The clustered Mallows model

Rankings represent preferences that arise from situations where assessors arrange items, for example, in decreasing order of utility. Orderings of the item set are permutations (pi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi $$\end{document}) that reflect strict preferences. However, strict preference relations can be unrealistic for real data. Common traits among items can justify equal ranks and there can also be different importance attribution to decisions that form pi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi $$\end{document}. In large item sets, assessors might prioritise certain items, rank others low, and express indifference towards the remaining. Rank aggregation may involve decisive judgments in some parts and ambiguity in others. In this paper, we extend the famous Mallows (Biometrika 44:114-130, 1957) model (MM) to accommodate item indifference. Grouping similar items motivates the proposed Clustered Mallows Model (CMM), a MM counterpart for tied ranks with ties learned from the data. The CMM provides the flexibility to combine strictness and indifferences, describing rank collections as ordered clusters. CMM Bayesian inference is a doubly-intractable problem since the normalised model is unavailable. We overcome this with a version of the exchange algorithm (Murray et al. in Proceedings of the 22nd annual conference on uncertainty in artificial intelligence (UAI-06), 2006) and provide a pseudo-likelihood approximation as a computationally cheaper alternative. Analysis of two real-world ranking datasets is presented, showcasing the practical application of the CMM and highlighting scenarios where it offers advantages over alternative models.