Bayesian Analysis and Prediction of Gaussian Random Fields

该项目开发了空间数据的客观贝叶斯分析方法，包括地理统计和点阵数据，这些数据出现在许多社会和地球科学中，如经济、流行病学、地理、地质和水文学，以及基于中等到大型空间数据集执行贝叶斯分析和预测的计算效率算法。在方法方面，研究者为不同类型高斯随机场的参数导出了新的自动先验分布，由它们的协方差矩阵或精度矩阵指定。研究了基于这些自动先验的贝叶斯推理的主要统计性质，如后验适当性条件、参数和预测推理的频率性以及预测摘要的存在性。特别有趣的一点是研究这些自动先验对抽样设计的依赖的利弊。在计算方面，研究者推导出近似这些自动先验分布的方法，因为这些先验的评估在大多数情况下计算成本很高，并开发了新的计算效率高的贝叶斯推理和空间数据预测算法，这将使基于中等到大型空间数据集的贝叶斯分析可行。本项目中提出的方法是向用于描述非高斯数据的空间层次模型的客观贝叶斯分析发展的第一步，因为大多数这些模型使用高斯随机场作为构建块。该项目开发的统计方法在许多社会和地球科学领域具有实际影响，如经济学、流行病学、地理学、地质学和水文学，在这些领域，空间数据的收集和分析已成为常见的任务。与传统的空间数据分析方法相比，统计范式贝叶斯方法具有一些概念和方法上的优势，但在实施过程中出现的技术和计算困难阻碍了其在实践者中更广泛的应用。对于某些类型的大型空间数据集的分析尤其如此，在这些数据集中，贝叶斯方法的当前实现过于繁琐或不可行而无法执行。本项目开发的统计方法将有助于克服其中一些技术和计算障碍，从而弥合贝叶斯空间数据分析的方法与实践之间的差距。研究生将参与该项目，为他们的统计培训做出贡献，并加强阿肯色大学的统计课程。

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 andpredictive 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 practicalimpacts 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, andconsequently 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.