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第925行：
第925行： − + − EI是一种度量因果机制中因果变量的因果联系强弱的一种指标。而在EI提出之前，已有多个因果度量指标被提出了。那么，EI和这些因果度量指标之间存在着什么样的联系呢？事实上，正如Comolatti与Hoel在2022年的文章中所指出的，包括EI在内的这些因果度量指标都可以统一表达为两个基本要素的组合<ref name=":0">Comolatti, R., & Hoel, E. (2022). Causal emergence is widespread across measures of causation. ''arXiv preprint arXiv:2202.01854''.</ref>。这两个基本要素被称为“因果元语”（Causal Primatives），分别代表了因果关系中的'''充分性'''和'''必要性'''。+ − + − + − + 第940行：
第940行： + − 其中<math>c</math>和<math>e</math>分别表示因事件（cause）和果事件（effect），<math>C</math>表示因事件的全部集合，<math>C \backslash c</math>则为因事件<math>c</math>的补集，即<math>c</math>之外的事件，也可记作<math>\lnot c</math>。'''充分性'''表明当因发生时，果发生的概率，当<math>suff = 1</math>时，因发生确定导致果发生；而'''必要性'''则衡量当因不发生时，果也不发生的概率；当<math>nec = 1</math>时，因不发生则果一定不发生。+ − − 有些因果指标中的必要性表现为以下的变型形式，在此也给出定义： − +
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===Distinction===
===Distinction===
It’s important to note that unlike EI calculations for Markov chains, the EI here measures the causal connections between two parts of the system, rather than the strength of causal connections across two different time points in the same system.
It’s important to note that unlike EI calculations for Markov chains, the EI here measures the causal connections between two parts of the system, rather than the strength of causal connections across two different time points in the same system.
==EI与其它因果度量指标==
==EI and Other Causal Metrics==
EI is a metric used to measure the strength of causal connections in a causal mechanism. Before the introduction of EI, several causal metrics had already been proposed. So, what is the relationship between EI and these causal measures? As Comolatti and Hoel pointed out in their 2022 paper, many causal metrics, including EI, can be expressed as combinations of two basic elements <ref name=":0">Comolatti, R., & Hoel, E. (2022). Causal emergence is widespread across measures of causation. ''arXiv preprint arXiv:2202.01854''.</ref>. These two basic elements are called "Causal Primitives", which represent '''Sufficiency'''和'''Necessity''' and in causal relationships.
===因果元语的定义===
===Definition of Causal Primitives===
<math>
<math>
\begin{aligned}
\begin{aligned}
\text{充分性：}~~~&suff(e,c) = P(e|c) \\
\text{Sufficiency：}~~~&suff(e,c) = P(e|c) \\
\text{必要性：}~~~&nec(e,c) = 1 - P(e|C \backslash c)
\text{Necessity：}~~~&nec(e,c) = 1 - P(e|C \backslash c)
\end{aligned}
\end{aligned}
</math>
</math>
In this context, <math>c</math> and <math>e</math> represent the cause and effect events, respectively. <math>C</math> denotes the complete set of cause events, and <math>C \backslash c</math> is the complement of the cause event <math>c</math>. Events other than <math>c</math> can also be written as <math>\lnot c</math>. '''Sufficiency''' indicates the probability that the effect occurs when the cause occurs, while '''Necessity''' measures the probability that the effect does not occur when the cause is absent. When <math>nec = 1</math>, the cause or absence of the cause determines the occurrence of the effect.
The necessity of some causal indicators manifests in the following variant forms, which are also defined here:
<math>
<math>
\begin{aligned}
\begin{aligned}
\text{必要性}'
\text{Necessity}'
\text{：}~~~nec'(e) = 1 - P(e|C)
\text{：}~~~nec'(e) = 1 - P(e|C)
\end{aligned}
\end{aligned}