Bayesian nonparametric Plackett–Luce models for the analysis of preferences for college degree programmes

Abstract

In this paper we propose a Bayesian nonparametric model for clustering partial ranking data. We start by developing a Bayesian nonparametric extension of the popular Plackett-Luce choice model that can handle an infinite number of choice items. Our framework is based on the theory of random atomic measures, with the prior specified by a completely random measure. We characterise the posterior distribution given data, and derive a simple and effective Gibbs sampler for posterior simulation. We then develop a Dirichlet process mixture extension of our model and apply it to investigate the clustering of preferences for college degree programmes amongst Irish secondary school graduates. The existence of clusters of applicants who have similar preferences for degree programmes is established and we determine that subject matter and geographical location of the third level institution characterise these clusters.

Exchangeable Random Measures for Sparse and Modular Graphs with Overlapping Communities

“…where a k > 0, b k > 0, k = 1, : : : , p. ρ 0 is set to be the mean measure of the jump part of a generalized gamma process (Hougaard, 1986;Brix, 1999), which has been extensively used in Bayesian non-parametric models because of its generality, the interpretability of its parameters and its Graph sampled from the model with three latent communities, ( , , ): for each node, the intensity of each colour is proportional to the value of the associated weight in that community; pure colours indicate that the node is only strongly affiliated with a single community; a mixture of those colours indicates balanced affiliations with different communities; the graph was generated with the software Gephi (Bastian et al, 2009) attractive conjugacy properties (James, 2002;Lijoi et al, 2007;Saeedi and Bouchard-Côté, 2011;Caron, 2012;Caron et al, 2014). The Lévy measure in this case is…”

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).…”

“…Regarding the parameter estimation of the Plackett-Luce model and ROL model, a number of efficient algorithms have been proposed in the literature. They include (a) Minorization-Maximization (MM) algorithm (Hunter & Lange, 2004); (b) Expectation-Maximization (EM) algorithm (Caron & Doucet, 2012;Caron & Teh, 2012); (c) Bayesian Inference via Markov Chain Monte Carlo (MCMC; Guiver & Snelson, 2009;Caron & Doucet, 2012;Caron & Teh, 2012;Caron, Teh, & Murphy, 2014, for Plackett-Luce model only); and (d) Stationary distribution of Markov Chain (Maystre & Grossglauser, 2015, for Plackett-Luce model only). Here, we will briefly introduce these estimation methods.…”

Adjusting background noise in cluster analyses of longitudinal data

“…To estimate the number of clusters automatically, sampling strategies such as the reversible jump Markov Chain Monte Carlo (MCMC) method [9] and the birth-death process [10] have been proposed. Dirichlet process (DP) [11], introduced as the prior distribution for the coefficients evaluating the strength of associations, has been used more often recently to estimate the number of clusters and infer the coefficients [12, 13, 14, 15]. …”