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

提案：DMS9505114 PI：Dave Higdon 院校：杜克大学 题目：马尔可夫链蒙特卡罗在空间应用中的鲁棒性和规模 关于Bayesian Inference 摘要： 所提出的研究开发灵活的先验分布空间系统的贝叶斯推理。 从建模的角度来看，空间先验是简单马尔可夫随机场（MRF）的混合物在两个不同的背景下开发。 在第一，混合物将被构造为允许强大的推理时，异常，和异质性存在于空间过程中。 在第二种情况下，将考虑不同尺度上的混合物，以便在进行推断时，可以严格地将MRF的尺度作为一个参数。 只有马尔可夫链蒙特卡罗（MCMC）的最新进展才使从这些模型中进行推断成为可能。 将考虑在农业、成像和环境监测方面的应用。 拟议的研究将开发用于分析空间数据的统计方法，这些数据结合了各种尺度的信息。 开发适用于农业的空间模型，成像（例如，遥感和医学成像），环境监测和评估环境趋势是这项研究的主要目标。 目前的统计方法通常需要限制性的假设，这些假设往往使形式分析 在这些领域是困难的或不可能的。 这种方法将放宽这些假设。 一个主要的动机，结合信息，从各种 比例尺用于处理环境监测所用的地理信息系统。 这些系统总是包含数据 在不同的尺度上，可以结合在分析中。

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.