Conditional independence and graphical models for rankings

Ranking models define probability distributions over rankings and are fundamental in applications such as preference learning and decision-making. Without structural assumptions, these models are intractable due to their factorial complexity. Although independence assumptions are key to reducing complexity and enabling interpretability, standard notions of conditional independence do not directly apply to rankings because they are subject to mutual exclusivity constraints. We introduce \emph{relative rank models}, a probabilistic framework based on structured coarsenings of the ranking space that relax these constraints while preserving relative order information. This representation enables a well-defined notion of conditional independence for rankings and allows rankings to be modeled using Bayesian networks. As a result, relative rank models support tractable learning, probabilistic inference, and interpretable representations. We further show that two prominent ranking models based on independence assumptions tailored to rankings arise as special cases, providing a unifying framework for independence in ranking models.