Bayesian inference for bivariate ranks

TL;DR

This paper introduces a Bayesian approach to predict user rankings based on expert and user partial rankings, using copula models and MCMC algorithms, with applications demonstrated on movie rating data.

Contribution

It develops a Bayesian framework for bivariate ranking prediction using copulas, including exact and approximate inference methods, and applies it to real-world data.

Findings

Exact predictive distribution for Farlie-Gumbel-Morgenstern copula.

MCMC algorithms effectively approximate the predictive distribution.

Application to MovieLens data demonstrates practical utility.

Abstract

A recommender system based on ranks is proposed, where an expert's ranking of a set of objects and a user's ranking of a subset of those objects are combined to make a prediction of the user's ranking of all objects. The rankings are assumed to be induced by latent continuous variables corresponding to the grades assigned by the expert and the user to the objects. The dependence between the expert and user grades is modelled by a copula in some parametric family. Given a prior distribution on the copula parameter, the user's complete ranking is predicted by the mode of the posterior predictive distribution of the user's complete ranking conditional on the expert's complete and the user's incomplete rankings. Various Markov chain Monte-Carlo algorithms are proposed to approximate the predictive distribution or only its mode. The predictive distribution can be obtained exactly for the Farlie-Gumbel-Morgenstern copula family, providing a benchmark for the approximation accuracy of the algorithms. The method is applied to the MovieLens 100k dataset with a Gaussian copula modelling dependence between the expert's and user's grades.