Item Preference Parameters from Grouped Ranking Observations

Given a set of rating data for a set of items, determining the values of items is a matter of importance and various probability models have been proposed. The Plackett-Luce model is one of such models, which parametrizes the value of each item by a real valued preference parameter. In this paper, the Plackett-Luce model is generalized to cope with the grouped ranking observations such as movies or restaurants ratings. Since the maximization of the likelihood of the proposed model is computationally intractable, the lower bound of the likelihood which is easy to evaluate is derived. The em algorithm is adopted to find the item preference parameter by maximizing the lower bound. Numerical experiments on synthetic and real-world data are carried out to confirm the appropriateness of the proposed model and algorithm.