[Paper Review] Probabilistic preference learning with the Mallows rank model

Valeria Vitelli, Øystein Sørensen+3 more|arXiv (Cornell University)|May 30, 2014

Game Theory and Voting Systems64 references18 citations

TL;DR

This paper proposes a computationally efficient Bayesian inference framework for the Mallows rank model using Markov Chain Monte Carlo (MCMC) methods, enabling probabilistic preference learning with any right-invariant distance. It supports full, partial, and pairwise rankings, quantifies uncertainty, and clusters heterogeneous assessors into subgroups with distinct consensus rankings, with theoretical guarantees that unranked items do not affect the consensus ranking.

ABSTRACT

Ranking and comparing items is crucial for collecting information about preferences in many areas, from marketing to politics. The Mallows rank model is among the most successful approaches to analyse rank data, but its computational complexity has limited its use to a particular form based on Kendall distance. We develop new computationally tractable methods for Bayesian inference in Mallows models that work with any right-invariant distance. Our method performs inference on the consensus ranking of the items, also when based on partial rankings, such as top-k items or pairwise comparisons. We prove that items that none of the assessors has ranked do not influence the maximum a posteriori consensus ranking, and can therefore be ignored. When assessors are many or heterogeneous, we propose a mixture model for clustering them in homogeneous subgroups, with cluster-specific consensus rankings. We develop approximate stochastic algorithms that allow a fully probabilistic analysis, leading to coherent quantifications of uncertainties. We make probabilistic predictions on the class membership of assessors based on their ranking of just some items, and predict missing individual preferences, as needed in recommendation systems. We test our approach using several experimental and benchmark datasets.

Motivation & Objective

• To develop computationally tractable Bayesian inference methods for the Mallows rank model beyond the standard Kendall distance.
• To enable probabilistic inference on consensus rankings from incomplete data, including top-k rankings and pairwise comparisons.
• To model heterogeneous assessors by clustering them into subgroups with cluster-specific consensus rankings.
• To quantify uncertainty in consensus rankings, class memberships, and missing preferences using posterior distributions.
• To prove that unranked items do not influence the maximum a posteriori consensus ranking, enabling data compression and efficiency.

Proposed method

• Uses a leap-and-shift Metropolis-Hastings MCMC algorithm to sample from the Mallows distribution with any right-invariant distance function.
• Employs a mixture model of Mallows distributions to cluster assessors into homogeneous subgroups, each with its own consensus ranking and dispersion parameter.
• Applies approximate stochastic algorithms to scale inference to large datasets while maintaining probabilistic coherence.
• Introduces a novel MCMC sampler that respects constraints from partial rankings and pairwise comparisons during proposal generation.
• Uses a Dirichlet prior over cluster proportions and jointly infers cluster assignments and rankings via Gibbs sampling.
• Implements thinning to store independent samples from the posterior, ensuring convergence and mixing in MCMC chains.

Experimental results

Research questions

• RQ1 Can Bayesian inference in the Mallows model be made computationally feasible for arbitrary right-invariant distances, not just Kendall distance?
• RQ2 How can uncertainty in consensus rankings be coherently quantified when data are incomplete or partial?
• RQ3 Can assessors be meaningfully clustered into subgroups with distinct preference patterns, even when they only rank subsets of items?
• RQ4 To what extent do unranked items influence the estimated consensus ranking in the posterior distribution?
• RQ5 Can the model predict missing preferences and classify new assessors based on partial ranking behavior?

Key findings

• The maximum a posteriori consensus ranking is invariant to items that are never ranked by any assessor, enabling efficient data pruning.
• The proposed MCMC algorithms achieve good mixing and convergence on both synthetic and real-world datasets, including benchmark preference learning tasks.
• The mixture model with cluster-specific consensus rankings successfully identifies homogeneous subgroups in heterogeneous preference data, improving predictive accuracy.
• The method enables full probabilistic prediction of missing preferences and assessors' cluster memberships, even from partial rankings or pairwise comparisons.
• The framework supports uncertainty quantification in consensus rankings and class memberships, which is critical for reliable decision-making in recommendation systems.
• Empirical evaluation on experimental and benchmark datasets confirms the method's robustness and scalability across different data types, including top-k and pairwise preference data.