Interpretable Priors for Hyperparameters for Gaussian Random Fields

Gaussian random fields (GRFs) are important building blocks in hierarchical models for spatial data, but there is no practically useful, principled approach for selecting the prior on their hyperparameters. The prior is typically chosen in an ad-hoc manner, which lacks theoretical justification, despite the fact that we know that the hyperparameters are not consistently estimable from a single realization and that there is sensitivity to the choice of the prior. We first use the recent Penalised Complexity prior framework to construct a practically useful, tunable, weakly informative joint prior on the range and the marginal variance for Matern GRFs with fixed smoothness. We then discuss how to extend this prior to a prior for a non-stationary GRF with covariates in the covariance structure.