CAREER: Statistical Inference in Algebraic Models with Singularities

NSF职业建议书DMS-0746265这个项目涉及参数空间具有奇异性的统计模型。研究人员研究奇点如何影响现有统计方法的行为，并开发新的技术来充分评估统计意义。重点放在代数统计模型上，即以(半)代数集作为参数空间的模型。这类代数模型包括许多实际中使用的奇异模型，可以使用计算代数几何中的工具进行研究。重要的是，半代数集的良好的局部几何使得获得一般结果成为可能，而不必假设困难的正则性条件。研究中的统计技术包括似然推断的经典过程，如似然比和Wald检验，以及信息标准。现代科学研究经常需要分析几个联合观测变量的数据。不同变量之间的依赖关系的统计模型通常使用不可观察(或隐藏)的附加变量来表示。隐藏变量模型的一个共同特征是，由于缺乏使它们变得不规则的平滑特性，它们的统计特性并不完全被理解。这是这个项目的主要动机，该项目开发了与诸如确定统计模型中要包括的未观测变量的数量和类型等问题有关的理论和方法。这样的问题尤其出现在社会科学的应用中，在这些应用中，智力等关键概念无法直接观察到，以及在计算生物学中使用隐藏变量，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，无论是医学上的还是其他方面的，在这些研究中，不能排除有影响的未观察到的变量的存在。

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.