State Space Methods for Efficient Inference in Student-t Process Regression

The added flexibility of Student-t processes (TPs) over Gaussian processes (GPs) robustifies inference in outlier-contaminated noisy data. The uncertainties are better accounted for than in GP regression, because the predictive covariances explicitly depend on the training observations. For an entangled noise model, the canonical-form TP regression problem can be solved analytically, but the naive TP and GP solutions share the same cubic computational cost in the number of training observations. We show how a large class of temporal TP regression models can be reformulated as state space models, and how a forward filtering and backward smoothing recursion can be derived for solving the inference analytically in linear time complexity. This is a novel finding that generalizes the previously known connection between Gaussian process regression and Kalman filtering to more general elliptical processes and non-Gaussian Bayesian filtering. We derive this connection, demonstrate the benefits of the approach with examples, and finally apply the method to empirical data.