In this paper, we introduce a new causal framework capable of dealing with probabilistic and non-probabilistic problems. Indeed, we provide a direct causal effect formula called Probabilistic vAriational Causal Effect (PACE) and its variations satisfying some ideas and postulates. Our formula of causal effect uses the idea of the total variation of a function integrated with probability theory. The probabilistic part is the natural availability of changing an exposure values given some variables. These variables interfere with the effect of the exposure on a given outcome. PACE has a parameter $d$ determining the degree of considering the natural availability of changing the exposure values. The lower values of $d$ refer to the scenarios for which rare cases are important. In contrast, with the higher values of $d$, our framework deals with the problems that are in nature probabilistic. Hence, instead of a single value for causal effect, we provide a causal effect vector by discretizing $d$. Further, we introduce the positive and negative PACE to measure the positive and the negative causal changes in the outcome while changing the exposure values. Furthermore, we provide an identifiability criterion for PACE to deal with observational studies. We also address the problem of computing counterfactuals in causal reasoning. We compare our framework to the Pearl, the mutual information, the conditional mutual information, and the Janzing et al. frameworks by investigating several examples.

翻译：本文提出了一种新的因果框架，能够处理概率性与非概率性问题。具体而言，我们给出了一种称为概率变分因果效应（Probabilistic vAriational Causal Effect, PACE）的直接因果效应公式及其满足某些思想与公设的变体。该因果效应公式融合了函数全变分思想与概率论。其概率部分表现为在给定某些变量条件下改变暴露值的自然可行性，这些变量会干扰暴露对特定结果的影响。PACE含有一个参数$d$，用于确定考虑改变暴露值自然可行性的程度。较低的$d$值对应罕见案例重要性的场景，而较高的$d$值则使本框架处理本质上具有概率性的问题。因此，我们通过离散化$d$值，以因果效应向量的形式替代单一因果效应值。此外，我们引入了正PACE与负PACE，用于度量改变暴露值时结果的正向与负向因果变化。进一步地，我们给出了PACE在观察性研究中的可识别性准则，并解决了因果推理中反事实计算问题。通过多个案例比较，我们将本框架与Pearl框架、互信息框架、条件互信息框架以及Janzing等人框架进行了对比分析。