Ji 2024 - Valid Standard Errors for Bayesian Quantile Regression - Overview
Summary
Bayesian quantile regression almost always uses the asymmetric Laplace (AL) distribution as a working likelihood because its maximiser equals the Koenker-Bassett estimator. Point estimates are consistent, but posterior standard deviations (as in brms and Stata
bayes:qreg) are wrong: they scale with an arbitrary AL scale parameter and give poor frequentist coverage. Ji, Lee and Rabe-Hesketh propose infinitesimal-jackknife (IJ) standard errors computed from one MCMC run, including a clustered version.
Overview
Question. When is a credible interval from AL-based Bayesian quantile regression trustworthy, and how can it be corrected cheaply?
Contribution. (1) Show the Yang-Wang-He (2016) sandwich adjustment is sensitive to the AL scale at small/moderate ; (2) adapt Giordano-Broderick infinitesimal-jackknife SEs to quantile regression, and to cluster data; (3) R package IJSE for brms/Stan.
Key result. IJ SEs have relative error near 0 across and 90% coverage near nominal for (plain posterior SDs can be up to 300% off); at extreme quantiles with large the point estimator is biased (skewed posterior), so even IJ SEs undercover.
Motivating PyMC examples
E83 “Bayesian quantile regression, and when it lies” (20,000 US singleton births, NCHS 2023, plus known-truth simulation): check loss and Frisch-Newton LP solver, the quantile process, AL working likelihood (mode right, spread wrong, units trap), coverage simulation, Yang-Wang-He sandwich, Gibbs posterior with a bootstrap-calibrated learning rate — at https://redam94.github.io/pymc-challenges/examples.html. The Yang-Wang-He (2016) paper itself is paywalled; this open paper reviews and tests its adjustment (concept summary: Ji 2024 - Yang-Wang-He Sandwich Adjustment). The Gibbs-posterior idea is in Ji 2024 - Gibbs Posterior and Learning-Rate Calibration (concept note, no raw source).
Main Content — Source Map
| Paper section | Covered in |
|---|---|
| Abstract, 1 Introduction | this note; Ji 2024 - Asymmetric Laplace Working Likelihood |
| 2.1 The AL likelihood (check loss, scale mixture, priors on ) | Ji 2024 - Asymmetric Laplace Working Likelihood |
| 2.2 Standard error adjustment (Yang et al.; Sriram) | Ji 2024 - Yang-Wang-He Sandwich Adjustment |
| 3 Infinitesimal jackknife SEs, 3.1 clustered data | Ji 2024 - Infinitesimal Jackknife Standard Errors |
| 4 Evaluation, independent data (Models 1-2; Figs. 1-4) | Ji 2024 - Simulation Results and Applications |
| 5 Evaluation, clustered data (Figs. 5-6) | Ji 2024 - Simulation Results and Applications |
| 6 Applications (Engel food expenditure; Project STAR) | Ji 2024 - Simulation Results and Applications |
| 7 Concluding remarks | Ji 2024 - Simulation Results and Applications, Ji 2024 - Gibbs Posterior and Learning-Rate Calibration |
Relevance to marketing measurement
Effects on the upper tail of customer value or lag times (the 90th percentile of time-to-convert, the 10th percentile of lift) are often reported with quantile regression. Treat default Bayesian credible intervals as unreliable unless calibrated; use bootstrap/IJ/sandwich SEs, especially with clustered markets or stores (geo experiments). See Geo-Experiment Methodology - Overview.
Connections
Quantile Regression (frequentist), Conformalized Quantile Regression (distribution-free intervals for quantile-based predictors), Stasinopoulos 2007 - GAMLSS in R - Overview (parametric alternative to estimate all quantiles coherently).