Bayesian Plackett–Luce Mixture Models for Partially Ranked Data

Abstract

The elicitation of an ordinal judgment on multiple alternatives is often required in many psychological and behavioral experiments to investigate preference/choice orientation of a specific population. The Plackett-Luce model is one of the most popular and frequently applied parametric distributions to analyze rankings of a finite set of items. The present work introduces a Bayesian finite mixture of Plackett-Luce models to account for unobserved sample heterogeneity of partially ranked data. We describe an efficient way to incorporate the latent group structure in the data augmentation approach and the derivation of existing maximum likelihood procedures as special instances of the proposed Bayesian method. Inference can be conducted with the combination of the Expectation-Maximization algorithm for maximum a posteriori estimation and the Gibbs sampling iterative procedure. We additionally investigate several Bayesian criteria for selecting the optimal mixture configuration and describe diagnostic tools for assessing the fitness of ranking distributions conditionally and unconditionally on the number of ranked items. The utility of the novel Bayesian parametric Plackett-Luce mixture for characterizing sample heterogeneity is illustrated with several applications to simulated and real preference ranked data. We compare our method with the frequentist approach and a Bayesian nonparametric mixture model both assuming the Plackett-Luce model as a mixture component. Our analysis on real datasets reveals the importance of an accurate diagnostic check for an appropriate in-depth understanding of the heterogenous nature of the partial ranking data.

Analysis of ranking data

“…They used the marginal probability for an item to be relevant as a measure of the importance of the item, and proposed a Bayesian approach to estimating the probabilities for all the items. Finally, the aggregated ranking is determined by ordering the probabilities for all the items. Unlike the supervised rank aggregation methods, the weighted rank aggregation method proposed by Desarkar et al () is an unsupervised method for rank aggregation by assigning different weights to different rankers according to their ranking qualities measured in terms of their own agreements with “majority” of rankers. Motivated from the fact that rankings can be transformed into pairwise preferences, Volkovs and Zemel () proposed the multinomial preference model (MPM) for unsupervised aggregation, a new score‐based model for pairwise preferences and extended MPM for supervised aggregation. Rank aggregation methods for heterogeneous ranking data include EM algorithm for mixtures of (weighted) distance‐based models (Lee & Yu, ; Murphy & Martin, ), Bayesian inference for Mallows Mixture model (Meilă & Chen, ; Vitelli et al, ), Bayesian inference for Mixtures of Plackett–Luce model (Caron et al, ; Mollica & Tardella, ).…”

“…7. Rank aggregation methods for heterogeneous ranking data include EM algorithm for mixtures of (weighted) distancebased models (Lee & Yu, 2012;Murphy & Martin, 2003), Bayesian inference for Mallows Mixture model (Meil a & Chen, 2010;Vitelli et al, 2018), Bayesian inference for Mixtures of Plackett-Luce model (Caron et al, 2014;Mollica & Tardella, 2017).…”

PLMIX: anRpackage for modelling and clustering partially ranked data

“…To account for the latent group structure z, Mollica and Tardella (2017) generalize Caron and Doucet (2012)'s approach with the following conjugate Bayesian model setup…”

“…, see Mollica and Tardella (2017) for more analytical details. The MAP solution represents a suitable starting point to initialize the GS algorithm.…”

Revealing Subgroup Structure in Ranked Data Using a Bayesian WAND

“…The basic premise is that the observed data should look like a plausible realisation from the (posterior) predictive distribution. Posterior predictive checks for models for ranked data have received relatively little attention in the literature, although Mollica and Tardella (2016) provide some guidance. Here, rather than focusing on particular test quantities, we aim to directly assess the full predictive distribution as follows.…”