New Methods and Theory of Statistical Inference for Non-Gaussian Graphical Models

无向图形模型（GM）是研究复杂系统中大量随机变量之间关系的强大工具，被广泛用于科学应用，包括图像分析、统计物理、天体物理学、金融和生物医学研究。随着最近的技术进步，可以为给定系统收集前所未有的大量信息，这使得对gm进行有意义的推理保证更具挑战性。尽管最近在开发高斯gm的方法和理论方面取得了成功，但对于一些重要的数据类型，连续和正态分布数据的基本假设被违反了。例如，序数、二进制和计数数据本质上都是离散的，不能天真地转换为高斯分布。在生物医学研究中，非高斯型数据的例子包括DNA拷贝数变异、突变和（单细胞）rna序列数据。与高斯GM的最新进展相比，非高斯数据类型的建模和理论基础研究相对落后。为了弥补这一差距，PI将确定一些主要的建模和推理挑战，并为非高斯数据提出几个新的图形模型。此外，PI将进一步开发，评估和改进新的统计和计算推理方法，为这些模型提供理论保证。本研究将通过三个方面的工作，对图形模型中非高斯数据的建模和统计推断进行基础性的理论理解。(1)采用协变量调整截断泊松图模型（TPGM）开发了新的两步推理程序，该模型为二进制和计数型数据的建模提供了统一的框架。推理过程充分尊重图的固有稀疏结构，使其更加可靠。提出了一种基于似然的非线性偏差校正方法。（II）提出了一种新的零膨胀TPGM，充分考虑数据中的零膨胀模式，在细胞水平上模拟单细胞RNA序列数据。基于EM算法的推理程序为更好地理解不同细胞类型的遗传网络铺平了道路，从而更好地理解各种疾病的机制。理论上，基于复合似然的电磁算法克服了计算困难。(3)建立了一种新的潜在半参数图模型，通过对有序型和连续型数据进行积分来推断图的内在结构。该方法考虑了潜在的混杂效应，以得出有意义的结论。除了统计建模和图形模型理论方面的基本进展之外，这项研究将对包括生物学、药学、金融和基因组学在内的许多科学学科的应用产生直接影响。研究结果将通过出版物、开放源码软件和在会议上的演讲来传播。该奖项反映了美国国家科学基金会的法定使命，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 undirected graphical model (GM), a powerful tool for investigating the relationship among a large number of random variables in a complex system, is used in a wide range of scientific applications, including image analysis, statistical physics, astrophysics, finance, and biomedical studies. With recent technological advances, unprecedented amounts of information can be collected for a given system, making meaningful inferential guarantees of GMs more challenging. Despite recent successes in development of methods and theory for Gaussian GMs, the underlying assumption of continuous and normally distributed data is violated for some important data types. For example, ordinal, binary and count data are all discrete in nature and cannot be naively transformed into Gaussian distributions. In biomedical studies, examples of non-Gaussian type data include DNA Copy Number Variation, mutation and (single cell) RNA-sequence data. Compared to recent advances in Gaussian GM, research in modeling and theoretical foundations for non-Gaussian data types has fallen behind. To bridge this gap, the PI will identify some of the major modeling and inferential challenges and propose several new graphical models for non-Gaussian data. In addition, the PI will further develop, evaluate and improve new statistical and computational inference methods for these models with theoretical guarantees. The proposed research will significantly advance fundamental theoretical understanding on modeling and statistical inference of non-Gaussian data in graphical models via three tasks. (I) Development of a new two-step inference procedure to employ the covariate-adjusted truncated Poisson graphical model (TPGM) which provides a unified framework for modeling both binary and count type data. The inferential procedure fully respects the intrinsic sparse structure of the graph making it more reliable. A novel likelihood-based non-linear score vector for bias correction will be developed. (II) A novel zero-inflated TPGM fully accounting for the zero-inflation pattern in the data is proposed to model single cell RNA sequence data at the cell level. The inferential procedure based on EM algorithms paves a road to better understanding of the genetic networks in different cell types, and thus a better understanding of the mechanisms of various diseases. Theoretically, a composite-likelihood-based EM algorithm is utilized to overcome computational difficulties. (III) Development of a novel latent semiparametric graphical model to draw inferences on intrinsic graph structure by integrating both ordinal and continuous type data. The method takes into account potential confounding effects to draw meaningful conclusions. Beyond fundamental advances in statistical modeling and theory of graphical models, the research will have immediate impact in applications from a number of scientific disciplines including biology, pharmacy, finance and genomics. The results will be disseminated through publications, open-source software and presentations at conferences.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.