Iterative Posterior Inference for Bayesian Kriging

TL;DR

This paper introduces an iterative method for estimating the posterior distribution in Bayesian geostatistics using normal mixture densities, improving approximation accuracy without ad hoc parameter bounds.

Contribution

It presents a novel iterative approach for posterior inference in Bayesian kriging that addresses tuning challenges and avoids ad hoc parameter treatments.

Findings

Method successfully approximates posterior distributions in geostatistical models.

Convergence of the iterative approximation is demonstrated in real data examples.

Applicable to models with parameters on the entire real line.

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.