Vecchia-Laplace approximations of generalized Gaussian processes for big non-Gaussian spatial data

Generalized Gaussian processes (GGPs) are highly flexible models that combine latent GPs with potentially non-Gaussian likelihoods from the exponential family. GGPs can be used in a variety of settings, including GP classification, nonparametric count regression, modeling non-Gaussian spatial data, and analyzing point patterns. However, inference for GGPs can be analytically intractable, and large datasets pose computational challenges due to the inversion of the GP covariance matrix. To achieve computationally feasible parameter inference and GP prediction for big spatial datasets, we propose a Vecchia-Laplace approximation for GGPs, which combines a Laplace approximation to the non-Gaussian likelihood with a computationally efficient Vecchia approximation to the GP. We examine the properties of the resulting algorithm, including its linear complexity in the data size. We also provide numerical studies and comparisons on simulated and real spatial data.