Bayesian estimation of Thurstonian ranking models based on the Gibbs sampler (original) (raw)

Bayesian analysis of order-statistics models for ranking data

Psychometrika, 2000

In this paper, a class of probability models for ranking data, the order-statistics models, is investigated. We extend the usual normal order-statistics model into one where the underlying random variables follow a multivariate normal distribution. Bayesian approach and the Gibbs sampling technique are used for parameter estimation. In addition, methods to assess the adequacy of model fit are introduced. Robustness of the model is studied by considering a multivariate-t distribution. The proposed method is applied to analyze the presidential election data of the American Psychological Association (APA).

Bayesian analysis of ranking data with the Extended Plackett–Luce model

Statistical Methods & Applications, 2020

Multistage ranking models, including the popular Plackett-Luce distribution (PL), rely on the assumption that the ranking process is performed sequentially, by assigning the positions from the top to the bottom one (forward order). A recent contribution to the ranking literature relaxed this assumption with the addition of the discrete-valued reference order parameter, yielding the novel Extended Plackett-Luce model (EPL). Inference on the EPL and its generalization into a finite mixture framework was originally addressed from the frequentist perspective. In this work, we propose the Bayesian estimation of the EPL in order to address more directly and efficiently the inference on the additional discrete-valued parameter and the assessment of its estimation uncertainty, possibly uncovering potential idiosyncratic drivers in the formation of preferences. We overcome initial difficulties in employing a standard Gibbs sampling strategy to approximate the posterior distribution of the EPL by combining the data augmentation procedure and the conjugacy of the Gamma prior distribution with a tuned joint Metropolis-Hastings algorithm within Gibbs. The effectiveness and usefulness of the proposal is illustrated with applications to simulated and real datasets.

Bayesian inference for Plackett-Luce ranking models

Proceedings of the 26th Annual International Conference on Machine Learning, 2009

This paper gives an efficient Bayesian method for inferring the parameters of a Plackett-Luce ranking model. Such models are parameterised distributions over rankings of a finite set of objects, and have typically been studied and applied within the psychometric, sociometric and econometric literature. The inference scheme is an application of Power EP (expectation propagation). The scheme is robust and can be readily applied to large scale data sets. The inference algorithm extends to variations of the basic Plackett-Luce model, including partial rankings. We show a number of advantages of the EP approach over the traditional maximum likelihood method. We apply the method to aggregate rankings of NASCAR racing drivers over the 2002 season, and also to rankings of movie genres.

Analysing multiattribute ranking data: Joint and conditional approaches

British Journal of Mathematical and Statistical Psychology, 1996

Rankings, in contrast to ratings, eliminate effects of individual differences in scale usage and avoid arbitrary definitions regarding the number of response categories and category labels. However, despite the appeal and popularity of this technique, few methods are available for the analysis of rankings on several attributes. This paper presents extensions of Thurstonian and logistic models for the joint analysis of multiattribute ranking responses and for conditional analyses where rankings on one of the attributes are modelled as a hnction of the rankings on the other attributes. These extensions are based on two approaches proposed to account for associations among the ranking responses. Empirical applications of the Thurstonian and logistic ranking models indicate that one of the two approaches appears particularly promising for the analysis of multiamibute ranking data.

Thurstonian modeling of ranking data via mean and covariance structure analysis.

Although Thurstonian models provide an attractive representation of choice behavior, they have not been extensively used in ranking applications since only recently efficient estimation methods for these models have been developed. These, however, require the use of special-purpose estimation programs, which limits their applicability. Here we introduce a formulation of Thurstonian ranking models that turns an idiosyncratic estimation problem into an estimation problem involving mean and covariance structures with dichotomous indicators. Well-known standard solutions for the latter can be readily applied to this specific problem, and as a result any Thurstonian model for ranking data can be fitted using existing general purpose software for mean and covariance structure analysis. Although the most popular programs for covariance structure analysis (e.g., LISREL and EQS) cannot be presently used to estimate Thurstonian ranking models, other programs such as Mplus already exist that can be straightforwardly used to estimate these models. See Maydeu-Olivares and Böckenholt (2005, Psychological Methods) for details.

New order-statistics-based ranking models and faster computation of outcome probabilities

Ima Journal of Management Mathematics, 2019

In sport, order-statistics-based models such as Henery's gamma model and the Thurstone-Mosteller type V model are useful in estimating competitor strengths from observed performance of players in competitions between 2 or more players. They can also be applied in many other areas, such as analysis of consumer preference data, which would be useful to marketing management. Two new families of such models derived from the exponentiated exponential and Pareto distributions are introduced. Use of order statistics-based models when there are more than 2 competitors has been hampered by lack of an efficient method of computation of outcome probabilities as a function of competitor strengths, and a fast method of computation of outcome probabilities is presented, that exploits the fact that the integral to be evaluated is an iterated integral.

Time-varying rankings with the Bayesian Mallows model

Stat, 2016

We present new statistical methodology for analysing rank data, where the rankings are allowed to vary in time. Such data arise, for example, when the assessments are based on a performance measure of the items, which varies in time, or if the criteria, according to which the items are ranked, change in time. Items can also be absent when the assessments are made, because of delayed entry or early departure, or purely randomly. In such situations, also the dimension of the rank vectors varies in time. Rank data in a time-dependent setting thus lead to challenging statistical problems. These problems are further complicated, from the perspective of computation, by the large dimension of the sample space consisting of all permutations of the items. Here, we focus on introducing and developing a Bayesian version of the Mallows rank model, suitable for situations in which the ranks vary in time and the assessments can be incomplete. The consequent missing data problems are handled by applying Bayesian data augmentation within Markov chain Monte Carlo. Our method is also adapted to the task of future rank prediction. The method is illustrated by analysing some aspects of a data set describing the academic performance, measured by a series of tests, of a class of high school students over a period of 4 years.

Probability models on rankings

Journal of Mathematical Psychology, 1991

This paper investigates many of the probability models on permutations that have been proposed in the statistical and psychological literature. The various models are categorized into the following general classes: (1) Thurstone order statistics models, (2) ranking models induced by paired comparisons, (3) ranking models based on distances between permutations, and (4) multistage ranking models. Several thematic properties of ranking models are introduced that provide the basis for a systematic study of each of the classes of models. These properties are label-invariance, reversibility, strong unimodality, complete consensus, and L-decomposability. Next, several important subclasses of the four general classes are explored, including a determination of the pairwise intersections of the different classes of models. The paper concludes with an illustration of many of the models on a set of ranked "word association" data.

Modelling Ranking Data with Wallenius Distribution

arXiv (Cornell University), 2017

The Wallenius distribution is a generalisation of the Hypergeometric distribution where weights are assigned to balls of different colours. This naturally defines a model for ranking categories which can be used for classification purposes. Since, in general, the resulting likelihood is not analytically available, we adopt an approximate Bayesian computational (ABC) approach for estimating the importance of the categories. We illustrate the performance of the estimation procedure on simulated datasets. Finally, we use the new model for analysing two datasets concerning movies ratings and Italian academic statisticians' journal preferences. The latter is a novel dataset collected by the authors.