In problems such as variable selection and graph estimation, models are characterized by Boolean logical structure such as presence or absence of a variable or an edge. Consequently, false positive error or false negative error can be specified as the number of variables/edges that are incorrectly included or excluded in an estimated model. However, there are several other problems such as ranking, clustering, and causal inference in which the associated model classes do not admit transparent notions of false positive and false negative errors due to the lack of an underlying Boolean logical structure. In this paper, we present a generic approach to endow a collection of models with partial order structure, which leads to a hierarchical organization of model classes as well as natural analogs of false positive and false negative errors. We describe model selection procedures that provide false positive error control in our general setting and we illustrate their utility with numerical experiments.

翻译：在变量选择和图估计等问题中，模型通过布尔逻辑结构（如变量或边的存在与否）进行表征。因此，误报错误或漏报错误可定义为估计模型中错误包含或排除的变量/边的数量。然而，在排序、聚类和因果推断等其他问题中，由于缺乏底层布尔逻辑结构，相关模型类无法提供直观的误报和漏报错误概念。本文提出一种通用方法，通过赋予模型集合偏序结构，从而形成模型类的层次化组织以及误报和漏报错误的自然对应概念。我们描述了能在一般设定下控制误报错误的模型选择程序，并通过数值实验验证了其实用性。