基于martingale后验的贝叶斯分位数估计与回归
Bayesian Quantile Estimation and Regression with Martingale Posteriors
AI总结：
本文提出基于martingale后验的非参数贝叶斯方法，用于分位数估计和回归，通过递归估计和渐近近似实现高效后验计算。
摘要：
在贝叶斯框架下进行分位数估计和回归具有挑战性，因为选择似然和先验并不明显。本文介绍了一种新的非参数贝叶斯方法，用于分位数估计和回归，基于最近引入的martingale后验（MP）框架。MP的核心思想是后验抽样等同于预测填补，这使我们能够摆脱严格的似然-先验规范。我们证明了对光滑分位数函数的递归估计，受martingale条件约束，完全足以进行非参数贝叶斯推断。我们将所得到的后验分布称为分位数martingale后验（QMP），它源于隐含的生成预测分布。与QMP相关的是一个便捷的、无需MCMC且可并行计算的后验计算方案，该方案可以进一步通过基于高斯过程的渐近近似来加速。此外，由于与贝叶斯bootstrap的联系，分位数估计中众所周知的单调性问题通过递增重排自然得到缓解。最后，QMP具有特别可处理的形式，允许全面的理论研究，这构成了本工作的重点。我们通过模拟和真实数据实验展示了后验计算的简便性。
英文摘要：
Quantile estimation and regression within the Bayesian framework is challenging as the choice of likelihood and prior is not obvious. In this paper, we introduce a novel Bayesian nonparametric method for quantile estimation and regression based on the recently introduced martingale posterior (MP) framework. The core idea of the MP is that posterior sampling is equivalent to predictive imputation, which allows us to break free of the stringent likelihood-prior specification. We demonstrate that a recursive estimate of a smooth quantile function, subject to a martingale condition, is entirely sufficient for full nonparametric Bayesian inference. We term the resulting posterior distribution as the quantile martingale posterior (QMP), which arises from an implicit generative predictive distribution. Associated with the QMP is an expedient, MCMC-free and parallelizable posterior computation scheme, which can be further accelerated with an asymptotic approximation based on a Gaussian process. Furthermore, the well-known issue of monotonicity in quantile estimation is naturally alleviated through increasing rearrangement due to the connections to the Bayesian bootstrap. Finally, the QMP has a particularly tractable form that allows for comprehensive theoretical study, which forms a main focus of the work. We demonstrate the ease of posterior computation in simulations and real data experiments.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
