We study how the response of a Bayesian posterior statistic to future observations changes as information accumulates. For a non-decreasing function $T$, define $\Pi_n^T=\E[T(\Theta)\vert \mathcal F_n]$, where $\Theta$ has an arbitrary prior and the observations come from a one-parameter exponential family. Conditioning on the same current value of $\Pi^T$, 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 L\'evy processes.
We consider the distribution-constrained optimal stopping problem $\sup_{\tau\sim \mu} \mathbb E[B^*_\tau]$, where $\mu$ is a probability distribution on $\mathbb R_+$, and $(B^*_t)$ denotes the running maximum of a standard Brownian motion. This problem was introduced in Beiglbock et al. (PTRF, 2018), where a monotoni...
Priors in Bayesian inverse problems are often approximated through discretization, hyperparameter estimation, or generative modeling. Understanding how prior approximation errors propagate to the posterior and subsequent predictions is therefore important. In this work, we study the stability of the prior-to-posterior...
This work establishes minimax-rate guarantees for nonparametric ABC under random-series priors with simulable finite-dimensional coordinates and proves stability of the zero-count-augmented trigonometric population summaries.
This work introduces a variant of the Thompson Sampling algorithm that uses a fractional or $\alpha$-posterior instead of the standard posterior, and identifies general regularity conditions on the prior and reward distributions that enable a regret analysis of $\alpha$-TS without assuming any tractable approximation o...
Prateek Jaiswal, D. Pati, A. Bhattacharya et al.· 0 citations
The results show that learning under an appropriate prior overcomes the curse of dimensionality with respect to the dependence on $n$ and introduces the class of sparse priors and defines the "Sparse Dimension" as a measure of sparsity of a prior over the space of all distributions.
E-values provide a flexible framework for statistical inference, but the universal threshold $1/\alpha$ can be conservative when the null distribution has additional structure. We study how structural restrictions limit the worst-case concentration underlying Markov's inequality, using a nondecreasing density model to...
Jia-Yi Shen, Hou-Zhen Liu, Qi-Run Zeng et al.· 0 citations
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