This work is devoted to the study of the probability of immunity, i.e. the effect occurs whether exposed or not. We derive necessary and sufficient conditions for non-immunity and $\epsilon$-bounded immunity, i.e. the probability of immunity is zero and $\epsilon$-bounded, respectively. The former allows us to estimate the probability of benefit (i.e., the effect occurs if and only if exposed) from a randomized controlled trial, and the latter allows us to produce bounds of the probability of benefit that are tighter than the existing ones. We also introduce the concept of indirect immunity (i.e., through a mediator) and repeat our previous analysis for it. Finally, we propose a method for sensitivity analysis of the probability of immunity under unmeasured confounding.

翻译：本文致力于研究免疫概率，即无论是否暴露于干预因素，效应均发生的情况。我们推导了非免疫性和ε-有界免疫性的充要条件，分别对应免疫概率为零和受ε约束的情形。前者使我们能够从随机对照试验中估算获益概率（即效应仅在暴露时发生），后者则能产生比现有方法更严格的获益概率边界。我们进一步引入间接免疫（即通过中介变量产生免疫）的概念，并对其重复上述分析。最后，我们提出了一种在未测量混杂因素下对免疫概率进行敏感性分析的方法。