New Directions in Bayesian Change-Point Analysis

自然界中几乎所有的动态和随机过程都会经历突然而重大的结构性变化。这种变化往往发生在可观察到的数量上，例如燃料价格或股票指数或犯罪活动因经济现象或公共政策变化或疾病暴发等不可观察到的潜在因素的变化而发生重大变化。在所有科学学科和应用领域，如经济学、流行病学、社会科学、网络安全和金融学，都经常观察到这样的“变化点”。具体的例子可以是改变回归，当观察到的变量通过随时间变化的平均结构依赖于预测值时，或者改变海量维度数据中的点，例如高分辨率成像数据或复杂的连通图。虽然有相当多的文献提出了在不同环境下检测变化点的详细方法，但对于具有复杂依赖或稀疏结构的分层模型中的变化点，贝叶斯方法的考虑有限。这项研究用新的统计工具填补了这一空白，新的统计工具受到特定现实生活应用的推动，通过开发理论框架，同时保持在当前应用中的效率和实用性。该项目将研究生教育和培训与统计研究相结合，强调维护社会和伦理考虑，以创造和促进一个包容和多样化的社区。在更高的层面上，由于组合计算的复杂性，发现变化点和变化的结构的问题往往变得极其困难。通过这项研究，绩效指标概述了变化点估计的理论和方法方面的全面框架，涵盖了不同应用领域可能出现的问题。特别是，PI从根本上构建了新的贝叶斯方法，该方法可以1)在具有一致性保证的变化的线性回归中执行稀疏信号恢复2)通过高斯图形模型中的变化来检测依赖结构中的变化点，以及3)通过随机投影来构建用于处理超高维对象的创新方法，以显著降低计算负担。将开发理论机制，以提供概率的严谨性和一致性保证。将开发计算效率高的算法，并将在R部署用户友好的软件工具，供科学界广泛使用。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Almost all dynamic and random processes in nature go through sudden and significant structural changes. Often the change is in the observable quantity, e.g. fuel prices or stock indices or crime activities changing significantly in response to a change in an unobservable, latent factor such as an economic phenomenon or a public policy change, or a disease outbreak. Such ‘change-points’ are routinely observed across all scientific disciplines and applications, such as economics, epidemiology, social sciences, cybersecurity and finance. Specific examples could be changing regression when the observed variable depends on predictors through a mean structure that changes with time, or change points in data with massive dimensions, such as high-resolution imaging data or complex connected graphs. While there is a substantial literature proposing elaborate methods for detecting change points in different settings, there has been limited consideration of Bayesian methods for change-points in hierarchical models with complex dependence or sparsity structures. This research fills this gap with new statistical tools motivated by specific real-life applications, by developing theoretical framework while retaining efficiency and usefulness in current applications. The project integrates graduate education and training with statistical research, and emphasizes upholding societal and ethical considerations that create and foster an inclusive and diverse community.In higher dimensions, the problem of detecting change-points and the changing structure is often rendered extremely difficult owing to a combinatorial computational complexity. Through this research, the PIs outline a comprehensive framework, both theoretical and methodological, in the context of change point estimation encompassing problems that may arise in different field of applications. In particular, the PIs build fundamentally new Bayesian methods that can 1) perform sparse signal recovery in a changing linear regression with consistency guarantees 2) detect change-points in dependence structure via changes in a Gaussian graphical model, and 3) build an innovative method for handling ‘ultra-high’-dimensional objects via random projections to drastically reduce the computational burden. Theoretical machinery will be developed to provide probabilistic rigor and consistency guarantee. Computationally efficient algorithms will be developed, and user-friendly software tools will be deployed in R for the usage of the developed methods by the scientific community at large.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.