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Generalized Bradley-Terry Models for Score Estimation from Paired Comparisons

Julien Fageot, Sadegh Farhadkhani, Lê Nguyên Hoang, Oscar Villemaud

TL;DR

A probabilistic model encompassing a broad variety of paired comparisons that can take discrete or continuous values, which is called the family of generalized Bradley-Terry (GBT) models, as it includes the classical Bradley-Terry model and many of its variants.

Abstract

Many applications, e.g. in content recommendation, sports, or recruitment, leverage the comparisons of alternatives to score those alternatives. The classical Bradley-Terry model and its variants have been widely used to do so. The historical model considers binary comparisons (victory or defeat) between alternatives, while more recent developments allow finer comparisons to be taken into account. In this article, we introduce a probabilistic model encompassing a broad variety of paired comparisons that can take discrete or continuous values. We do so by considering a well-behaved subset of the exponential family, which we call the family of generalized Bradley-Terry (GBT) models, as it includes the classical Bradley-Terry model and many of its variants. Remarkably, we prove that all GBT models are guaranteed to yield a strictly convex negative log-likelihood. Moreover, assuming a Gaussian prior on alternatives' scores, we prove that the maximum a posteriori (MAP) of GBT models, whose existence, uniqueness and fast computation are thus guaranteed, varies monotonically with respect to comparisons (the more A beats B, the better the score of A) and is Lipschitz-resilient with respect to each new comparison (a single new comparison can only have a bounded effect on all the estimated scores). These desirable properties make GBT models appealing for practical use. We illustrate some features of GBT models on simulations.

Generalized Bradley-Terry Models for Score Estimation from Paired Comparisons

TL;DR

A probabilistic model encompassing a broad variety of paired comparisons that can take discrete or continuous values, which is called the family of generalized Bradley-Terry (GBT) models, as it includes the classical Bradley-Terry model and many of its variants.

Abstract

Many applications, e.g. in content recommendation, sports, or recruitment, leverage the comparisons of alternatives to score those alternatives. The classical Bradley-Terry model and its variants have been widely used to do so. The historical model considers binary comparisons (victory or defeat) between alternatives, while more recent developments allow finer comparisons to be taken into account. In this article, we introduce a probabilistic model encompassing a broad variety of paired comparisons that can take discrete or continuous values. We do so by considering a well-behaved subset of the exponential family, which we call the family of generalized Bradley-Terry (GBT) models, as it includes the classical Bradley-Terry model and many of its variants. Remarkably, we prove that all GBT models are guaranteed to yield a strictly convex negative log-likelihood. Moreover, assuming a Gaussian prior on alternatives' scores, we prove that the maximum a posteriori (MAP) of GBT models, whose existence, uniqueness and fast computation are thus guaranteed, varies monotonically with respect to comparisons (the more A beats B, the better the score of A) and is Lipschitz-resilient with respect to each new comparison (a single new comparison can only have a bounded effect on all the estimated scores). These desirable properties make GBT models appealing for practical use. We illustrate some features of GBT models on simulations.
Paper Structure (47 sections, 20 theorems, 103 equations, 1 figure, 1 table)

This paper contains 47 sections, 20 theorems, 103 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Works
  4. Outline
  5. Generalized Bradley-Terry Models
  6. The Setting
  7. GBT Models as an Exponential Subfamily
  8. Cumulant-Generating Functions
  9. Discrete and Continuous GBT Models
  10. Discrete GBT models.
  11. Continuous GBT models.
  12. Historical Bradley-Terry as Bernoulli-GBT
  13. MAP Scores Based on GBT Models
  14. Bayesian Model for Score Estimation
  15. Maximum A Posteriori Estimator
  16. ...and 32 more sections

Key Result

Proposition 1

Under the $f$-GBT model, the comparisons $r_{ab}$ are independent conditionally to $\Theta \in\mathbb{R}^A$ and $r_{ab}| \Theta = r_{ab} |\theta_{ab}$. Moreover, we have, for any $(\mathrm{R},\Theta)$

Figures (1)

  • Figure 1: Left: Normalized mean-square error with respect to the sparsity parameter $p_c$ for Erdös-Rényi comparison graphs. Middle: Normalized mean-square errors $\textsc{NormError}_K$ using $K$-nary-GBT MAP estimators on the data generated via the Uniform-GBT model. Blue: $K \mapsto \textsc{NormError}_K$; Red: $\textsc{NormError}$ for the Uniform-GBT model. Right: Normalized mean-square error with respect to the regularization scale $\frac{1}{\sigma^2}$.

Theorems & Definitions (43)

  • Definition 1
  • Proposition 1
  • Definition 2
  • Theorem 1: dembo2009large
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Definition 3
  • Definition 4
  • ...and 33 more