The Entropy and Crossentropy of Generalized Mallows Models

Abstract

The Generalized Mallows Model (GMM) is a well known family of models for ranking data. A GMM is a distribution over $\mathbb{S}_n$, the set of permutations of n objects, characterized by a location parameter $\sigma \in \mathbb{S}_n$, known as central permutation and a set of dispersion parameters $\theta_{1:n-1}\in(0,1]$. The GMM shares many properties, such as having sufficient statistics, with exponential models, thus it can be seen as an exponential family with a discrete parameter $\sigma$. This paper shows that computing entropy, crossentropy and Kullback-Leibler divergence in the the class of GMM is tractable, paving the way for a better understanding of this exponential family.