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Defining causal emergence is a key issue. Several representative works include the method based on EI by Hoel et al. <ref name=":0" /><ref name=":1" />, and the [[information decomposition]] method by Rosas et al. <ref name=":5">Rosas F E, Mediano P A, Jensen H J, et al. Reconciling emergences: An information-theoretic approach to identify causal emergence in multivariate data[J]. PLoS computational biology, 2020, 16(12): e1008289.</ref>. Zhang et al. <ref name=":2" />, also proposed a new causal emergence theory based on [[singular value decomposition]].

Defining causal emergence is a key issue. Several representative works include the method based on EI by Hoel et al. <ref name=":0" /><ref name=":1" />, and the [[information decomposition]] method by Rosas et al. <ref name=":5">Rosas F E, Mediano P A, Jensen H J, et al. Reconciling emergences: An information-theoretic approach to identify causal emergence in multivariate data[J]. PLoS computational biology, 2020, 16(12): e1008289.</ref>. Zhang et al. <ref name=":2" />, also proposed a new causal emergence theory based on [[singular value decomposition]].

==== Erik Hoel's causal emergence theory ====

==== Erik Hoel's theory of causal emergence ====

Hoel et al. (2013) <ref name=":0" /><ref name=":1" />  proposed the theory of causal emergence. The following figure is an abstract framework for this theory. The horizontal axis represents time  and the vertical axis represents scale. This framework describes the same dynamical system at both microscopic and macroscopic scales. Among them, [math]f_m[/math] represents microscopic dynamics, [math]f_M[/math] represents macroscopic dynamics, and the two are connected by a coarse-graining function [math]\phi[/math]. In a discrete-state Markov dynamical system, both [math]f_m[/math] and [math]f_M[/math] are Markov chains. By performing [[coarse-graining of the Markov chain]] on [math]f_m[/math], [math]f_M[/math] can be obtained. <math> EI </math> is a measure of EI. Since the microscopic state may have greater randomness, which leads to relatively weak [[causality]] of microscopic dynamics, by performing reasonable coarse-graining on the microscopic state at each moment, it is possible to obtain a macroscopic state with stronger causality. Causal emergence refers to the phenomenon where coarse-graining a microscopic state increases the EI of macroscopic dynamics. The difference in EI between the macroscopic and the microscopic states is defined as the strength of causal emergence.  

Hoel et al. (2013) <ref name=":0" /><ref name=":1" />  proposed the theory of causal emergence. The following figure is an abstract representation of the whole framework for this theory. The horizontal axis represents time  and the vertical axis represents scale. This framework describes the same dynamical system at both microscopic and macroscopic scales. Among them, [math]f_m[/math] represents the microscopic dynamics, [math]f_M[/math] represents the macroscopic dynamics, and the two are connected by a coarse-graining function [math]\phi[/math]. In a discrete-state Markov dynamical system, both [math]f_m[/math] and [math]f_M[/math] are transitional probability matrices (TPMs) of Markov chains. By performing [[coarse-graining of the Markov chain]] on [math]f_m[/math], [math]f_M[/math] can be obtained. <math> EI </math> is a measure of effective information. Since the microscopic state may have greater randomness, which leads to relatively weak [[causal effect]] in microscopic dynamics, by performing reasonable coarse-graining on the microscopic state at each moment, it is possible to obtain a macroscopic state with stronger causality. Causal emergence refers to the phenomenon where coarse-graining a microscopic state can lead to a macroscopic dynamics with higher EI. The difference in EI between the macroscopic and the microscopic dynamics is defined as the strength of causal emergence.  

[[文件:CE.png|替代=因果涌现示意图|500x500像素]]

[[文件:CE.png|替代=因果涌现示意图|500x500像素]]

===== Effective information =====

===== Effective information =====

[[Tononi]] et al. first introduced [[Effective Information]] (<math> EI </math>) in their study on [[integrated information theory]] <ref>Tononi G, Sporns O. Measuring information integration[J]. BMC neuroscience, 2003, 41-20.</ref>. In causal emergence research, [[Erik Hoel]] and colleagues use this [[measures of causal effects]] index to quantify the causal strength of a [[causal mechanism]].

[[Tononi]] et al. first introduced [[Effective Information]] (<math> EI </math>) in their study on [[integrated information theory]] <ref>Tononi G, Sporns O. Measuring information integration[J]. BMC neuroscience, 2003, 41-20.</ref>. In the study of causal emergence, [[Erik Hoel]] and colleagues use this [[measures of causal effects]] index to quantify the strength of causal effect for a [[causal mechanism]].

