Collaborative Research: Aggregated Monte Carlo: A General Framework for Distributed Bayesian Inference in Massive Spatiotemporal Data

随着空间参考技术的巨大进步，如全球定位系统，它可以用一个简单的手持设备识别地理坐标，各个学科的研究人员已经收集了前所未有的各种地理编码的时间数据。因此，在过去的十年中，使用灵活的统计模型对时空数据进行建模已经成为许多学科（包括环境科学、健康科学和海洋学等）的一个非常活跃的研究领域。在所有这些应用中，研究人员需要高效的数据建模工具，这些工具可以适应现代时空数据的复杂性和规模，使他们能够快速适应各种科学模型，解释关联的复杂性。该研究项目开发了一类新的分布式贝叶斯统计算法，即聚合蒙特卡罗（AMC），它可以在前所未有的规模上有效地建模海量时空数据。虽然PI的动机主要来自复杂的建模和大量时空数据的不确定性量化，但所提出的算法足够通用，可以在机器学习和计算机实验的相关文献中留下重要的足迹。总体目标还包括开发软件工具包，以便更好地为相关学科的从业人员服务。时空索引数据的规模、复杂性和可用性都在爆炸式增长。这一事件已经超过了贝叶斯统计方法的发展，因为除非施加限制性假设，否则基于随机过程分析时空点参考和点过程数据的最新方法的拟合速度非常慢。主要的问题是马尔可夫链蒙特卡罗（MCMC）方法中用于拟合这些模型的蒙特卡罗（MC）计算与数据的大小不匹配。为了解决这个问题，PI开发了一个称为聚合蒙特卡罗（AMC）的通用框架，用于使用分治技术在基于随机过程的海量时空数据建模中扩展MC计算。AMC有三个阶段，包括将数据划分为较小的子集，使用MCMC获得所有子集中未知参数和潜变量的后验样本，以及组合所有子集的MCMC样本。AMC经过调整，可以提高任何基于随机过程的最先进模型的可扩展性，并使用分治技术。在计算方面，主要的创新包括为具有不同时空结构的数据开发一般的划分和组合方案。从理论上讲，该项目提供了子集数量的界限，使得使用AMC估计的后验分布在后验风险和收缩率的衰减方面提供了完整数据后验分布的近最佳近似。从概念上讲，AMC提供了一个自然的扩展，现有的结果组合使用的重心的子集后验分布的参数模型的非参数模型与复杂的时空结构。AMC的最吸引人的特点是它利用并行计算机体系结构对海量时空数据进行高效灵活的建模，并提供具有理论保证的后验推理和不确定性估计。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

With tremendous advancements in spatial referencing technologies such as Global Positioning Systems that can identify geographical coordinates with a simple hand-held device, researchers in various disciplines have gathered an unprecedented variety of geo-coded temporal data. Consequently, modeling spatiotemporal data with flexible statistical models has become an enormously active area of research over the last decade in many disciplines including the environmental sciences, health sciences and oceanography, among others. In all these applications, researchers require efficient data modeling tools that can adapt to the complexity and size of modern spatiotemporal data, empowering them to quickly fit a variety of scientific models that explain the intricate nature of associations. This research project develops a new class of distributed Bayesian statistical algorithms, the Aggregated Monte Carlo (AMC), that enables efficient modeling of massive spatiotemporal data on an unprecedented scale. While the motivation of the PIs comes primarily from complex modeling and uncertainty quantification of massive spatiotemporal data, the proposed algorithm is general enough to set important footprints in the related literature of machine learning and computer experiments. The overarching goal also includes the development of software toolkits to better serve practitioners in related disciplines. There has been an explosion in the size, complexity, and availability of spatiotemporally indexed data. This event has outpaced the development in Bayesian statistical methodology in that the fitting of state-of-the-art methods based on stochastic processes for analyzing spatiotemporal point referenced and point process data is prohibitively slow unless restrictive assumptions are imposed. The main problem is that the Monte Carlo (MC) computations in Markov chain Monte Carlo (MCMC) methods for fitting these models scale poorly with the size of the data. Solving this problem, the PIs develop a general framework, called Aggregated Monte Carlo (AMC), for scaling MC computations in the stochastic process-based modeling of massive space-time data using a divide-and-conquer technique. AMC has three stages that involve dividing the data into smaller subsets, obtaining posterior samples of the unknown parameters and latent variables across all the subsets using MCMC, and combining the MCMC samples from all the subsets. AMC is tuned to boost the scalability of any state-of-the-art model based on a stochastic process using a divide-and-conquer technique. Computationally, the main innovations include the development of general division and combination schemes for data with diverse spatiotemporal structures. Theoretically, the project provides bounds on the number of subsets such that the posterior distribution estimated using AMC provides a near optimal approximation of the full data posterior distribution in terms of decay of the posterior risks and contraction rates. Conceptually, AMC provides a natural extension of the existing results for combination using the barycenter of subset posterior distributions in parametric models to non-parametric models with complex spatiotemporal structures. The most appealing features of AMC are that it exploits parallel computer architecture for efficient and flexible modeling of massive spatiotemporal data and it provides posterior inference and uncertainty estimates with theoretical guarantees.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.