Bayesian Analysis and Prediction of Gaussian Random Fields

该项目开发了对经济、流行病学、地理学、地质学和水文学等许多社会科学和地球科学中出现的空间数据进行客观贝叶斯分析的方法，以及基于大中型空间数据集进行贝叶斯分析和预测的计算高效算法。在方法论方面，研究者为不同类型的高斯随机场的参数推导出新的自动先验分布，由它们的协方差矩阵或它们的精度矩阵指定。该研究探索了基于这些自动先验的贝叶斯推断的主要统计特性，例如后验合适性条件、参数和预测推理的频率特性以及预测总和的存在。一个特别感兴趣的点是研究这些自动先验依赖于抽样设计的利弊。在计算方面，调查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是计算昂贵的，并开发了新的计算高效的空间数据贝叶斯推理和预测算法，使基于中大型空间数据集的贝叶斯分析变得可行。由于大多数空间层次模型都使用高斯随机场作为构建块，因此本项目提出的方法是发展用于描述非高斯数据的空间层次模型的客观贝叶斯分析的第一步。本项目期间开发的统计方法在许多社会科学和地球科学中具有实际影响，如经济学、流行病学、地理学、地质学和水文学，空间数据的收集和分析已经成为常见的任务。与传统的空间数据分析方法相比，被称为贝叶斯方法的统计学范式具有几个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它在实践者中的更广泛应用。尤其是对于某些类型的大型空间数据集的分析，目前的贝叶斯方法过于繁琐或不可行。在这个项目中开发的统计方法将有助于克服这些技术和计算障碍，从而弥合空间数据的贝叶斯分析方法和实践之间的差距。研究生将参与该项目，为他们的统计培训以及加强阿肯色大学的统计课程做出贡献。

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.