Abstract
This thesis is dedicated to advancing computationally efficient Bayesian inference for Gaussian random fields, particularly those that can be represented as solutions to a class of fractional-order stochastic partial differential equations (SPDEs).
First, we develop a novel Gaussian Markov random field (GMRF) representation for Gaussian fields specified via SPDEs with fractional orders greater than a threshold determined by the spatial domain’s dimensionality. This representation is crucial for computational efficiency, overcoming the limitations of previous methods that either do not permit inference on the smoothness parameter or fail to provide a GMRF representation.
Second, we address key limitations of the recently proposed Penalizing Complexity (PC) priors by introducing Wasserstein Complexity Penalization (WCP) priors. The WCP priors offer a more robust framework, particularly in scenarios where PC priors are inadequate.
Third, we construct WCP priors for all parameters of Gaussian Matern random fields, a task that is not feasible with PC priors. This development significantly enhances the applicability of the SPDE approach in statistical modeling.
Brief Biography
Zhen Xiong is a PhD student working on Bayesian statistics and the SPDE approach in Professor David Bolin's group. Prior to Kaust, he studied applied mathematics at Shandong University and University of Pennsylvania.
