Iterative Posterior Inference for Bayesian Kriging

Abstract: We propose a method for estimating the posterior distribution of a standard geostatistical model. After choosing the model formulation and specifying a prior, we use normal mixture densities to approximate the posterior distribution. The approximation is improved iteratively. Some difficulties in estimating the normal mixture densities, including determining tuning parameters concerning bandwidth and localization, are addressed. The method is applicable to other model formulations as long as all the parameters, or transforms thereof, are defined on the whole real line, $(-\infty, \infty)$. Ad hoc treatments in the posterior inference such as imposing bounds on an unbounded parameter or discretizing a continuous parameter are avoided. The method is illustrated by two examples, one using digital elevation data and the other using historical soil moisture data. The examples in particular examine convergence of the approximate posterior distributions in the iterations.