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第951行： − 根据定义，当<math>c</math>为极小概率事件时，<math>nec(e,c) \approx nec'(e)</math>。当<math>C</math>为连续状态空间时，可认为两者等价。 − − 注意：<math>nec'(e)</math>的定义与文献<ref name=":0" />中定义的<math>nec^\dagger(e) = P(e|C)</math>不同，两者关系为<math>net'(e) = 1 - nec^\dagger(e)</math>。 − ===因果元语与确定性和简并性=== − 如前所述，EI可被分解为确定性与简并性两项，这两项分别对应充分性和必要性的因果元语表达： + + + + 第965行：
第964行： − 可以看到，充分性和确定性之间，以及必要性和简并性之间存在单调映射关系。充分性越高，确定性也越高；必要性越高，简并性则越小。+ − + + − + − + − + − + − + − + 第983行：
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第1,028行： − + − 正如示例{{EquationNote|example}}中的马尔科夫链所示，当概率转移矩阵呈现为一种[[排列置换矩阵]]（Permutation matrix）的时候，EI会更大。+ − 可以证明，[[排列置换矩阵]]是唯一一种能同时满足如下两个条件的矩阵：+ + − 1、矩阵是可逆的； 2、矩阵满足马尔科夫链的归一化条件，也就是对于任意的[math]i\in[1,N][/math]来说，[math]|P_i|_1=1[/math]+ − 我们将这一性质称为[[动力学可逆性]]。因此，从某种程度上说，EI衡量的是马尔科夫链的一种[[动力学可逆性]]。+ − 需要注意的是，这里所说的马尔科夫链的[[动力学可逆性]]与通常意义下的[[马尔科夫链的可逆性]]是不等同的。前者的可逆性体现为马尔科夫概率转移矩阵的可逆性，也就是它针对状态空间中的每一个确定性状态的运算都是可逆的，所以也称其为动力学可逆的。但是，文献中通常意义下的可逆的马尔科夫链并不要求转移矩阵是可逆的，而是要以稳态分布为时间反演对称轴，使得在动力学P作用构成的演化下的正向时间形成的状态分布序列和逆向状态分布序列完全相同。+
无编辑摘要
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 and Other Causal Metrics==
==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.
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''' and '''Necessity''' and in causal relationships.
===Definition of Causal Primitives===
===Definition of Causal Primitives===
<math>
<math>
</math>
</math>
According to the definition, when <math>c</math> is a very low probability event,<math>nec(e,c) \approx nec'(e)</math>. When <math>C</math> is a continuous state space, the two can be considered equivalent.
Note: The definition of <math>nec'(e)</math> is different from the definition of <math>nec^\dagger(e) = P(e|C)</math> in the literature<ref name=":0" />. The relationship between the two is <math>net'(e) = 1 - nec^\dagger(e)</math>.
===Causal Primitives, Determinism, and Degeneracy===
As previously mentioned, EI (Effective Information) can be decomposed into two terms: determinism and degeneracy, which correspond to the causal primitives of sufficiency and necessity, respectively.
<math>
<math>
Determinism\quad Coef= \sum_{e \in E, c \in C}{P(e,c)\cdot \left[1 - \frac{\log{\frac{1}{suff(e,c)}}}{\log{N}}\right]}
Determinism\quad Coef= \sum_{e \in E, c \in C}{P(e,c)\cdot \left[1 - \frac{\log{\frac{1}{suff(e,c)}}}{\log{N}}\right]}
</math>
</math>
There is a monotonic relationship between sufficiency and determinism, as well as between necessity and degeneracy. The higher the sufficiency, the higher the determinism; the higher the necessity, the lower the degeneracy.
