Consensus ranking under the exponential model

TL;DR

This paper studies the generalized Mallows model for ranking data, showing exact estimation methods for the central ranking and parameters, and introducing a conjugate prior, with experiments validating the approach.

Contribution

It provides new exact search algorithms for estimating the central ranking and parameters in the generalized Mallows model, and introduces a conjugate prior for this exponential model.

Findings

Exact estimation of central ranking and parameters is feasible when the distribution is concentrated.

The generalized Mallows model is jointly exponential in its parameters.

Preliminary experiments support the theoretical results and compare algorithms.

Abstract

We analyze the generalized Mallows model, a popular exponential model over rankings. Estimating the central (or consensus) ranking from data is NP-hard. We obtain the following new results: (1) We show that search methods can estimate both the central ranking pi0 and the model parameters theta exactly. The search is n! in the worst case, but is tractable when the true distribution is concentrated around its mode; (2) We show that the generalized Mallows model is jointly exponential in (pi0; theta), and introduce the conjugate prior for this model class; (3) The sufficient statistics are the pairwise marginal probabilities that item i is preferred to item j. Preliminary experiments confirm the theoretical predictions and compare the new algorithm and existing heuristics.