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We propose a novel combinatorial inference framework to conduct general uncertainty quantification in ranking problems. We consider the widely adopted Bradley-Terry-Luce (BTL) model, where each item is assigned a positive preference score…

Machine Learning · Statistics 2021-10-04 Yue Liu , Ethan X. Fang , Junwei Lu

The Bradley-Terry model is widely used for the analysis of pairwise comparison data and, in essence, produces a ranking of the items under comparison. We embed the Bradley-Terry model within a stochastic block model, allowing items to…

Methodology · Statistics 2025-11-06 Lapo Santi , Nial Friel

The question of aggregating pair-wise comparisons to obtain a global ranking over a collection of objects has been of interest for a very long time: be it ranking of online gamers (e.g. MSR's TrueSkill system) and chess players, aggregating…

Machine Learning · Computer Science 2015-11-13 Sahand Negahban , Sewoong Oh , Devavrat Shah

Inspired by applications in sports where the skill of players or teams competing against each other varies over time, we propose a probabilistic model of pairwise-comparison outcomes that can capture a wide range of time dynamics. We…

Machine Learning · Statistics 2019-05-20 Lucas Maystre , Victor Kristof , Matthias Grossglauser

We describe the application of the Bradley-Terry model to NCAA Division I Men's Ice Hockey. A Bayesian construction gives a joint posterior probability distribution for the log-strength parameters, given a set of game results and a choice…

Applications · Statistics 2023-03-21 John T. Whelan , Adam Wodon

A number of applications (e.g., AI bot tournaments, sports, peer grading, crowdsourcing) use pairwise comparison data and the Bradley-Terry-Luce (BTL) model to evaluate a given collection of items (e.g., bots, teams, students, search…

Machine Learning · Computer Science 2019-06-12 Jingyan Wang , Nihar B. Shah , R. Ravi

Comparative Judgement is an assessment method where item ratings are estimated based on rankings of subsets of the items. These rankings are typically pairwise, with ratings taken to be the estimated parameters from fitting a Bradley-Terry…

Methodology · Statistics 2024-05-22 Ian Hamilton , Nick Tawn

With the advent of highly capable instruction-tuned neural language models, benchmarking in natural language processing (NLP) is increasingly shifting towards pairwise comparison leaderboards, such as LMSYS Arena, from traditional global…

Computation and Language · Computer Science 2025-09-24 Georgii Levtsov , Dmitry Ustalov

Frequently in sporting competitions it is desirable to compare teams based on records of varying schedule strength. Methods have been developed for sports where the result outcomes are win, draw, or loss. In this paper those ideas are…

Applications · Statistics 2021-12-22 Ian Hamilton , David Firth

A principled approach to cyclicality and intransitivity in paired comparison data is developed. The proposed methodology enables more precise estimation of the underlying preference profile and facilitates the identification of all cyclic…

Methodology · Statistics 2025-10-08 Rahul Singh , Ori Davidov

In Generalised Bayesian Inference (GBI), the learning rate and hyperparameters of the loss must be estimated. These inference-hyperparameters can't be estimated jointly with the other parameters, from the data, by giving them a prior.…

Methodology · Statistics 2026-05-18 Jeong Eun Lee , Sitong Liu , Geoff K. Nicholls

Doubly intractable problems occur when both the likelihood and the posterior are available only in unnormalised form, with computationally intractable normalisation constants. Bayesian inference then typically requires direct approximation…

We propose a time-varying generalization of the Bradley-Terry model that allows for nonparametric modeling of dynamic global rankings of distinct teams. We develop a novel estimator that relies on kernel smoothing to pre-process the…

Statistics Theory · Mathematics 2020-03-03 Heejong Bong , Wanshan Li , Shamindra Shrotriya , Alessandro Rinaldo

We revisit the problem of inferring the overall ranking among entities in the framework of Bradley-Terry-Luce (BTL) model, based on available empirical data on pairwise preferences. By a simple transformation, we can cast the problem as…

Machine Learning · Computer Science 2016-05-10 Vivek S. Borkar , Nikhil Karamchandani , Sharad Mirani

The Bradley-Terry-Luce (BTL) model is a popular statistical approach for estimating the global ranking of a collection of items using pairwise comparisons. To ensure accurate ranking, it is essential to obtain precise estimates of the model…

Statistics Theory · Mathematics 2022-06-24 Wanshan Li , Shamindra Shrotriya , Alessandro Rinaldo

Optimization is widely used in statistics, and often efficiently delivers point estimates on useful spaces involving structural constraints or combinatorial structure. To quantify uncertainty, Gibbs posterior exponentiates the negative loss…

Methodology · Statistics 2025-07-23 Cheng Zeng , Eleni Dilma , Jason Xu , Leo L Duan

The Bradley-Terry-Luce (BTL) model is one of the most widely used models for ranking a collection of items or agents based on pairwise comparisons among them. Given $n$ agents, the BTL model endows each agent $i$ with a latent skill score…

Machine Learning · Computer Science 2025-12-03 Anuran Makur , Japneet Singh

For nonbalanced paired comparisons, a wide variety of ranking methods have been proposed. One of the best popular methods is the Bradley-Terry model in which the ranking of a set of objects is decided by the maximum likelihood estimates…

Methodology · Statistics 2016-11-07 Ting Yan

We provide an algorithm with constant running time that given a weighted tournament $T$, distinguishes with high probability of success between the cases that $T$ can be represented by a Bradley--Terry model, or cannot even be approximated…

Combinatorics · Mathematics 2016-09-19 Agelos Georgakopoulos , Konstantinos Tyros

Bayesian Likelihood-Free Inference (LFI) approaches allow to obtain posterior distributions for stochastic models with intractable likelihood, by relying on model simulations. In Approximate Bayesian Computation (ABC), a popular LFI method,…

Methodology · Statistics 2022-02-08 Lorenzo Pacchiardi , Ritabrata Dutta