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，效应均发生的情形。我们推导了非免疫性和$epsilon$有界免疫的充要条件，其中非免疫性对应免疫概率为零，$epsilon$有界免疫对应免疫概率受$epsilon$限制。前者使我们能够从随机对照试验中估计获益概率（即当且仅当暴露时效应发生），后者则能产生比现有方法更紧的获益概率界限。我们还引入了间接免疫（即通过中介变量）的概念，并对其重复了前述分析。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。