# Uncertainty quantification in the Bradley-Terry-Luce model arXiv: https://arxiv.org/abs/2110.03874 · HTML: https://mlsanity.com/paper/2110.03874/ ## Key facts - Cited by 29 papers in the map (data through 14 Oct 2026). - Builds on 11 of its 47 references that are in the map. - First posted on arXiv 8 Oct 2021. ## Builds on (most cited first, 11 of 11) - [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015) - [A modern maximum-likelihood theory for high-dimensional logistic regression](https://mlsanity.com/paper/1803.06964/) (2018) - [Confidence Intervals and Hypothesis Testing for High-Dimensional Regression](https://mlsanity.com/paper/1306.3171/) (2013) - [Deep Neural Networks for Estimation and Inference](https://mlsanity.com/paper/1809.09953/) (2018) - [Spectral Method and Regularized MLE Are Both Optimal for Top-K Ranking](https://mlsanity.com/paper/1707.09971/) (2017) - [Inference and Uncertainty Quantification for Noisy Matrix Completion](https://mlsanity.com/paper/1906.04159/) (2019) - [Spectral MLE: Top-K Rank Aggregation from Pairwise Comparisons](https://mlsanity.com/paper/1504.07218/) (2015) - [A General Theory of Hypothesis Tests and Confidence Regions for Sparse High Dimensional Models](https://mlsanity.com/paper/1412.8765/) (2014) - [Optimal Full Ranking from Pairwise Comparisons](https://mlsanity.com/paper/2101.08421/) (2021) - [Asymptotic normality and optimalities in estimation of large Gaussian graphical models](https://mlsanity.com/paper/1309.6024/) (2013) - [A Unified Theory of Confidence Regions and Testing for High Dimensional Estimating Equations](https://mlsanity.com/paper/1510.08986/) (2015) ## Built on by (most cited first, 29 of 29) - [Ranking Inferences Based on the Top Choice of Multiway Comparisons](https://mlsanity.com/paper/2211.11957/) (2022) - [Uncertainty Quantification of MLE for Entity Ranking with Covariates](https://mlsanity.com/paper/2212.09961/) (2022) - [Spectral Ranking Inferences based on General Multiway Comparisons](https://mlsanity.com/paper/2308.02918/) (2023) - [Dropping Just a Handful of Preferences Can Change Top Large Language Model Rankings](https://mlsanity.com/paper/2508.11847/) (2025) - [Estimating Semantic Alphabet Size for LLM Uncertainty Quantification](https://mlsanity.com/paper/2509.14478/) (2025) - [Contextual Online Uncertainty-Aware Preference Learning for Human Feedback](https://mlsanity.com/paper/2504.19342/) (2025) - [A Judge-Aware Ranking Framework for Evaluating Large Language Models without Ground Truth](https://mlsanity.com/paper/2601.21817/) (2026) - [Ranking from Pairwise Comparisons in General Graphs and Graphs with Locality](https://mlsanity.com/paper/2304.06821/) (2023) - [Statistical ranking with dynamic covariates](https://mlsanity.com/paper/2406.16507/) (2024) - [Prompt-Dependent Ranking of Large Language Models with Uncertainty Quantification](https://mlsanity.com/paper/2603.03336/) (2026) - [Reinforcement Learning from Human Feedback: A Statistical Perspective](https://mlsanity.com/paper/2604.02507/) (2026) - [Statistical inference for pairwise comparison models](https://mlsanity.com/paper/2401.08463/) (2024) - [Top-K ranking with a monotone adversary](https://mlsanity.com/paper/2402.07445/) (2024) - [Minimax Hypothesis Testing for the Bradley-Terry-Luce Model](https://mlsanity.com/paper/2410.08360/) (2024) - [What Does Preference Learning Recover from Pairwise Comparison Data?](https://mlsanity.com/paper/2602.10286/) (2026) - [Improved theoretical guarantee for rank aggregation via spectral method](https://mlsanity.com/paper/2309.03808/) (2023) - [Transductive Conformal Inference for Full Ranking](https://mlsanity.com/paper/2501.11384/) (2025) - [A Unified Perturbation Framework for Analyzing Leaderboard Stability and Manipulation](https://mlsanity.com/paper/2605.15761/) (2026) - [Combinatorial Inference on the Optimal Assortment in Multinomial Logit Models](https://mlsanity.com/paper/2301.12254/) (2023) - [Inferences on Mixing Probabilities and Ranking in Mixed-Membership Models](https://mlsanity.com/paper/2308.14988/) (2023) - [Hypothesis Testing for Generalized Thurstone Models](https://mlsanity.com/paper/2512.02912/) (2025) - [Entrywise Error Bounds for Spectral Ranking with Semi-Random Adversaries](https://mlsanity.com/paper/2605.23854/) (2026) - [Convergence analysis of a family of Zermelo-type iterations for the Bradley--Terry model](https://mlsanity.com/paper/2607.22221/) (2026) - [Random pairing MLE for estimation of item parameters in Rasch model](https://mlsanity.com/paper/2406.13989/) (2024) - [Preference-Based Dynamic Ranking Structure Recognition](https://mlsanity.com/paper/2509.24493/) (2025) - [Privacy utility trade offs for parameter estimation in degree heterogeneous higher order networks](https://mlsanity.com/paper/2602.03948/) (2026) - [The Trust Paradox: How CS Researchers Engage LLM Leaderboards](https://mlsanity.com/paper/2605.28966/) (2026) - [Bradley-Terry Rankings for Recommender Systems Across Dataset Taxonomies](https://mlsanity.com/paper/2606.07492/) (2026) - [Bradley-Terry model under general comparison graphs](https://mlsanity.com/paper/2610.06237/) (2026) ## Abstract (arXiv) The Bradley-Terry-Luce (BTL) model is a benchmark model for pairwise comparisons between individuals. Despite recent progress on the first-order asymptotics of several popular procedures, the understanding of uncertainty quantification in the BTL model remains largely incomplete, especially when the underlying comparison graph is sparse. In this paper, we fill this gap by focusing on two estimators that have received much recent attention: the maximum likelihood estimator (MLE) and the spectral estimator. Using a unified proof strategy, we derive sharp and uniform non-asymptotic expansions for both estimators in the sparsest possible regime (up to some poly-logarithmic factors) of the underlying comparison graph. These expansions allow us to obtain: (i) finite-dimensional central limit theorems for both estimators; (ii) construction of confidence intervals for individual ranks; (iii) optimal constant of ℓ2 estimation, which is achieved by the MLE but not by the spectral estimator. Our proof is based on a self-consistent equation of the second-order remainder vector and a novel leave-two-out analysis.