Convex order and preservation of convexity for Bayesian posterior updates
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 ΠnT=E[T(Θ)| ℱn], where Θ has an arbitrary prior and the observations come from a one-parameter exponential family. Conditioning on the same current value of ΠT, 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.
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BibTeX
@misc{bayraktar2026convex,
title = {{Convex order and preservation of convexity for Bayesian posterior updates}},
author = {Erhan Bayraktar and Yuqiong Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05065},
eprint = {2609.05065},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05065},
}
RIS
TY - UNPB
TI - Convex order and preservation of convexity for Bayesian posterior updates
AU - Erhan Bayraktar
AU - Yuqiong Wang
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05065
UR - https://arxiv.org/abs/2609.05065
AB - 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)| 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.
ER -