Classical statistical methods have theoretical justification when the sample size is predetermined. In applications, however, it's often the case that sample sizes aren't predetermined; instead, they're often data-dependent. Since those methods designed for static sample sizes aren't reliable when sample sizes are dynamic, there's been recent interest in e-processes and 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. The present paper offer a new, regularized e-process framework that features a knowledge-based, imprecise-probabilistic regularization with 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, the proposed regularized e-processes facilitate possibility-theoretic uncertainty quantification with strong frequentist-like calibration properties and other desirable 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，而是依赖于数据本身。由于针对静态样本量设计的方法在动态样本量场景下并不可靠，近期学界对e-过程及其对应的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，其统计依据适用于任意动态数据收集方案。但若研究者对目标参数具备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此类信息。本文提出一种新型正则化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，我们证明了基于正则化e-过程的推断在新型知识依赖意义上仍保持任意时间有效性。此外，所提出的正则化e-过程支持可能性理论的不确定性量化，兼具强频率学派校准特性与其他贝叶斯式优良性质：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。