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Here, P is the microstate Markov transition matrix with dimensions N×N, where N is the number of microstates. P′ is the macro-state transition matrix obtained after coarse-graining, with dimensions M×M, where M<N represents the number of macrostates.
Here, P is the microstate Markov transition matrix with dimensions N×N, where N is the number of microstates. P′ is the macro-state transition matrix obtained after coarse-graining, with dimensions M×M, where M<N represents the number of macrostates.
The process of coarse-graining a Markov transition matrix typically involves two steps: 1) grouping N microstates into M macrostates, and 2) reducing the Markov transition matrix accordingly. For more details on the specific methods for coarse-graining a Markov chain, refer to the topic of Markov chain coarse-graining.
If the computed CE>0, the system is said to exhibit causal emergence; otherwise, it does not.
Below, we demonstrate a specific example of causal emergence:
{|
{|
|+马尔科夫链示例
|+Example of Causal Emergence in a Markov Chain
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|[math]\begin{aligned}&Det(P_m)=0.81\ bits,\\&Deg(P_m)=0\ bits,\\&EI(P_m)=0.81\ bits\end{aligned}[/math]||[math]\begin{aligned}&Det(P_M)=1\ bits,\\&Deg(P_M)=0\ bits,\\&EI(P_M)=1\ bits\end{aligned}[/math]
|[math]\begin{aligned}&Det(P_m)=0.81\ bits,\\&Deg(P_m)=0\ bits,\\&EI(P_m)=0.81\ bits\end{aligned}[/math]||[math]\begin{aligned}&Det(P_M)=1\ bits,\\&Deg(P_M)=0\ bits,\\&EI(P_M)=1\ bits\end{aligned}[/math]
|}在这个例子中，微观态的转移矩阵是一个4*4的矩阵，其中前三个状态彼此以1/3的概率相互转移，这导致该转移矩阵具有较小的确定性，因此EI也不是很大为0.81。然而，当我们对该矩阵进行粗粒化，也就是把前三个状态合并为一个状态a，而最后一个状态转变为一个宏观态b。这样所有的原本三个微观态彼此之间的转移就变成了宏观态a到a内部的转移了。因此，转移概率矩阵也就变成了[math]P_M[/math]，它的EI为1。在这个例子中，可以计算它的[[因果涌现度量]]为：
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In this example, the microstate transition matrix is a 4x4 matrix, where the first three states transition to each other with a probability of 1/3. This leads to a transition matrix with relatively low determinism, and thus, the EI is not very high, with a value of 0.81. However, when we coarse-grain this matrix—merging the first three states into one macrostate a, and the last state becomes another macrostate b—all transitions between the original three microstates now become internal transitions within macrostate a. Thus, the transition probability matrix becomes PM, with an EI of 1. In this case, the causal emergence can be measured as:
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That is, there is 0.19 bits of causal emergence.
Sometimes, causal emergence is calculated using the normalized EI, defined as:
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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。-->
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。-->
==计算EI的源代码==
==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>).
python:<syntaxhighlight lang="python3">
python:<syntaxhighlight lang="python3">
tpm_ei(mi_states)
tpm_ei(mi_states)
</syntaxhighlight>
</syntaxhighlight>
=连续变量的EI=
=EI for Continuous Variables=
In reality, most systems need to be considered in continuous spaces, so it is necessary to extend the concept of EI to continuous systems.
The core idea of this extension is to simplify the causal mechanism in continuous space into a deterministic function mapping f(X), combined with a noise variable ξ. In the cases listed below, ξ∼N(0,Σ), meaning it follows a Gaussian distribution, allowing us to obtain an analytical expression for EI. For more general cases, there is no literature available for discussion yet.
==Random Function Mapping==
==随机函数映射==
Initially, Erik Hoel considered this and proposed the framework of causal geometry. This framework not only discusses the calculation of EI for random function mappings but also introduces the concepts of intervention noise and causal geometry. It defines the local form of EI and draws analogies and comparisons with information geometry. Below, we will discuss one-dimensional and multi-dimensional function mappings and the local form of EI.
===一维函数映射===
===一维函数映射===
首先，我们考虑最简单的情况：
首先，我们考虑最简单的情况：