Exponential Family Models for Rich Preference Ranking Data

Abstract

Preferences can be found in a wide array of contexts, from recommender systems, to opinion polls, consumer habits, and elections. The specific method of data collection, and the types of data collected can greatly vary the tools available for analysis. We seek to expand the class of exponential family ranking models by considering two types of more rich preference data. We first look at the Recursive Inversion Model, a highly flexible exponential ranking model that can reflect high level trends in ranking data with informative parameters for inference. We expand these models for partial rankings, rankings that more accurately reflect the true opinions of most individuals by allowing for non-strict orderings of preference. While this addition of partial rankings accounts for increased overhead in algorithmic and computational complexity of maximum likelihood estimation, we detail methods and algorithms that ensure tractability. We also utilize this same theory to provide algorithms for calculating conditional and marginal probabilities for the Recursive Inversion Model. Using this new theory, we demonstrate the usefulness of expression ratings and rankings, highlighting a novel method of data analysis for preference data expressed as ratings. We also expand on this further by proposing a new data structure, rankings with landmarks, which combine the relative and absolute preferences expressed in rankings and ratings into one. This new class of rankings requires the construction of new ranking models, of which the Landmark Generalized Mallows Model (L-GMMs) appears the most promising. We detail algorithms for maximum likelihood estimation of the L-GMMs, providing a solution to creating exponential ranking models containing non-invertible subsets, and demonstrate them on real world data.