Non-Parametric Bayesian Inference for Partial Orders with Ties from Rank Data observed with Mallows Noise

Partial orders may be used for modeling and summarising ranking data when the underlying order relations are less strict than a total order. They are a natural choice when the data are lists recording individuals' positions in queues in which queue order is constrained by a social hierarchy, as it may be appropriate to model the social hierarchy as a partial order and the lists as random linear extensions respecting the partial order. In this paper, we set up a new prior model for partial orders incorporating ties by clustering tied actors using a Poisson Dirichlet process. The family of models is projective. We perform Bayesian inference with different choices of noisy observation model. In particular, we propose a Mallow's observation model for our partial orders and give a recursive likelihood evaluation algorithm. We demonstrate our model on the 'Royal Acta' (Bishop) list data where we find the model is favored over well-known alternatives which fit only total orders.