Model Selection over Partially Ordered Sets

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.