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===EI and Φ===
===EI and Φ===
The integrated information (or the degree of integration) <math>\Phi</math>, can be defined as the minimum value of EI between any two bipartitions of a system. Suppose the system is 𝑋, and 𝑆 is a subset of 𝑋, that is partitioned into two parts, 𝐴 and 𝐵. There are causal interactions between 𝐴, 𝐵, and the rest of 𝑋. [[文件:OriginalEI.png|350x350px|整合信息论中的划分|替代=|缩略图|链接=https://wiki.swarma.org/index.php/%E6%96%87%E4%BB%B6:OriginalEI.png]] In this scenario, we can measure the strength of these causal interactions. First, we calculate the EI from 𝐴 to 𝐵, i.e., we intervene on 𝐴 such that it follows the maximum entropy distribution, then measure the mutual information between 𝐴 and 𝐵:
The integrated information (or the degree of integration) <math>\Phi</math>, can be defined as the minimum value of EI between any two bipartitions of a system. Suppose the system is 𝑋, and 𝑆 is a subset of 𝑋, that is partitioned into two parts, 𝐴 and 𝐵. There are causal interactions between 𝐴, 𝐵, and the rest of 𝑋. [[文件:OriginalEI.png|350x350px|整合信息论中的划分|替代=|缩略图|链接=https://wiki.swarma.org/index.php/%E6%96%87%E4%BB%B6:OriginalEI.png]] In this scenario, we can measure the strength of these causal interactions. First, we calculate the EI from 𝐴 to 𝐵, i.e., we intervene on 𝐴 such that it follows the maximum entropy distribution, then measure the mutual information between 𝐴 and 𝐵:
<math>
<math>
EI(A\rightarrow B) = I(A^{H^{max}}: B)
EI(A\rightarrow B) = I(A^{H^{max}}: B)
</math>
</math>
Here, <math>A^{H^{max}}</math> denotes the maximum entropy distribution on 𝐴, which is implicitly a uniform distribution. This equation implies a causal mechanism from 𝐴 to 𝐵, denoted as
[math]Pr(B|A)[/math], which remains unchanged during the intervention. If different states of 𝐴 cause significantly different changes in 𝐵, the EI will be high; conversely, if changes in 𝐴 have little effect on 𝐵, the EI will be low. This measure is directional, so EI from 𝐴 to 𝐵 may differ from EI from 𝐵 to 𝐴. We can sum these two directions to obtain the total EI for the partition:
<math>
<math>
</math>
</math>
Traversing various partitions, if there exists a partition S that makes EI 0, it indicates that S can be seen as two causally independent parts, so the degree of integration should also be 0. From this particular example, we can see that we should focus on the one with the least significant information among all partitions. Of course, different partitions will result in different state spaces for A and B, so a normalization process should be performed by dividing by the smaller of the maximum entropy values of A and B. So, we can have a minimum information partition (MIB). The degree of integration <math>\Phi</math> is defined as follows:
<math>
<math>
</math>
</math>
This defines the relationship between [[Integrated Information Ability]] and EI.
===区别===
===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.
==EI与其它因果度量指标==
==EI与其它因果度量指标==
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'''。
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'''。