A Bayesian nonparametric approach for generalized Bradley-Terry models in random environment

TL;DR

This paper introduces a Bayesian nonparametric method for estimating hidden variable distributions in generalized Bradley-Terry models within random environments, enabling better predictions of outcomes like championships from limited pairwise comparison data.

Contribution

It establishes contraction rates for the posterior, proposes a MCMC sampling algorithm, and evaluates its performance in predicting championship outcomes.

Findings

Posterior contraction rates are established.

A MCMC algorithm for sampling from the posterior is developed.

The algorithm's predictions are validated against actual team outcomes.

Abstract

This paper deals with the estimation of the unknown distribution of hidden random variables from the observation of pairwise comparisons between these variables. This problem is inspired by recent developments on Bradley-Terry models in random environment since this framework happens to be relevant to predict for instance the issue of a championship from the observation of a few contests per team. This paper provides three contributions on a Bayesian nonparametric approach to solve this problem. First, we establish contraction rates of the posterior distribution. We also propose a Markov Chain Monte Carlo algorithm to approximately sample from this posterior distribution inspired from a recent Bayesian nonparametric method for hidden Markov models. Finally, the performance of this algorithm are appreciated by comparing predictions on the issue of a championship based on the actual values of the teams and those obtained by sampling from the estimated posterior distribution.