CAREER: Statistical Inference in Algebraic Models with Singularities

本课题研究参数空间具有奇异性的统计模型。研究者研究奇点如何影响现有统计方法的行为，并开发新技术，以充分评估统计显著性。重点是代数统计模型，即具有（半）代数集作为参数空间的模型。这类代数模型包括许多在实践中使用的奇异模型，可以使用计算代数几何的工具来研究。重要的是，半代数集具有良好的局部几何特性，可以在不需要假定难以验证的正则性条件的情况下得到一般结果。所研究的统计技术包括来自似然推断的经典程序，如似然比和沃尔德检验以及信息标准。现代科学研究常常需要对几个共同观测变量的数据进行分析。不同变量之间依赖关系的统计模型通常使用不可观察（或隐藏）的附加变量来制定。隐变量模型的一个共同特征是，由于缺乏使其不规则的平滑特性，它们的统计特性不能完全理解。这是本项目发展理论和方法的主要动机，这些理论和方法对诸如确定统计模型中未观察变量的数量和类型等问题有影响。这类问题尤其出现在社会科学的应用中，如智力等关键概念不能直接观察到，以及计算生物学中使用隐藏变量的应用中，例如，当使用现代物种的DNA来验证涉及灭绝物种的进化理论时。更广泛地说，这项工作与任何研究有关，无论是医学研究还是其他研究，其中不能排除存在有影响的未观察到的变量。

NSF CAREER proposal DMS-0746265This project is concerned with statistical models whose parameter spaces have singularities. The investigator studies how singularities impact the behavior of existing statistical methods and develops new techniques for adequate assessment of statistical significance. The focus is on algebraic statistical models, that is, models that have (semi-)algebraic sets as parameter spaces. The class of algebraic models comprises many of the singular models employed in practice and can be studied using tools from computational algebraic geometry. Importantly, the well-behaved local geometry of semi-algebraic sets makes it possible to obtain general results without having to assume difficult to verify regularity conditions. The statistical techniques under study include classical procedures from likelihood inference such as likelihood ratio and Wald tests as well as information criteria.Modern scientific studies often require analysis of data on several jointly observed variables. Statistical models of dependence relationships among the different variables are often formulated using additional variables that are not observable (or hidden). A common feature of hidden variable models is that their statistical properties are not entirely understood because of a lack of smoothness properties that makes them irregular. This is the primary motivation for this project that develops theory and methods that have a bearing on problems such as determining the number and type of unobserved variables to be included in a statistical model. Such problems arise in particular in applications in the social sciences where key concepts such as intelligence are not directly observable, and in computational biology where hidden variables are employed, for example, when DNA of present-day species is used to validate evolutionary theories that involve extinct species. More broadly, the work is relevant for any study, medical or otherwise, in which the existence of influential unobserved variables cannot be excluded.