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 $Π_n^{T=\E[T(Θ)\vert} \mathcal F_n]$, where $Θ$ has an arbitrary prior and the observations come from a one-parameter exponential family. Conditioning on the same current value of $Π<sup>T$, 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.