Presenter: Pulong Ma, Assistant Professor, Department of Statistics, Iowa State University

Title: Revisiting Objective Bayesian Analysis For Spatially Correlated Data

Abstract: Gaussian processes have been widely used for modeling spatially correlated data through their mean function and covariance function. The Mat\'ern covariance function has become standard practice for modeling the spatial covariance structures in the past few decades due to its flexible control over the smoothness of the process. However, one limitation of the Mat\'ern  model is that it only allows short-range dependence which may not be desirable. This can be remedied by the recently proposed confluent hypergeometric (CH) covariance function with a tail-decay parameter controlling the degree of long-range dependence. Moreover, it is also widely recognized that parameter estimation for covariance parameters are notoriously challenging. This is partly due to the following reasons: first, there is no closed form derivative of the Mat\'ern covariance function with respect to the smoothness in general; hindering its deployment for likelihood-based inference using gradient-based optimization algorithms; second, the likelihood function can be shown to be maximized either at zero or infinity, resulting in degraded prediction. To address these issues, we revisit the existing objective Bayes framework and construct the reference prior for Gaussian process models under the CH covariance function. Next, we show that the reference prior can result in the so-called robust estimation that avoids degraded prediction. Ongoing work is focused on investigating posterior propriety resulting from the reference prior and direct computation of Bayes factors to compare different correlation functions. Numerical illustrations will be given.