# Uncertainty Quantification of MLE for Entity Ranking with Covariates

arXiv: https://arxiv.org/abs/2212.09961 · HTML: https://mlsanity.com/paper/2212.09961/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 46 references that are in the map.
- First posted on arXiv 20 Dec 2022.

## Builds on (most cited first, 16 of 16)

- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Spectral Methods for Data Science: A Statistical Perspective](https://mlsanity.com/paper/2012.08496/) (2020)
- [Simple, Robust and Optimal Ranking from Pairwise Comparisons](https://mlsanity.com/paper/1512.08949/) (2015)
- [Noisy Matrix Completion: Understanding Statistical Guarantees for Convex Relaxation via Nonconvex Optimization](https://mlsanity.com/paper/1902.07698/) (2019)
- [Spectral Method and Regularized MLE Are Both Optimal for Top-K Ranking](https://mlsanity.com/paper/1707.09971/) (2017)
- [Spectral MLE: Top-K Rank Aggregation from Pairwise Comparisons](https://mlsanity.com/paper/1504.07218/) (2015)
- [Stochastically Transitive Models for Pairwise Comparisons: Statistical and Computational Issues](https://mlsanity.com/paper/1510.05610/) (2015)
- [Uncertainty quantification in the Bradley-Terry-Luce model](https://mlsanity.com/paper/2110.03874/) (2021)
- [Minimax-optimal Inference from Partial Rankings](https://mlsanity.com/paper/1406.5638/) (2014)
- [Ranking Inferences Based on the Top Choice of Multiway Comparisons](https://mlsanity.com/paper/2211.11957/) (2022)
- [Lagrangian Inference for Ranking Problems](https://mlsanity.com/paper/2110.00151/) (2021)
- [Optimal Full Ranking from Pairwise Comparisons](https://mlsanity.com/paper/2101.08421/) (2021)
- [Rank Aggregation via Heterogeneous Thurstone Preference Models](https://mlsanity.com/paper/1912.01211/) (2019)
- [Top-K Ranking from Pairwise Comparisons: When Spectral Ranking is Optimal](https://mlsanity.com/paper/1603.04153/) (2016)
- [Competitive analysis of the top-K ranking problem](https://mlsanity.com/paper/1605.03933/) (2016)
- [Worst-case vs Average-case Design for Estimation from Fixed Pairwise Comparisons](https://mlsanity.com/paper/1707.06217/) (2017)

## Built on by (most cited first, 16 of 16)

- [Spectral Ranking Inferences based on General Multiway Comparisons](https://mlsanity.com/paper/2308.02918/) (2023)
- [Low-Rank Contextual Reinforcement Learning from Heterogeneous Human Feedback](https://mlsanity.com/paper/2412.19436/) (2024)
- [Covariate Assisted Entity Ranking with Sparse Intrinsic Scores](https://mlsanity.com/paper/2407.08814/) (2024)
- [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)
- [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)
- [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)
- [A Unified Perturbation Framework for Analyzing Leaderboard Stability and Manipulation](https://mlsanity.com/paper/2605.15761/) (2026)
- [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)
- [Random pairing MLE for estimation of item parameters in Rasch model](https://mlsanity.com/paper/2406.13989/) (2024)
- [Pairwise Comparisons without Stochastic Transitivity: Model, Theory and Applications](https://mlsanity.com/paper/2501.07437/) (2025)
- [Bradley-Terry Rankings for Recommender Systems Across Dataset Taxonomies](https://mlsanity.com/paper/2606.07492/) (2026)

## Abstract (arXiv)

This paper concerns with statistical estimation and inference for the ranking problems based on pairwise comparisons with additional covariate information such as the attributes of the compared items. Despite extensive studies, few prior literatures investigate this problem under the more realistic setting where covariate information exists. To tackle this issue, we propose a novel model, Covariate-Assisted Ranking Estimation (CARE) model, that extends the well-known Bradley-Terry-Luce (BTL) model, by incorporating the covariate information. Specifically, instead of assuming every compared item has a fixed latent score θi^*i=1n, we assume the underlying scores are given by αi^*+xi^topβ^*i=1n, where αi^* and xi^topβ^* represent latent baseline and covariate score of the i-th item, respectively. We impose natural identifiability conditions and derive the ℓ∞- and ℓ2-optimal rates for the maximum likelihood estimator of αi^*i=1n and β^* under a sparse comparison graph, using a novel `leave-one-out' technique (Chen et al., 2019) . To conduct statistical inferences, we further derive asymptotic distributions for the MLE of αi^*i=1n and β^* with minimal sample complexity. This allows us to answer the question whether some covariates have any explanation power for latent scores and to threshold some sparse parameters to improve the ranking performance. We improve the approximation method used in (Gao et al., 2021) for the BLT model and generalize it to the CARE model. Moreover, we validate our theoretical results through large-scale numerical studies and an application to the mutual fund stock holding dataset.