Bayesian inference for partial orders from random linear extensions: power relations from 12th Century Royal Acta

In the eleventh and twelfth centuries in England, Wales and Normandy, Royal Acta were legal documents in which witnesses were listed in order of social status. Any bishops present were listed as a group. For our purposes, each witness-list is an ordered permutation of bishop names with a known date or date-range. Changes over time in the order bishops are listed may reflect changes in their authority. Historians would like to detect and quantify these changes. There is no reason to assume that the underlying social order which constrains bishop-order within lists is a complete order. We therefore model the evolving social order as an evolving partial ordered set or {\it poset}. We construct a Hidden Markov Model for these data. The hidden state is an evolving poset (the evolving social hierarchy) and the emitted data are random total orders (dated lists) respecting the poset present at the time the order was observed. This generalises existing models for rank-order data such as Mallows and Plackett-Luce. We account for noise via a random ``queue-jumping'' process. Our latent-variable prior for the random process of posets is marginally consistent. A parameter controls poset depth and actor-covariates inform the position of actors in the hierarchy. We fit the model, estimate posets and find evidence for changes in status over time. We interpret our results in terms of court politics. Simpler models, based on Bucket Orders and vertex-series-parallel orders, are rejected. We compare our results with a time-series extension of the Plackett-Luce model. Our software is publicly available.