Approximate Reference Prior for Gaussian Random Fields

TL;DR

This paper introduces a new class of approximate reference priors for Gaussian random fields that are computationally efficient, stable, and proper, facilitating Bayesian analysis and model selection in geostatistics.

Contribution

The authors derive a novel approximation to reference priors based on spectral methods, improving computational stability and ensuring properness for correlation parameters.

Findings

Approximate priors are more stable and computationally less demanding.

The marginal approximate prior for correlation is always proper.

Application to pollution data demonstrates practical utility.

Abstract

Reference priors are theoretically attractive for the analysis of geostatistical data since they enable automatic Bayesian analysis and have desirable Bayesian and frequentist properties. But their use is hindered by computational hurdles that make their application in practice challenging. In this work, we derive a new class of default priors that approximate reference priors for the parameters of some Gaussian random fields. It is based on an approximation to the integrated likelihood of the covariance parameters derived from the spectral approximation of stationary random fields. This prior depends on the structure of the mean function and the spectral density of the model evaluated at a set of spectral points associated with an auxiliary regular grid. In addition to preserving the desirable Bayesian and frequentist properties, these approximate reference priors are more stable, and their computations are much less onerous than those of exact reference priors. Unlike exact reference priors, the marginal approximate reference prior of correlation parameter is always proper, regardless of the mean function or the smoothness of the correlation function. This property has important consequences for covariance model selection. An illustration comparing default Bayesian analyses is provided with a data set of lead pollution in Galicia, Spain.