerge as natural extensions of GPs, retaining elliptical symmetry while offering heavier tails that provide robust performance in the presence of outliers or non-stationary data.
3. Analytical Framework: The work provides closed-form expressions for both the marginal likelihood and predictive distributions of TPs, facilitating easy integration into existing GP frameworks without added computational demand.
4. Predictive Dependence on Observations: A significant distinction of TPs is that their predictive covariances explicitly depend on the observed data, unlike traditional GPs, which is crucial in applications such as Bayesian optimization where this adaptability enhances performance.
5. Empirical Validation: Through comprehensive experiments, the Student-$t$ process demonstrates its superiority in scenarios with changing covariance structures and robust performance in Bayesian optimization tasks compared to GPs.

Implications and Future Directions

This research offers notable implications for the practical application of nonparametric processes in machine learning. The enhanced flexibility of Student-$t$ processes can be leveraged to address more complex structures in data that traditional Gaussian processes may fail to model effectively. Furthermore, the potential to apply TPs as direct replacements for GPs across various domains signifies a step forward in nonparametric modeling practices.

The future of research may focus on exploring various kernels along with TP priors to further enhance their applicability and examining the performance improvement in real-world, high-dimensional datasets. Further investigations into computational optimizations specific to TPs could drive broader adoption in large-scale applications.

In conclusion, this paper provides a compelling argument for substituting Gaussian processes with Student-$t$ processes in many instances, offering the benefits of nonparametric modeling with additional robustness and adaptability, crucial for tackling evolving patterns in data-driven fields.