Non-parametric Bayesian inference for partial orders with ties from rank data observed with Mallows noise

A social hierarchy is a set of power relations between social actors. These relations form a partial order, a transitively-closed directed acyclic graph, with a vertex for each actor and directed edges indicating power relations. Partial orders are more general than total orders (simple rankings) because pairs of actors in a partial order may have no order relation. A partial order can be estimated using sampled linear extensions. These are lists of actors which respect the partial order, with higher status actors first: actors ordered in the partial order must appear in the same order in each list; unordered actors swap randomly across lists. Examples of list-data include records of queues (higher status individuals at the front) and lists of length two recording outcomes of agonistic social interactions. This paper develops non-parametric Bayesian inference for partial orders. Tied actors are clustered using a Poisson-Dirichlet process, the partial order dimension is modeled and estimated using reversible-jump MCMC, and errors in list-data are modeled using a Mallow’s observation model. The new framework is used to estimate social hierarchies for humans and chimpanzees. Statistical tests select partial-order-based models over well-known alternatives which fit total orders.