Developing Conjugate Models for Exact MCMC free Bayesian Inference with Application to High-Dimensional Spatio-Temporal Data

数据生产的大规模扩张导致了不确定性量化的自然计算挑战。特别是，贝叶斯方法可以解释不确定性的来源，但需要已知的计算要求高的技术。当数据在空间和/或时间上相关时，这些困难加剧。目前的解决方案主要使用近似或低效的迭代方法，如马尔可夫链蒙特卡罗（MCMC）。该项目解决了不确定性量化的计算挑战，新的统计方法，不需要近似和MCMC。大数据已经影响了几乎所有的科学领域，因此，可扩展的、精确的、不含MCMC的贝叶斯方法的方法论开发和软件将产生重大影响。所提出的方法和软件不仅将是统计学的进步，而且将在处理复杂时空过程的广泛学科中发挥作用，如神经科学，气候学，人口学，计量经济学，生态学，气象学，海洋学和官方统计。研究人员将教育和培训研究生，并通过期刊出版物，公共使用的软件和会议演示来传播项目成果。该项目的目标是为可扩展的贝叶斯分层模型开发共轭分布理论，为统计学家和主题科学家创建一个更大的框架，以执行MCMC自由贝叶斯推断而无需近似后验分布。特别是，该项目将开发和扩展广义共轭多变量（GCM）分布，它允许人们直接从特定大类混合效应模型的精确后验分布进行模拟。这个精确的样本被称为精确后验回归（EPR）。在目标1中，研究人员将开发GCM和EPR对新设置的扩展，包括有序和名义数据、某些超参数的精确MCMC自由推断以及与现有统计模型的理论联系。 目标2涉及EPR的扩展到多变量时空和多尺度空间数据，允许利用几个来源的依赖，以改善预测和执行空间变化的支持（COS），而不使用MCMC或近似贝叶斯方法。为了实现可扩展性，在目标3中，研究人员将开发一个精确的贝叶斯分层模型，该模型以信息丰富的方式反复对数据进行子集化，而不会对数据施加额外的假设。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The massive expansion in the production of data has led to natural computational challenges in uncertainty quantification. In particular, Bayesian methodology can account for sources of uncertainty, but requires techniques known to be computationally demanding. These difficulties are exacerbated when data are spatially and/or temporally correlated. The current solutions predominantly use either approximations or inefficient iterative methods such as Markov chain Monte Carlo (MCMC). This project resolves the computational challenges in uncertainty quantification with novel statistical methodology that does not require approximations and MCMC. Big data has impacted nearly every area of science, and as a result, methodological development and software for scalable, exact, MCMC free Bayesian methodology will have a substantial effect. Not only will the proposed methodology and software be an advancement in statistics, but it will be useful across a broad range of disciplines that deal with complex spatio-temporal processes such as neuroscience, climatology, demography, econometrics, ecology, meteorology, oceanography, and official statistics. The investigator will educate and train graduate students, and disseminate project findings through journal publications, public-use software, and conference presentations.The objective of this project is to develop conjugate distribution theory for scalable Bayesian hierarchical models that create a larger framework for statisticians and subject matter scientist to perform MCMC free Bayesian inference without approximating the posterior distribution. In particular, this project will develop and extend the generalized conjugate multivariate (GCM) distribution, which allows one to simulate directly from the exact posterior distribution for a particular large class mixed effects models. This exact sample is referred to as Exact Posterior Regression (EPR). In Aim 1, the investigator will develop extensions of GCM and EPR to new settings including ordinal and nominal data, exact MCMC free inference for certain hyperparameters, and theoretical connections to existing statistical models. Aim 2 involves extensions of EPR to multivariate spatio-temporal and multiscale spatial data, allowing one to leverage several sources of dependence to improve predictions and perform spatial change of support (COS) without the use of MCMC or approximate Bayesian methods. To achieve scalability, in Aim 3, the investigator will develop an exact Bayesian hierarchical model that repeatedly subsets the data in an informative manner that does not impose additional assumptions on the data.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.