In this work, the goal is to detect closely-linked entities within a data set. The entities of interest have a tie causing them to be similar, such as a shared origin or a channel of influence. Given a collection of people or other entities with their attributes or behavior, we identify unusually similar pairs, and we pose the question: Are these two people linked, or can their similarity be explained by chance? Computing similarities is a core operation in many domains, but two constraints differentiate our version of the problem. First, the score assigned to a pair should account for the probability of a coincidental match. Second, no training data is provided; we must learn about the system from the unlabeled data and make reasonable assumptions about the linked pairs. This problem has applications to social network analysis, where it can be valuable to identify implicit relationships among people from indicators of coordinated activity. It also arises in situations where we must decide whether two similar observations correspond to two different entities or to the same entity observed twice. This dissertation explores how to assess such ties and, in particular, how the similarity scores should depend on not only the two entities in question but also properties of the entire data set. We develop scoring functions that incorporate both the similarity and rarity of a pair. Then, using these functions, we investigate the statistical power of a data set to reveal (or conceal) such pairs. In the dissertation, we develop generative models of linked pairs and independent entities and use them to derive scoring functions for pairs in three different domains: people with job histories, Gaussian-distributed points in Euclidean space, and people (or entities) in a bipartite affiliation graph. For the first, we present a case study in fraud detection that highlights the potential, as well as the complexities, of using these methods to address real-world problems. In the latter two domains, we develop an inference framework to estimate whether two entities were more likely generated independently or as a pair. In these settings, we analyze how the scoring function works in terms of similarity and rarity; how well it can detect pairs as a function of the data set; and how it differs from existing similarity functions when applied to real data.