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 and false negative errors can be specified as the number of variables or edges that are incorrectly included/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，模型通常通过布尔逻辑结构（如变量或边的存在与否）来刻画。因此，假阳性和假阴性错误可以定义为估计模型中错误包含/排除的变量或边的数量。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，相关的模型类别无法获得明确的假阳性和假阴性错误概念。本文提出了一种通用方法，旨在为模型集合赋予偏序结构，从而形成模型类别的层次化组织以及假阳性和假阴性错误的自然类比。我们描述了在一般设置下能够控制假阳性错误的模型选择程序，并通过数值实验展示了其有效性。