Classical statistical methods have theoretical justification when the sample size is predetermined by the data-collection plan. In applications, however, it's often the case that sample sizes aren't predetermined; instead, investigators might use the data observed along the way to make on-the-fly decisions about when to stop data collection. Since those methods designed for static sample sizes aren't reliable when sample sizes are dynamic, there's been a recent surge of interest in e-processes and the corresponding tests and confidence sets that are anytime valid in the sense that their justification holds up for arbitrary dynamic data-collection plans. But if the investigator has relevant-yet-incomplete prior information about the quantity of interest, then there's an opportunity for efficiency gain, but existing approaches can't accommodate this. Here I build a new, regularized e-process framework that features a knowledge-based, imprecise-probabilistic regularization that offers improved efficiency. A generalized version of Ville's inequality is established, ensuring that inference based on the regularized e-process remains anytime valid in a novel, knowledge-dependent sense. In addition to anytime valid hypothesis tests and confidence sets, the proposed regularized e-processes facilitate possibility-theoretic uncertainty quantification with strong frequentist-like calibration properties and other Bayesian-like features: satisfies the likelihood principle, avoids sure-loss, and offers formal decision-making with reliability guarantees.

翻译：经典统计方法在样本量由数据收集计划预先确定时具有理论依据。然而在实际应用中，样本量往往并非预先确定；研究者可能根据观测过程中的数据动态决定何时停止数据收集。由于为静态样本量设计的方法在动态样本量下不可靠，近年来学界对e过程及相应检验与置信集产生了浓厚兴趣——这些方法具有任意时间有效性，即其统计依据适用于任意的动态数据收集方案。但若研究者对目标参数具有相关却不完整的先验信息，则存在提升统计效率的空间，而现有方法无法利用这种信息。本文构建了一个新的正则化e过程框架，该框架采用基于知识的非精确概率正则化机制以实现效率提升。通过建立维勒不等式的广义形式，确保了基于正则化e过程的推断在新型的、知识依赖意义上保持任意时间有效性。除任意时间有效的假设检验与置信集外，所提出的正则化e过程还能支持可能性理论的不确定性量化，其具备类频率学派的强校准特性以及类贝叶斯学派的特征：满足似然原理、避免必然损失，并提供具有可靠性保证的形式化决策框架。