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

摘要基于无环有向图（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，计算方便。20世纪80年代末，ADG模型被推广到包含有向边和无向边的二环图或链图，因此可以同时表示一些有向边和一些关联边的依赖关系。研究人员将研究一类新的链图模型的马尔可夫和统计性质，这些模型保留了许多ADG模型的理想性质。研究的问题包括这些新模型的完备性和可靠性，它们的局部马尔可夫性质的确定，以及用适当的本质图来表征它们的马尔可夫等价类。研究人员还将研究齐次锥上的一类非常一般的Wishart分布，其中传递无环有向图（TADGs）起着核心作用。E. B. Vinberg对齐次锥的经典描述揭示了满足TADG马尔可夫条件的正态模型与这类广义的Wishart分布之间的基本关系。这个类包含了多元统计分析中已知的所有Wishart分布，包括超Wishart分布和与正规格条件独立（LCI）模型相关的Wishart分布，以及许多新的Wishart分布。要研究的其他主题包括多参数假设检验和估计问题的内曼-皮尔逊、似然比和最大似然标准的局限性，以及检验顺序限制和多变量单侧替代方案的似然比检验的有效性。统计科学最核心的思想之一是评估一组随机变量之间的相关性。我们熟悉的相关、回归和预测等概念都是这一思想的体现，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，即网络，无论是无向的，有向的，还是混合的，以可视化和计算效率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关系来构建，即图的节点，根据其直接邻居，父节点或两者，但可以通过图的全局结构表示高度变化和复杂的多元依赖系统。本地规范允许在建模、推理和概率计算方面提高效率。在其众多应用中，gmm已在统计科学中变得普遍，用于分析列联表中的分类数据，用于模拟依赖空间的过程，例如流行病在人类和动物种群中的传播，以及用于开发恶劣天气条件的早期预警系统；在计算机科学（如贝叶斯网络）中，用于信息处理和检索，用于机器人，计算机视觉和模式识别，用于复杂程序的调试（如Windows 98），以及用于医学诊断的专家系统的表示；在决策科学中（如影响图），作为信息流和控制的模型，以及结合许多决策者的意见的模型。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”现实世界问题的软件的开发。

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.