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Uncertainty Quantification of MLE for Entity Ranking with Covariates

Jianqing Fan, Jikai Hou, Mengxin Yu

TL;DR

A novel model, Covariate-Assisted Ranking Estimation (CARE) model, that extends the well-known Bradley-Terry-Luce model by incorporating the covariate information, and improves the approximation method used in (Gao et al., 2021) for the BLT model and generalize it to the CARE model.

Abstract

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 $\{θ_i^*\}_{i=1}^n$, we assume the underlying scores are given by $\{α_i^*+{x}_i^\topβ^*\}_{i=1}^n$, where $α_i^*$ and ${x}_i^\topβ^*$ represent latent baseline and covariate score of the $i$-th item, respectively. We impose natural identifiability conditions and derive the $\ell_{\infty}$- and $\ell_2$-optimal rates for the maximum likelihood estimator of $\{α_i^*\}_{i=1}^{n}$ 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 $\{α_i^*\}_{i=1}^n$ 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.

Uncertainty Quantification of MLE for Entity Ranking with Covariates

TL;DR

A novel model, Covariate-Assisted Ranking Estimation (CARE) model, that extends the well-known Bradley-Terry-Luce model by incorporating the covariate information, and improves the approximation method used in (Gao et al., 2021) for the BLT model and generalize it to the CARE model.

Abstract

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 , we assume the underlying scores are given by , where and represent latent baseline and covariate score of the -th item, respectively. We impose natural identifiability conditions and derive the - and -optimal rates for the maximum likelihood estimator of and 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 and 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.
Paper Structure (47 sections, 33 theorems, 310 equations, 3 figures, 2 tables, 2 algorithms)

This paper contains 47 sections, 33 theorems, 310 equations, 3 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Prior Arts
  3. Notation
  4. Roadmap
  5. Model Formulation
  6. Rate of Convergence of Maximum Likelihood Estimator
  7. Uncertainty Quantification of MLE
  8. Numerical Results
  9. Rate of Convergence
  10. Distributional Results
  11. Comparison with BTL Model Without Covariates
  12. Application to Pokemon Challenge Dataset
  13. Conclusion and Discussion
  14. Proof Outline of Estimation Results
  15. Preliminaries and Basic Results
  16. ...and 32 more sections

Key Result

Proposition 1

CARE model Eq. CARE with parameter space $\Theta = \{ (\boldsymbol{\alpha},\boldsymbol{\beta}):\bar{\boldsymbol{X}}^\top\boldsymbol{\alpha} = \boldsymbol{0} \}$ is identifiable.

Figures (3)

  • Figure 1: Statistical rates of $\Vert \widehat{\boldsymbol{\alpha}}_M-\boldsymbol{\alpha}^*\Vert_\infty$ and $\Vert \widehat{\boldsymbol{\beta}}_M-\boldsymbol{\beta}^*\Vert_2/\Vert \boldsymbol{\beta}^*\Vert_2$ for three simulated instances (realization of simulated models). The solid red lines and light areas represent the averaged $\Vert \widehat{\boldsymbol{\alpha}}_M-\boldsymbol{\alpha}^*\Vert_\infty$, $\Vert \widehat{\boldsymbol{\beta}}_M-\boldsymbol{\beta}^*\Vert_2/\Vert \boldsymbol{\beta}^*\Vert_2$ and their associated standard errors based on 200 Monte Carlo simulations.
  • Figure 2: Q-Q Plots for checking the normality of $(\widehat{\boldsymbol{\alpha}}_M)_1$ based on 250 simulations.
  • Figure 3: Histograms of standardized random variables $A$ and $B$ in \ref{['AB']} along with the density of the standardized Gaussian random variable. The first row is based on $(p,L) = (1.25/n_a, 2)$ and the second row is based on $(p,L) = (2/n_a, 20)$.

Theorems & Definitions (38)

  • Proposition 1
  • Theorem 4
  • Remark 5
  • Remark 6
  • Theorem 7
  • Remark 8
  • Remark 9
  • Theorem 10
  • Corollary 11
  • Corollary 12
  • ...and 28 more