An Exponential Model for Infinite Rankings

This paper presents a statistical model for expressing preferences through rankings, when the number of alternatives (items to rank) is large. A human ranker will then typically rank only the most preferred items, and may not even examine the whole set of items, or know how many they are. Similarly, a user presented with the ranked output of a search engine, will only consider the highest ranked items. We model such situations by introducing a stagewise ranking model that operates with finite ordered lists called top-t orderings over an infinite space of items. We give algorithms to estimate this model from data, and demonstrate that it has sufficient statistics, being thus an exponential family model with continuous and discrete parameters. We describe its conjugate prior and other statistical properties. Then, we extend the estimation problem to multimodal data by introducing an Exponential-Blurring-Mean-Shift nonparametric clustering algorithm. The experiments highlight the properties of our model and demonstrate that infinite models over permutations can be simple, elegant and practical.