Mathematical Sciences: Robustness and Scale in Spatial Applications of Markov Chain Monte Carlo for Bayesian Inference

提案：DMS9505114 PI: Dave Higdon机构：Duke University题目：马尔可夫链蒙特卡罗在贝叶斯推理空间应用中的鲁棒性和规模摘要：提出的研究为空间系统的贝叶斯推理开发了灵活的先验分布。从建模的角度来看，空间先验是在简单马尔可夫随机场（MRF）上的混合物，在两种不同的背景下发展。首先，将构建混合物，以便在空间过程中存在异常和非均质时进行稳健推断。在第二种情况下，将考虑不同尺度上的混合，以便在进行推理时可以严格地将MRF的尺度视为参数。只有最近在马尔可夫链蒙特卡罗（MCMC）方面取得的进展，才能从这些模型中进行推断。将考虑在农业、成像和环境监测方面的应用。拟议的研究将发展用于分析空间数据的统计方法，这些数据结合了不同尺度的信息。发展适合农业的空间模型、成像(如：遥感和医学成像)、环境监测和评估环境趋势是本研究的主要目标。目前的统计方法通常需要限制性的假设，这往往使这些领域的正式分析变得困难或不可能。这种方法将放松这些假设。将不同尺度的信息结合起来的一个主要动机是处理用于环境监测的地理信息系统。这样的系统总是包含可以纳入分析的不同规模的数据。

Proposal: DMS9505114 PI: Dave Higdon Institution: Duke University Title: Robustness and Scale in Spatial Applications of Markov Chain Monte Carlo for Bayesian Inference Abstract: The proposed research develops flexible prior distributions for Bayesian inference for spatial systems. From the modeling standpoint, spatial priors that are mixtures over simple Markov random fields (MRF's) are developed in two distinct contexts. In the first, mixtures will be constructed to allow robust inference when anomalies, and heterogeneity are present in the spatial process. In the second context, mixtures over different scales will be considered, so that the scale of the MRF can be rigorously treated as a parameter when making inference. Only very recent advances in Markov chain Monte Carlo (MCMC) make inference from such models possible. Applications in agriculture, imaging and environmental monitoring will be considered. The proposed research will develop statistical methodologies for analyzing spatial data which combines information at various scales. Development of spatial models suitable for agriculture, imaging (eg. remote sensing and medical imaging), environmental monitoring and assessing environmental trends are main goals of this research. Current statistical methods typically require restrictive assumptions which often make formal analyses in these areas difficult or impossible. This methodology will relax these assumptions. A primary motivation for combining information from various scales is for dealing with geographical information systems used for environmental monitoring. Such systems invariably contain data at varying scales which can be incorporated in the analysis.