发表机构
University of Michigan（密歇根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对单参数指数族观测的贝叶斯后验统计量，证明其满足凸序与凸性保持性质，可用于推导动态贝叶斯决策等问题的时间单调性结果，还能扩展到含无限可分分布的连续时间模型。
AI 中文摘要
我们研究随着信息积累，贝叶斯后验统计量对未来观测的响应如何变化。对于非递减函数$T$，定义$\u03a0_n^T = \mathbb{E}[T(\Theta) \mid \mathcal{F}_n]$，其中$\Theta$具有任意先验，观测来自单参数指数族。在条件为当前$\Pi^T$值相同时，我们证明当当前后验基于更少观测时，额外观测后的后验统计量在凸序意义上更大。我们还证明了凸性保持性质：未来后验统计量的凸函数的期望关于当前后验统计量是凸的。这两个性质共同为动态贝叶斯决策和最优停止问题中的时间单调性结果提供了结构工具。若指数族包含无限可分分布，结果可通过一族Lévy过程扩展到连续时间观测模型。
英文摘要
We study how the response of a Bayesian posterior statistic to future observations changes as information accumulates. For a non-decreasing function $T$, define $Π_n^T=\E[T(Θ)\vert \mathcal F_n]$, where $Θ$ has an arbitrary prior and the observations come from a one-parameter exponential family. Conditioning on the same current value of $Π^T$, we show that the posterior statistic after additional observations is larger in convex order when the current posterior is based on fewer observations. We also prove preservation of convexity: the expected value of a convex function of the future posterior statistic is convex in the current posterior statistic. Together, these two properties provide structural tools for establishing time-monotonicity results in dynamic Bayesian decision and optimal stopping problems. If the exponential family contains an infinitely divisible distribution, the results extend to a continuous-time observation model through a family of Lévy processes.
