Sparse Spatio-temporal Gaussian Processes with General Likelihoods

In this paper, we consider learning of spatio-temporal processes by formulating a Gaussian process model as a solution to an evolution type stochastic partial differential equation. Our approach is based on converting the stochastic infinite-dimensional differential equation into a finite dimensional linear time invariant (LTI) stochastic differential equation (SDE) by discretizing the process spatially. The LTI SDE is time-discretized analytically, resulting in a state space model with linear-Gaussian dynamics. We use expectation propagation to perform approximate inference on non-Gaussian data, and show how to incorporate sparse approximations to further reduce the computational complexity. We briefly illustrate the proposed methodology with a simulation study and with a real world modelling problem.