Efficient estimation of finite mixtures of Mallows models with the Spearman distance

The class of Mallows models (MMs) occupy a central role in the literature for the analysis and learning of preferences from a sample of ranking data. The MMs rely on the distance notion over the set of permutations but, despite the wide range of possible metrics, the choice is typically limited to the Kendall or Cayley distances, due to the related analytical simplifications. We go beyond these conventional few options and explore the formal properties of the MM with the Spearman distance, also referred to as the theta-model. The attractive feature of this model is its correspondence with the restriction of the normal distribution over the permutation set such that, similarly to the Gaussian density, the theta model enjoys a convenient closed-form expression for the critical estimation of the modal ranking. This means that, differently from the MMs with the other metrics, an efficient and accurate inferential procedure can be developed, where the computational burden of inferring the discrete parameter is significantly reduced. Additionally, an efficient estimation within the finite mixture framework is realized via the EM algorithm, for enlarging the applicability of theta-models to samples of rankings characterized by a group structure. Finally, an application to a real-world dataset endorsing our proposals in the comparison with competing mixtures of ranking models is provided.