In this paper, we generalize the Pearl and Neyman-Rubin methodologies in
causal inference by introducing a generalized approach that incorporates fuzzy
logic. Indeed, we introduce a fuzzy causal inference approach that consider
both the vagueness and imprecision inherent in data, as well as the subjective
human perspective characterized by fuzzy terms such as 'high', 'medium', and
'low'. To do so, we introduce two fuzzy causal effect formulas: the Fuzzy
Average Treatment Effect (FATE) and the Generalized Fuzzy Average Treatment
Effect (GFATE), together with their normalized versions: NFATE and NGFATE. When
dealing with a binary treatment variable, our fuzzy causal effect formulas
coincide with classical Average Treatment Effect (ATE) formula, that is a
well-established and popular metric in causal inference. In FATE, all values of
the treatment variable are considered equally important. In contrast, GFATE
takes into account the rarity and frequency of these values. We show that for
linear Structural Equation Models (SEMs), the normalized versions of our
formulas, NFATE and NGFATE, are equivalent to ATE. Further, we provide
identifiability criteria for these formulas and show their stability with
respect to minor variations in the fuzzy subsets and the probability
distributions involved. This ensures the robustness of our approach in handling
small perturbations in the data. Finally, we provide several experimental
examples to empirically validate and demonstrate the practical application of
our proposed fuzzy causal inference methods.

本文介绍了一种模糊因果推理方法，利用模糊逻辑综合了 Pearl 和 Neyman-Rubin 方法，考虑了数据的模糊性和不确定性以及模糊术语如 “高”、“中” 和 “低” 所表达的主观人类观点。通过引入两个模糊因果效应公式：Fuzzy Average Treatment Effect（FATE）和 Generalized Fuzzy Average Treatment Effect（GFATE），以及它们的归一化版本：NFATE 和 NGFATE，对于二元处理变量，这些模糊因果效应公式与经典的平均处理效应（ATE）公式相一致。同时，提供了这些公式的可识别性标准，并展示了它们对模糊子集和概率分布的微小变化的稳定性，确保了方法在处理数据中的小扰动时的鲁棒性。最后，通过多个实验示例验证和展示了所提出的模糊因果推理方法的实际应用。

将模糊逻辑与因果推断相结合：增强 Pearl 和 Neyman-Rubin 方法

Integrating Fuzzy Logic with Causal Inference: Enhancing the Pearl and  Neyman-Rubin Methodologies

Let $X$ and $Z$ be random vectors, and $Y=g(X,Z)$. In this paper, on the one
hand, for the case that $X$ and $Z$ are continuous, by using the ideas from the
total variation and the flux of $g$, we develop a point of view in causal
inference capable of dealing with a broad domain of causal problems. Indeed, we
focus on a function, called Probabilistic Easy Variational Causal Effect
(PEACE), which can measure the direct causal effect of $X$ on $Y$ with respect
to continuously and interventionally changing the values of $X$ while keeping
the value of $Z$ constant. PEACE is a function of $d\ge 0$, which is a degree
managing the strengths of probability density values $f(x|z)$. On the other
hand, we generalize the above idea for the discrete case and show its
compatibility with the continuous case. Further, we investigate some properties
of PEACE using measure theoretical concepts. Furthermore, we provide some
identifiability criteria and several examples showing the generic capability of
PEACE. We note that PEACE can deal with the causal problems for which
micro-level or just macro-level changes in the value of the input variables are
important. Finally, PEACE is stable under small changes in $\partial
g_{in}/\partial x$ and the joint distribution of $X$ and $Z$, where $g_{in}$ is
obtained from $g$ by removing all functional relationships defining $X$ and
$Z$.

本文研究了因果推断中连续和离散情况下的概率易变因果效应（PEACE），并探讨了 PEACE 的可识别性准则及其对输入变量值的微小变化的稳定性。