Bayesian analysis of wandering vector models for displaying ranking data

Abstract

In a process of examining k objects, each judge provides a ranking of them. The aim of this paper is to investigate a probabilistic model for ranking data - the wandering vector model. The model represents objects by points in a d-dimensional space, and the judges are represented by latent vectors emanating from the origin in the same space. Each judge samples a vector from a multivariate normal distribution; given this vector, the judge's utility assigned to an object is taken to be the length of the orthogonal projection of the object point onto the judge vector, plus a normally distributed random error. The ordering of the k utilities given by the judge determines the judge's ranking. A Bayesian approach and the Gibbs sampling technique are used for parameter estimation. The method of computing the marginal likelihood proposed by Chib (1995) is used to select the dimensionality of the model. Simulations are done to demonstrate the proposed estimation and model selection method. We then analyze the Goldberg data, in which 10 occupations are ranked according to the degree of social prestige.