Specifically, <math> EI </math> is calculated by [[intervening]] on the independent variable and measuing the [[mutual information]] between the cause and effect variables. This mutual information represents EI, quantifying the causal effect of the mechanism.  

Specifically, <math> EI </math> is calculated by [[intervening]] on the cause variable and measuring the [[mutual information]] between the cause and effect variables. This mutual information represents EI, quantifying the causal effect of the mechanism.  

In a [[Markov chain]], the state variable [math]X_{t}[/math] represents the cause, while [math]X_{t + 1}[/math] represents the effect. Thus, the [math]X_{t + 1}[/math] at the next time can be regarded as the result. Thus, the [[state transition matrix]] of a [[Markov chain]] represents its [[causal mechanism]]. Therefore, the calculation formula for <math>EI</math> for a [[Markov chain]] is as follows:

In a [[Markov chain]], the state variable [math]X_{t}[/math] represents the cause, while [math]X_{t + 1}[/math] represents the effect. Thus, the [math]X_{t + 1}[/math] at the next time can be regarded as the result. Thus, the [[state transition matrix]] of a [[Markov chain]] represents its [[causal mechanism]]. Therefore, the calculation formula for <math>EI</math> for a [[Markov chain]] is as follows:

</math>

</math>

<math>f</math> is the state transition matrix of a Markov chain, [math]U(\mathcal{X})[/math] represents the uniform distribution over the value space [math]\mathcal{X}[/math] of the state variable [math]X_{t}[/math]. [math]X_{t}[/math], [math]X_{t+1}[/math] are the states at two consecutive moments following the [[intervention]] of [math]X_{t}[/math] at time <math>t</math> into a [[uniform distribution]]. <math>p_{ij}</math> denots the transition probability from state <math>i</math> to state <math>j</math>. The formula shows that <math> EI </math> depends solely on the probability transition matrix [math]f[/math]. The intervention operation is ensures that EI measures the causal effects of the dynamics objectively. This approach avoids influence from the original data distribution.  

<math>f</math> is the state transition matrix of a Markov chain, [math]U(\mathcal{X})[/math] represents the uniform distribution over the value space [math]\mathcal{X}[/math] of the state variable [math]X_{t}[/math]. [math]X_{t}[/math], [math]X_{t+1}[/math] are the states at two consecutive moments following the [[intervention]] of [math]X_{t}[/math] at time <math>t</math> into a [[uniform distribution]]. <math>p_{ij}</math> denots the transition probability from state <math>i</math> to state <math>j</math>. The formula shows that <math> EI </math> depends solely on the probability transition matrix [math]f[/math]. The intervention operation is to ensure that EI measures the causal effects of the dynamics objectively. This approach avoids influence from the original data distribution.  

EI can be decomposed into two parts: '''determinism''' and '''degeneracy'''. It is also possible to eliminate the influence of the size of the state space by introducing normalization. For more detailed information about EI, please refer to the entry: [[Effective Information]].

EI can be decomposed into two parts: '''determinism''' and '''degeneracy'''. It is also possible to eliminate the influence of the size of the state space by introducing normalization. For more detailed information about EI, please refer to the entry: [[Effective Information]].

The effective coefficient is introduced to eliminate the influence of the number of states on EI. The formula is shown as follows:  

The effective coefficient(or the effectiveness) is introduced to eliminate the influence of the number of states on EI. The formula is shown as follows:  

<math>

<math>

where <math>N</math> represents the number of states.

where <math>N</math> represents the number of states.

=====The measurement of causal emergence=====

=====The degree of causal emergence=====

Causal emergence can be defined by comparing the EI of macroscopic and microscopic  dynamics in the system:

The degree of causal emergence can be defined by comparing the EI of macroscopic and microscopic  dynamics in the system:

<math>

<math>

</math>

</math>

Here <math>CE</math>  is the strength of causal emergence. Macroscopic dynamics exhibit causal emergence if their EI exceeds that of microscopic dynamics <math>CE>0</math>.

Here <math>CE</math>  is the degree of causal emergence. Macroscopic dynamics exhibit causal emergence if their EI exceeds that of microscopic dynamics, i.e., <math>CE>0</math>.

