In Bayesian quantile regression, the most commonly used likelihood is the asymmetric Laplace (AL) likelihood. The reason for this choice is not that it is a plausible data-generating model but that the corresponding maximum likelihood estimator is identical to the classical estimator by Koenker and Bassett (1978), and in that sense, the AL likelihood can be thought of as a working likelihood. AL-based quantile regression has been shown to produce good finite-sample Bayesian point estimates and to be consistent. However, if the AL distribution does not correspond to the data-generating distribution, credible intervals based on posterior standard deviations can have poor coverage. Yang, Wang, and He (2016) proposed an adjustment to the posterior covariance matrix that produces asymptotically valid intervals. However, we show that this adjustment is sensitive to the choice of scale parameter for the AL likelihood and can lead to poor coverage when the sample size is small to moderate. We therefore propose using Infinitesimal Jackknife (IJ) standard errors (Giordano & Broderick, 2023). These standard errors do not require resampling but can be obtained from a single MCMC run. We also propose a version of IJ standard errors for clustered data. Simulations and applications to real data show that the IJ standard errors have good frequentist properties, both for independent and clustered data. We provide an R-package that computes IJ standard errors for clustered or independent data after estimation with the brms wrapper in R for Stan.

翻译：在贝叶斯分位数回归中，最常用的似然函数是非对称拉普拉斯（AL）似然。选择该似然并非因其是合理的数据生成模型，而是因为其对应的最大似然估计量与 Koenker 和 Bassett（1978）提出的经典估计量完全一致。从这个意义上说，AL 似然可被视为一种工作似然。基于 AL 的分位数回归已被证明能产生良好的有限样本贝叶斯点估计，并具有一致性。然而，若 AL 分布与真实数据生成分布不符，基于后验标准差构造的置信区间可能产生较差的覆盖率。Yang、Wang 和 He（2016）提出了一种对后验协方差矩阵的调整方法，可产生渐近有效的置信区间。但我们发现，该调整对 AL 似然尺度参数的选择非常敏感，在中小样本量下可能导致较差的覆盖率。为此，我们建议采用无穷折刀（IJ）标准误（Giordano & Broderick, 2023）。这类标准误无需重复抽样，仅通过单次 MCMC 运行即可获得。我们还提出了适用于聚类数据的 IJ 标准误版本。模拟实验和实际数据应用表明，无论对于独立数据还是聚类数据，IJ 标准误均具有良好的频率性质。我们开发了一个 R 软件包，可在通过 R 中的 brms 封装接口调用 Stan 完成估计后，为聚类或独立数据计算 IJ 标准误。