有限时域分位数鞅后验：原始瓮定律与矩阵增益回归
Finite-horizon quantile martingale posteriors: raw-urn laws and matrix-gain regression
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中文总结 AI 辅助
研究有限时域分位数鞅后验，推导其停止状态定律，分析方差因子及增益校准，得出共享瓮创新的联合定律，还表明特定条件分位数回归的平滑鞅后验满足相关定理，标量或对角增益无法匹配协方差过程。
中文摘要 AI 辅助
鞅后验通过从一步预测分布向前推算观测值来量化不确定性，但在有限次推算后停止。本文推导了分位数的经验波利亚瓮后验停止状态的定律。停止瓮测度的分位数保持熟悉的鞅尾和方差分数；而具有固定增益\(c\)的随机近似跟踪器则不然。其方差带有明确因子\(G_a\)，其中\(a = cf_0(q_\tau)\)，可能低于或超过尾分数，密度适应增益通过无密度膨胀恢复校准。共享瓮创新产生有限多个分位数水平的联合定律。对于条件分位数回归，从普通分位数回归估计器开始、具有全逆雅可比矩阵增益的平滑鞅后验满足具有校准有限时域带的过程伯恩斯坦 - 冯·米塞斯定理；标量或对角增益无法匹配三明治协方差过程。
英文摘要
Martingale posteriors quantify uncertainty by forward-imputing observations from one-step-ahead predictive distributions, but implementations stop after finitely many imputations. For the empirical Pólya-urn posterior of a quantile the law of the stopped state is derived. The quantile of the stopped urn measure keeps the familiar martingale tail-sum variance fraction; the deployed stochastic-approximation tracker with frozen gain $c$ does not. Its variance carries an explicit factor $G_a$ with $a=cf_0(q_τ)$, which may fall below or exceed the tail fraction, and a density-adapted gain restores calibration through a density-free inflation. Shared urn innovations yield the joint law of finitely many quantile levels. For conditional quantile regression, a smoothed martingale posterior started at the ordinary quantile-regression estimator with a full inverse-Jacobian matrix gain satisfies a process Bernstein--von Mises theorem with calibrated finite-horizon bands; scalar or diagonal gains cannot match the sandwich covariance process.