===因果度量指标的因果元语表示===
===Causal Metrics in Terms of Causal Primitives===
{| class="wikitable"
{| class="wikitable"
|+因果度量指标的因果元语表现形式
|+Causal metalinguistic expressions of causal measurement indicators
!因果指标
!Causal indicators
!概率定义
!Definition of Probability
!因果元语定义
!Definition of causal metalanguage
!参考文献
!References
|-
|-
|有效信息EI
|Effective Information EI
|<math>
|<math>
\log_2{\frac{P(e|c)}{P(e|C)}}
\log_2{\frac{P(e|c)}{P(e|C)}}
|<ref name="hoel_2013">{{cite journal|last1=Hoel|first1=Erik P.|last2=Albantakis|first2=L.|last3=Tononi|first3=G.|title=Quantifying causal emergence shows that macro can beat micro|journal=Proceedings of the National Academy of Sciences|volume=110|issue=49|page=19790–19795|year=2013|url=https://doi.org/10.1073/pnas.1314922110}}</ref>
|<ref name="hoel_2013">{{cite journal|last1=Hoel|first1=Erik P.|last2=Albantakis|first2=L.|last3=Tononi|first3=G.|title=Quantifying causal emergence shows that macro can beat micro|journal=Proceedings of the National Academy of Sciences|volume=110|issue=49|page=19790–19795|year=2013|url=https://doi.org/10.1073/pnas.1314922110}}</ref>
|-
|-
|Galton度量
|Galton Metric
|<math>
|<math>
P(c)P(C\backslash c)(P(e|c) - P(e|C\backslash c))
P(c)P(C\backslash c)(P(e|c) - P(e|C\backslash c))
|<ref>Fitelson, B., & Hitchcock, C. (2011). ''Probabilistic measures of causal strength'' (pp. 600-627). na.</ref>
|<ref>Fitelson, B., & Hitchcock, C. (2011). ''Probabilistic measures of causal strength'' (pp. 600-627). na.</ref>
|-
|-
|Suppes度量
|Suppes Metric
|<math>
|<math>
P(e|c) - P(e|C)
P(e|c) - P(e|C)
|<ref>Suppes, P. (1973). A probabilistic theory of causality. ''British Journal for the Philosophy of Science'', ''24''(4).</ref>
|<ref>Suppes, P. (1973). A probabilistic theory of causality. ''British Journal for the Philosophy of Science'', ''24''(4).</ref>
|-
|-
|Eells度量(等同于Judea Pearl的充要概率PNS)
|Eells Metric(Equivalent to Judea Pearl's necessary and sufficient probability PNS)
|<math>
|<math>
P(e|c) - P(e|C\backslash c)
P(e|c) - P(e|C\backslash c)
|<ref name="pearl_causality">{{cite book|title=因果论——模型、推理和推断|author1=Judea Pearl|author2=刘礼|author3=杨矫云|author4=廖军|author5=李廉|publisher=机械工业出版社|year=2022|month=4}}</ref><ref>Eells, E. (1991). ''Probabilistic causality'' (Vol. 1). Cambridge University Press.</ref>
|<ref name="pearl_causality">{{cite book|title=因果论——模型、推理和推断|author1=Judea Pearl|author2=刘礼|author3=杨矫云|author4=廖军|author5=李廉|publisher=机械工业出版社|year=2022|month=4}}</ref><ref>Eells, E. (1991). ''Probabilistic causality'' (Vol. 1). Cambridge University Press.</ref>
|-
|-
|Cheng度量(等同于Judea Pearl的充分概率PS)
|Cheng Metric (Equivalent to Judea Pearl's sufficient probability PS)
|<math>
|<math>
\frac{P(e|c) - P(e|C\backslash c)}{1 - P(e|C\backslash c)}
\frac{P(e|c) - P(e|C\backslash c)}{1 - P(e|C\backslash c)}
|<ref name="pearl_causality" /><ref>Cheng, P. W., & Novick, L. R. (1991). Causes versus enabling conditions. ''Cognition'', ''40''(1-2), 83-120.</ref>
|<ref name="pearl_causality" /><ref>Cheng, P. W., & Novick, L. R. (1991). Causes versus enabling conditions. ''Cognition'', ''40''(1-2), 83-120.</ref>
|-
|-
|Lewis度量(等同于Judea Pearl的必要概率PN)
|Lewis Metric (Equivalent to Judea Pearl's necessary probability PN)
|<math>
|<math>
\frac{P(e|c) - P(e|C\backslash c)}{P(e|c)}
\frac{P(e|c) - P(e|C\backslash c)}{P(e|c)}
|<ref name="pearl_causality" /><ref>Lewis, D. (1973). Causation. ''The journal of philosophy'', ''70''(17), 556-567.</ref>
|<ref name="pearl_causality" /><ref>Lewis, D. (1973). Causation. ''The journal of philosophy'', ''70''(17), 556-567.</ref>
|}
|}
==EI与动力学可逆性==
==EI and Dynamical Reversibility==
As demonstrated in the example {{EquationNote|example}} of the Markov chain, EI increases when the probability transition matrix is a [[Permutation Matrix]].
