Beyond Gaussian processes: Flexible Bayesian modeling and inference for geostatistical processes

This paper proposes a novel family of geostatistical models to account for features that cannot be properly accommodated by traditional Gaussian processes. The family is specified hierarchically and combines the infinite dimensional dynamics of Gaussian processes to that of any multivariate continuous distribution. This combination is stochastically defined through a latent Poisson process and the new family is called the Poisson-Gaussian Mixture Process - POGAMP. Whilst the attempt of defining a geostatistical process by assigning some arbitrary continuous distributions to be the finite-dimension distributions usually leads to non-valid processes, the finite-dimensional distributions of the POGAMP can be arbitrarily close to any continuous distribution and still define a valid process. Formal results to establish the existence and some important properties of the POGAMP, such as absolute continuity with respect to a Gaussian process measure, are provided. Also, a MCMC algorithm is carefully devised to perform Bayesian inference when the POGAMP is discretely observed in some space domain. Simulations are performed to empirically investigate the modelling properties of the POGAMP and the efficiency of the MCMC algorithm.