This project develops methodology for objective Bayesian analysis of spatial data, both geostatistical and lattice data, that arise in many of the social and earth sciences, such as economy, epidemiology, geography, geology and hydrology, as well as computationally efficient algorithms to perform Bayesian analysis and prediction based on moderate to large spatial datasets. On the methodological side, the investigator derives new automatic prior distributions for the parameters of different kinds of Gaussian random fields, specified either by their covariance matrices or by their precision matrices. The research explores the main statistical properties of Bayesian inferences based on these automatic priors, such as conditions for posterior propriety, frequentist properties of parameter and predictive inferences, and existence of predictive summaries. A point of particular interest is the study of the pros and cons of the dependence of these automatic priors on the sampling design. On the computational side, the investigator derives methods to approximate these automatic priors distributions, since evaluation of these priors is in most cases computationally expensive, and develops new computationally efficient algorithms for Bayesian inference and prediction of spatial data that would make feasible Bayesian analysis based on moderate to large spatial datasets. The methodology proposed in this project serves as an initial step toward the development of objective Bayesian analysis for spatial hierarchical models used to describe non-Gaussian data, since most of these models use Gaussian random fields as building blocks.
The statistical methodology developed during this project has practical impacts in many social and earth sciences, such as economy, epidemiology, geography, geology and hydrology, where the collection and analysis of spatial data have become common tasks. A paradigm of statistics called the Bayesian approach possesses several conceptual and methodological advantages when compared to traditional approaches for the analysis of spatial data, but technical and computational difficulties that arise during implementation have hindered its more widespread use among practitioners. This is particularly so for the analysis of some types large spatial datasets where current implementations of the Bayesian approach are too cumbersome or unfeasible to be carried out. The statistical methodology developed in this project would contribute to overcome some of these technical and computational hurdles, and consequently to bridge the gap between methodology and practice for Bayesian analysis of spatial data. Graduate students would be engaged in the project, contributing to their statistical training as well as the enhancement of the Statistics program at the University of Arkansas.
