Ranking items based on pairwise comparisons is common, from using match outcomes to rank sports teams to using purchase or survey data to rank consumer products. Statistical inference-based methods such as the Bradley-Terry model, which extract rankings based on an underlying generative model, have emerged as flexible and powerful tools to tackle ranking in empirical data. In situations with limited and/or noisy comparisons, it is often challenging to confidently distinguish the performance of different items based on the evidence available in the data. However, most inference-based ranking methods choose to assign each item to a unique rank or score, suggesting a meaningful distinction when there is none. Here, we develop a principled nonparametric Bayesian method, adaptable to any statistical ranking method, for learning partial rankings (rankings with ties) that distinguishes among the ranks of different items only when there is sufficient evidence available in the data. We develop a fast agglomerative algorithm to perform Maximum A Posteriori (MAP) inference of partial rankings under our framework and examine the performance of our method on a variety of real and synthetic network datasets, finding that it frequently gives a more parsimonious summary of the data than traditional ranking, particularly when observations are sparse.

翻译：基于成对比较对项目进行排序是常见的做法，从利用比赛结果对运动队进行排名，到利用购买或调查数据对消费品进行排序。基于统计推断的方法，例如Bradley-Terry模型，它基于一个潜在的生成模型来提取排序，已成为处理经验数据中排序问题的灵活而强大的工具。在比较数据有限和/或存在噪声的情况下，通常难以根据数据中可用的证据来有把握地区分不同项目的性能。然而，大多数基于推断的排序方法选择为每个项目分配一个唯一的排名或分数，这在没有实质性区别时暗示了有意义的区分。本文中，我们开发了一种原则性的非参数贝叶斯方法，可适应于任何统计排序方法，用于学习部分排序（允许并列的排序），该方法仅在数据中存在足够证据时才区分不同项目的排名。我们开发了一种快速的凝聚算法，以在我们的框架下执行部分排序的最大后验概率推断，并在各种真实和合成的网络数据集上检验了我们方法的性能，发现与传统排序方法相比，它通常能给出更简约的数据摘要，尤其是在观测数据稀疏的情况下。