It can be shown that [[Permutation Matrix]] is the only matrix that simultaneously meet the following two conditions:
1、The matrix is invertible； 2、The matrix satisfies the Markov chain normalization condition, meaning for any [math]i\in[1,N][/math], [math]|P_i|_1=1[/math]
This property is referred to as [[Dynamical Reversibility]]. Therefore, in a certain sense, EI measures a form of [[Dynamical Reversibility]] in the Markov chain.
需要注意的是，这里所说的马尔科夫链的[[动力学可逆性]]与通常意义下的[[马尔科夫链的可逆性]]是不等同的。前者的可逆性体现为马尔科夫概率转移矩阵的可逆性，也就是它针对状态空间中的每一个确定性状态的运算都是可逆的，所以也称其为动力学可逆的。但是，文献中通常意义下的可逆的马尔科夫链并不要求转移矩阵是可逆的，而是要以稳态分布为时间反演对称轴，使得在动力学P作用构成的演化下的正向时间形成的状态分布序列和逆向状态分布序列完全相同。
It's important to note that the [[Dynamical Reversibility]] of a Markov chain described here is different from the commonly understood [[the reversibility of Markov chains]] in the literature. Dynamical reversibility refers to the invertibility of the Markov transition matrix itself, meaning the operations on each deterministic state in the state space are reversible. However, conventional reversible Markov chains do not require the transition matrix to be invertible but rather exhibit time-reversal symmetry with respect to the steady-state distribution. This symmetry implies that the state distribution sequence formed under the evolution of the dynamics 𝑃 remains the same in both forward and reverse time.
由于[[排列置换矩阵]]过于特殊，我们需要能够衡量一般的马尔科夫概率转移矩阵与排列置换矩阵的靠近程度，以度量其[[近似动力学可逆性]]。在文献<ref name="zhang_reversibility">{{cite journal|author1=Jiang Zhang|author2=Ruyi Tao|author3=Keng Hou Leong|author4=Mingzhe Yang|author5=Bing Yuan|year=2024|title=Dynamical reversibility and a new theory of causal emergence|url=https://arxiv.org/abs/2402.15054|journal=arXiv}}</ref>中，作者们提出了一种用矩阵的类[[Schatten范数]]来度量一个马尔科夫概率转移矩阵的[[近似动力学可逆性]]的方法，定义为：
由于[[排列置换矩阵]]过于特殊，我们需要能够衡量一般的马尔科夫概率转移矩阵与排列置换矩阵的靠近程度，以度量其[[近似动力学可逆性]]。在文献<ref name="zhang_reversibility">{{cite journal|author1=Jiang Zhang|author2=Ruyi Tao|author3=Keng Hou Leong|author4=Mingzhe Yang|author5=Bing Yuan|year=2024|title=Dynamical reversibility and a new theory of causal emergence|url=https://arxiv.org/abs/2402.15054|journal=arXiv}}</ref>中，作者们提出了一种用矩阵的类[[Schatten范数]]来度量一个马尔科夫概率转移矩阵的[[近似动力学可逆性]]的方法，定义为：