Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Authors: Bayraktar, Erhan, Wang, Yuqiong
Institution: University of Michigan
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W7211871477