Bayesian analysis of hierarchical random fields for material modeling

In probabilistic assessments, spatially variable material properties are modeled with random fields. These random fields can be learned from spatial data by means of Bayesian analysis. This paper presents analytical expressions for the Bayesian analysis of hierarchical Gaussian random fields. We model the prior spatial distribution by a Gaussian random field with normal-gamma distributed mean and precision and make use of the conjugacy of prior distribution and likelihood function to find the posterior distribution of the random field parameters. We present closed-form expressions for the spatial mean and precision function of the posterior predictive Student's t-random field. Furthermore, we discuss the application of the hierarchical model to non-Gaussian random fields (translation random fields) and show the connection of the methodology to the Bayesian approachof EN 1990 for estimating characteristic values for material parameters. The method is illustrated on two spatial data sets of concrete and soil strength parameters.