Abstract

We analyze a popular exponential model over rankings called the generalized Mallows model. Estimating the central ranking (or consensus ranking) of this model from data is NP-hard. We obtain the following new results: (1) We show that a standard search method can estimate both the central ranking π0 and the model parameters θ exactly. The search is n! in the worst case, but is tractable when the true distribution is concentrated around its mode. (2) From a statistical point of view, we show that the generalized Mallows model is jointly exponential in (π0, θ), and introduce the conjugate prior for this model class. (3) The sufficient statistics are the pairwise marginal probabilities that item i is prefered to item j. These probabilities are of interest in various applications. This paper provides the first exact tractable algorithm for their evaluation. Preliminary experiments confirm the theoretical predictions and compare the new algorithm and existing heuristics.