AI 中文总结
研究提出EM-INLA算法用于经验贝叶斯分层分位数回归，结合EM与INLA，利用ALD混合表示重述M步为加权高斯回归，避免MCMC采样，可扩展至大型复杂数据集，模拟及实际应用中展现出与HMC相当的准确性和显著加速。
AI 中文摘要
我们提出了EM-INLA算法，这是一种用于经验贝叶斯分层分位数回归的可扩展算法，它将期望最大化（EM）算法与集成嵌套拉普拉斯近似（INLA）相结合。该方法利用非对称拉普拉斯分布（ALD）的正态-指数混合表示，将每个M步重新表述为INLA处理的加权高斯回归，在每次迭代中以封闭形式更新ALD尺度和随机效应方差。尺度和方差超参数通过EM算法的边际最大似然估计，回归和随机效应参数的后验边际通过最终的INLA调用获得，条件是收敛的超参数估计。结果是一种避免所有MCMC采样的算法，同时可扩展到对标准全贝叶斯方法来说难以处理的大型数据集和复杂分层结构。在四个误差场景的模拟研究中，EM-INLA恢复真实参数的准确性与哈密顿蒙特卡罗（HMC）相当，同时根据场景实现了15倍到50倍以上的加速。该方法应用于哥伦比亚2023年的Prueba Saber 11标准化考试，在学校和市嵌套学生的分层结构下，将学生成绩的条件分位数建模为社会经济协变量的函数，有超过400,000个观测值。
英文摘要
We propose EM-INLA, a scalable algorithm for empirical-Bayes hierarchical quantile regression that combines the Expectation-Maximization (EM) algorithm with Integrated Nested Laplace Approximations (INLA). The method exploits the normal-exponential mixture representation of the Asymmetric Laplace Distribution (ALD) to reformulate each M-step as a weighted Gaussian regression handled by INLA, with the ALD scale and random-effect variances updated in closed form at each iteration. The scale and variance hyperparameters are estimated by marginal maximum likelihood via the EM algorithm, and posterior marginals for the regression and random-effect parameters are obtained from a final INLA call conditional on the converged hyperparameter estimates. The result is an algorithm that avoids all MCMC sampling while scaling to large datasets and complex hierarchical structures that are intractable for standard fully Bayesian approaches. In a simulation study across four error scenarios, EM-INLA recovers the true parameters with accuracy comparable to Hamiltonian Monte Carlo (HMC) while achieving speedups ranging from 15x to over 50x depending on the scenario. The method is applied to the 2023 Prueba Saber 11 standardized test in Colombia to model the conditional quantiles of student scores as a function of socioeconomic covariates under a hierarchical structure of students nested within schools and municipalities, with over 400,000 observations.
Comments28 pages, 5 figures, 9 tables
