Constructing Priors That Penalize the Complexity of Gaussian Random Fields

Priors are important for achieving proper posteriors with physicallymeaningful covariance structures for Gaussian random fields (GRFs) since thelikelihood typically only provides limited information about the covariancestructure under in-fill asymptotics. We extend the recent Penalised Complexityprior framework and develop a principled joint prior for the range and themarginal variance of one-dimensional, two-dimensional and three-dimensionalMatérn GRFs with fixed smoothness. The prior is weakly informative andpenalises complexity by shrinking the range towards infinity and the marginalvariance towards zero. We propose guidelines for selecting the hyperparameters,and a simulation study shows that the new prior provides a principledalternative to reference priors that can leverage prior knowledge to achieveshorter credible intervals while maintaining good coverage. We extend the prior to a non-stationary GRF parametrized through local rangesand marginal standard deviations, and introduce a scheme for selecting thehyperparameters based on the coverage of the parameters when fitting simulatedstationary data. The approach is applied to a dataset of annual precipitationin southern Norway and the scheme for selecting the hyperparameters leads toconcervative estimates of non-stationarity and improved predictive performanceover the stationary model.