CAREER: Statistical Inference in Algebraic Models with Singularities

NSF CAREER proposal DMS-0746265这个项目关注的是参数空间具有奇异性的统计模型。 研究人员研究奇异性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，即模型，有（半）代数集作为参数空间。 代数模型的类包括许多奇异模型在实践中使用，可以使用计算代数几何的工具进行研究。 重要的是，半代数集的良好的局部几何使得有可能获得一般的结果，而不必假设难以验证的正则性条件。 研究中的统计技术包括似然比、Wald检验等经典的似然推断方法和信息准则。现代科学研究往往需要对多个联合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，由于缺乏平滑特性，使得它们不规则，因此它们的统计特性并不完全被理解。 这是该项目的主要动机，该项目开发的理论和方法对确定统计模型中未观察变量的数量和类型等问题具有影响。 这些问题特别出现在社会科学的应用中，其中智能等关键概念无法直接观察到，而在计算生物学中，隐藏变量被使用，例如，当现代物种的DNA被用来验证涉及灭绝物种的进化理论时。 更广泛地说，这项工作是相关的任何研究，医学或其他方面，其中存在的有影响力的未观察到的变量不能排除。

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.