Collaborative Research on Graphical Markov Models and Related Topics in Multivariate Statistical Analysis

多元统计分析中图马尔可夫模型及相关主题的协作研究

• 批准号: 0071920
• 负责人: Steen Andersson
• 金额: $ 8万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-08-15 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0071920&HistoricalAwards=false
• 关键词: Collaborative; Research; Graphical; Markov; Models

Perlman 0071818Andersson 0071920AbstractStatistical models based on acyclic directed graphs (ADGs) (also called directed acyclic graphs (DAGs), Bayesian networks, or influence diagrams) are particularly well behaved, easily interpretable, and computationally convenient. In the late 1980s, ADG models were generalized to adicyclic graphs or chain graphs, which include both directed and undirected edges, hence can simultaneously represent dependences some of which are directional and some associative. The investigators will study the Markov and statistical properties of a new class of chain graph models that retains many of the desirable properties of ADG models. Problems to be investigated include the completeness and faithfulness of these new models, determination of their local Markov property, and characterization of their Markov equivalence classes by means of an appropriate essential graph. The investigators will also study a very general class of Wishart distributions on homogeneous cones, in which transitive acyclic directed graphs (TADGs) play a central role. E. B. Vinberg's classical characterization of homogeneous cones has been found to reveal a fundamental relationship between normal models satisfying TADG Markov conditions and this general class of Wishart distributions. This class includes all Wishart distributions previously known in multivariate statistical analysis, including the hyper-Wishart distributions and Wishart distributions associated with normal lattice conditional independence (LCI) models, as well as a great many new ones. Additional topics to be investigated include the limitations of the Neyman-Pearson, likelihood ratio, and maximum likelihood criteria for multiparameter hypothesis-testing and estimation problems, and the efficacy of the likelihood ratio test for testing order-restricted and multivariate one-sided alternatives.One of the most central ideas of statistical science is the assessment of dependences among a set of stochastic variables. The familiar concepts of correlation, regression, and prediction are manifestations of this idea, and many aspects of causal relationships ultimately rest on representations of multivariate dependence. Graphical Markov models (GMM) use graphs i.e. networks, either undirected, directed, or mixed, to represent multivariate dependencies in a visual and computationally efficient manner. A GMM is usually constructed by specifying local dependences for each variable, i.e. node of the graph, in terms of its immediate neighbors, parents, or both, yet can represent a highly varied and complex system of multivariate dependences by means of the global structure of the graph. The local specification permits efficiencies in modeling, inference, and probabilistic calculations. Among their many applications, GMMs have become prevalent in statistical science for the analysis of categorical data in contingency tables, for the modeling of spatially-dependent processes such as the spread of epidemics in human and animal populations, and for the development of early warning systems for severe weather conditions; in computer science (as Bayesian networks) for information processing and retrieval, for robotics, computer vision, and pattern recognition, for the debugging of complex programs (such as Windows 98), and for the representation of expert systems for medical diagnosis; and in decision science (as influence diagrams) as models for information flow and control and for combining the opinions of many decision-makers. A crucial feature of these models is that they are designed for fast computational implementation, thereby facilitating the development of software that can "reason" about real world problems.

基于无环有向图（ADG）（也称为有向无环图（DAG）、贝叶斯网络或影响图）的统计模型表现得特别好，易于解释，并且计算方便。 在20世纪80年代后期，ADG模型被推广到自环图或链图，其中包括有向和无向边，因此可以同时表示其中一些是有向的和一些是关联的依赖。 研究人员将研究一类新的链图模型的马尔可夫和统计特性，该模型保留了ADG模型的许多理想特性。 要研究的问题包括这些新模型的完整性和忠实性，确定其局部马尔可夫性质，并通过适当的基本图表征其马尔可夫等价类。 研究人员还将研究均匀锥上的一类非常一般的Wishart分布，其中传递无环有向图（TADG）发挥着核心作用。E. B。温伯格的经典特征的齐次锥已被发现揭示了一个基本的正常模型之间的关系，满足TADG马尔可夫条件和这类一般的Wishart分布。 这个类别包括所有以前在多元统计分析中已知的Wishart分布，包括超Wishart分布和与正态格点条件独立（LCI）模型相关的Wishart分布，以及许多新的Wishart分布。 其他主题将被调查包括奈曼皮尔逊的局限性，似然比，和最大似然准则的多参数假设检验和估计问题，以及效率的似然比检验的顺序限制和多变量单侧alternatives.One of the most central ideas of statistical science is the assessment of dependencies among a set of stochastic variables. 相关性、回归和预测等熟悉的概念都是这一思想的表现，因果关系的许多方面最终都依赖于多元依赖的表示。 图形马尔可夫模型（GMM）使用图形，即网络，无向，有向或混合，以可视化和计算高效的方式表示多变量依赖关系。 GMM通常通过指定每个变量的局部依赖来构造，即图的节点，根据其直接邻居，父母或两者，但可以通过图的全局结构来表示高度变化和复杂的多变量依赖系统。 本地规范允许建模、推断和概率计算的效率。 在其许多应用中，GARCH在统计科学中已变得普遍，用于分析列联表中的分类数据，用于模拟与空间有关的过程，如流行病在人类和动物群体中的传播，以及用于开发恶劣天气条件的预警系统;计算机科学（如贝叶斯网络）用于信息处理和检索，用于机器人技术，计算机视觉和模式识别，用于复杂程序的调试（如Windows 98），并用于医疗诊断专家系统的表示;在决策科学中（如影响图），作为信息流和控制的模型，并结合许多决策者的意见。 这些模型的一个关键特征是，它们是为快速计算实现而设计的，从而促进了能够“推理”真实的世界问题的软件的开发。