Efficient and accurate inference for mixtures of Mallows models with Spearman distance

TL;DR

This paper introduces an efficient EM algorithm for mixtures of Mallows models with Spearman distance, enabling better clustering of ranking data, especially with large item sets and partial rankings.

Contribution

It develops a novel EM algorithm and a new approximation for the normalizing constant specific to Spearman distance in Mallows models, expanding their applicability.

Findings

The EM algorithm is efficient and accurate based on simulation studies.

The approximation supports clustering with many items effectively.

Application to real datasets demonstrates improved performance over existing models.

Abstract

The Mallows model occupies a central role in parametric modelling of ranking data to learn preferences of a population of judges. Despite the wide range of metrics for rankings that can be considered in the model specification, the choice is typically limited to the Kendall, Cayley or Hamming distances, due to the closed-form expression of the related model normalizing constant. This work instead focuses on the Mallows model with Spearman distance. An efficient and accurate EM algorithm for estimating finite mixtures of Mallows models with Spearman distance is developed, by relying on a twofold data augmentation strategy aimed at i) enlarging the applicability of Mallows models to samples drawn from heterogeneous populations; ii) dealing with partial rankings affected by diverse forms of censoring. Additionally, a novel approximation of the model normalizing constant is introduced to support the challenging model-based clustering of rankings with a large number of items. The inferential ability of the EM scheme and the effectiveness of the approximation are assessed by extensive simulation studies. Finally, we show that the application to three real-world datasets endorses our proposals also in the comparison with competing mixtures of ranking models.