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.

翻译：在贝叶斯框架下进行分位数估计与回归颇具挑战性，因为似然函数与先验分布的选择并不直观。本文基于最新提出的鞅后验（MP）框架，引入了一种新颖的贝叶斯非参数方法用于分位数估计与回归。鞅后验的核心思想在于：后验采样等价于预测性插补，这使得我们能够突破严格的似然-先验规范限制。我们证明：在鞅条件下，对平滑分位数函数进行递归估计，足以实现完全非参数贝叶斯推断。由此产生的后验分布被命名为分位数鞅后验（QMP），它源于隐式生成预测分布。与QMP相伴的是一种便捷、无需马尔可夫链蒙特卡洛且可并行化的后验计算方案，该方案还可通过基于高斯过程的渐近近似进一步加速。此外，由于与贝叶斯自助法的联系，分位数估计中众所周知的单调性问题可通过递增重排自然缓解。最后，QMP具有特别易处理的形式，便于开展全面的理论研究，这也是本文的重点。我们在模拟实验和真实数据实验中展示了后验计算的便捷性。