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{{#seo:

{{#seo:

|keywords=有效信息,因果涌现,互信息,Effective Information,EI,确定性,简并性

|keywords=Effective information, Causal Emergence, Mutual Information,Effective Information,EI,Certainty, Degeneracy

|description=本文是目前互联网世界中关于因果涌现理论和整合信息论中的有效信息（Effective information,简称EI）的少数的全面的介绍文章，包括有效信息的来源，有效信息的定义与分解、实际例子，以及如何扩展到连续变量、与因果度量指标、动力学可逆性的关系是什么等问题进行了全面的讨论。

|description=This paper is one of the few comprehensive introductory articles on the causal emergence theory and effective information (EI) in the integrated information theory in the current Internet world, including the source of effective information, the definition and decomposition of effective information, practical examples, and how to expand to continuous variables, what is the relationship with causal metrics and dynamic reversibility.

}}

}}

Since normalized EI eliminates the effect of system size, the measure of causal emergence becomes larger.<!--[[文件:Example1.png|815x815px|无框|居中]]

Since normalized EI eliminates the effect of system size, the measure of causal emergence becomes larger.<!--[[文件:Example1.png|815x815px|无框|居中]]

上图展示了几种马尔科夫链的转移概率矩阵，其中（a）是确定性高，简并性低，所以整体eff比较高。（b）则是确定性和简并性都比较高，所以eff是0。（c）相比于（a）确定性更低，（d）也是确定性和简并性都较高导致eff较低，它们都可以通过同一种粗粒化策略（将前4个状态合并为一个状态）来得到（e）。此时（e）确定性很高，无简并性，所以（e）的eff比（c）（d）要高。-->

The above figure shows the transition probability matrices of several Markov chains, where (a) has high determinacy and low degeneracy, resulting in a relatively high overall eff. (b) Then both determinacy and degeneracy are relatively high, so eff is 0. (c) Compared to (a) which has lower certainty, (d) also has higher certainty and degeneracy, resulting in lower eff. Both can be obtained through the same coarsening strategy (merging the first four states into one) to obtain (e). At this point, the certainty of (e) is high and there is no degeneracy, so the eff of (e) is higher than that of (c) and (d).-->

==Python Source Code for Calculating EI==

==Python Source Code for Calculating EI==

Below is the Python source code for calculating EI for a Markov transition matrix. The input <code>tpm</code> is a Markov transition matrix that satisfies the row normalization condition. The returned values are <code>ei_all</code>, which is the EI, and other parameters such as effectiveness (<code>eff</code>), determinism (<code>det</code>), degeneracy (<code>deg</code>), determinism coefficient (<code>det_c</code>), and degeneracy coefficient (<code>deg_c</code>).

Below is the Python source code for calculating EI for a Markov transition matrix. The input <code>tpm</code> is a Markov transition matrix that satisfies the row normalization condition. The returned values are <code>ei_all</code>, which is the EI, and other parameters such as effectiveness (<code>eff</code>), determinism (<code>det</code>), degeneracy (<code>deg</code>), determinism coefficient (<code>det_c</code>), and degeneracy coefficient (<code>deg_c</code>).

def tpm_ei(tpm, log_base = 2):

def tpm_ei(tpm, log_base = 2):

     '''

     '''

     tpm: 输入的概率转移矩阵，可以是非方阵

     tpm: The input probability transition matrix can be a non square matrix

     log_base：对数的底

     log_base：Base of a logarithm

     ei_all：EI的值

     ei_all：The value of EI

     det：EI中确定性的部分

     det：The deterministic part of EI

     deg：EI中简并性的部分

     deg：The degeneracy part of EI

     eff：有效性

     eff：Effectiveness

     det_c：确定性系数

     det_c：Coefficient of certainty

     deg_c：简并性系数

     deg_c：Coefficient of degeneracy

     '''

     '''

     # marginal distribution of y given x ~ Unifrom Dist

     # marginal distribution of y given x ~ Unifrom Dist

<math>

<math>

EI\approx \ln(\frac{L}{\sqrt{2\pi e}})+\frac{1}{2L}\int_{-\frac{L}{2}}^{\frac{L}{2}}\ln \left(\frac{f'(x)}{\sigma}\right)^2dx

EI\approx \ln(\frac{L}{\sqrt{2\pi e}})+\frac{1}{2L}\int_{-\frac{L}{2}}^{\frac{L}{2}}\ln \left(\frac{f'(x)}{\sigma}\right)^2dx

</math><!--如果同时考虑两种噪声，并且如果干预空间大小为<math>L

</math><!--If two types of noise are considered simultaneously and the intervention space size is <math>L

</math>和<math>\epsilon\ll 1

</math> and <math>\epsilon\ll 1

</math>，EI数学形式如下所示

</math>，The mathematical form of EI is as follows

<math>EI\approx -\frac{1}{2}\int_{-L/2}^{L/2}\ln\left[\left(\frac{\epsilon}{f'(x)}\right)^2+\delta^2\right]dx.

<math>EI\approx -\frac{1}{2}\int_{-L/2}^{L/2}\ln\left[\left(\frac{\epsilon}{f'(x)}\right)^2+\delta^2\right]dx.