Furthermore, <math>CE</math> can be decomposed into the sum of two terms, <math>CE = \Delta I_{Eff} + \Delta I_{Size}</math>, where <math>\Delta I_{Eff} = (Eff(f_M) - Eff(f_m))\cdot \log_2(M)</math>, <math>\Delta I_{Size} = Eff(f_m)\cdot(\log_2(M) - \log_2(m))</math>. Here, <math>M</math> and <math>m</math> represent the sizes of the macroscopic and microscopic states respectively. Because coarse-graining reduces the macroscopic state, <math>\Delta I_{Size}</math> is necessarily less than 0. Causal emergence occurs if the increase in <math>\Delta I_{Eff}</math> exceeds the decrease in <math>\Delta I_{Size}</math>.

Furthermore, <math>CE</math> can be decomposed into the sum of two terms, <math>CE = \Delta I_{Eff} + \Delta I_{Size}</math>, where <math>\Delta I_{Eff} = (Eff(f_M) - Eff(f_m))\cdot \log_2(M)</math> is the causal emergence induced by the increase of effectiveness, and <math>\Delta I_{Size} = Eff(f_m)\cdot(\log_2(M) - \log_2(m))</math> is the causal emergence induced by the shrink of state space. Here, <math>M</math> and <math>m</math> represent the sizes of the macroscopic and microscopic states respectively. Because coarse-graining reduces the macroscopic state, <math>\Delta I_{Size}</math> is necessarily less than 0. Causal emergence occurs if the increase in <math>\Delta I_{Eff}</math> exceeds the decrease in <math>\Delta I_{Size}</math>.

=====Markov chain example=====

=====An Example of Markov chain=====

In the literature <ref name=":0"/>, Hoel gives an example of a state transition matrix ([math]f_m[/math]) of a Markov chain with 8 states, as shown in the left figure below. Among them, the first 7 states transfer with equal probability, and the last state is independent and can only transition to its own state.

In the literature <ref name=":0"/>, Hoel gives an example of a state transition matrix ([math]f_m[/math]) of a Markov chain with 8 states, as shown in the left figure below. Among them, the first 7 states transfer with equal probability, and the last state is independent and can only transition to its own state.

First, merge the first 7 states into one macroscopic state, tentatively called A, and then sum the probability values in the first 7 rows and the first 7 columns of [math]f_m[/math] and divide by 7, thereby obtaining the state transition probability from macroscopic state A to A, while leaving the other values of the [math]f_m[/math] matrix unchanged. The new probability transition matrix after merging is shown in the right figure, denoted as [math]f_M[/math]. This definite macroscopic Markov transition matrix ensures that the future state is entirely determined by the current state. At this time <math>EI(f_M)>EI(f_m)</math>, and causal emergence occurs in the system.  

First, we can use a coarse-graining strategy that merges the first 7 states into one macroscopic state, tentatively called A, and then sum the probability values in the first 7 rows and the first 7 columns of [math]f_m[/math] and divide by 7, thereby obtaining the state transition probability from macroscopic state A to A, while leaving the other values of the [math]f_m[/math] matrix unchanged to obtain a coarsen TPM which is shown in the right figure, denoted as [math]f_M[/math]. This definite macroscopic Markov transition matrix ensures that the future state is entirely determined by the current state. At this time <math>EI(f_M)>EI(f_m)</math>, and causal emergence occurs in the system.  

[[文件:Causal emergence in the state space3.png|居左|600x600像素]]

[[文件:Causal emergence in the state space3.png|居左|600x600像素]]

However, for more general Markov chains and more general state groupings, this simple operation of averaging probabilities is not always feasible. This occurs when the merged transition matrix does not meet Markov chain conditions, such as row normalization or element values within [0, 1]. Refer to the “Reduction of Markov Chains” section or the “[[Coarse-graining of Markov Chains]]” entry for details on feasible macroscopic Markov chains.

However, for more general Markov chains and more general state groupings, this simple operation of averaging probabilities is not always feasible. This occurs when the merged transition matrix does not meet Markov chain conditions, such as row normalization or element values within [0, 1]. Refer to the “Reduction of Markov Chains” section or the “[[Coarse-graining of Markov Chains]]” entry for details on feasible macroscopic Markov chains.

=====Boolean network example=====

=====An Example of Boolean network=====

Another instance of causal emergence in the literature <ref name=":0"/> involves a [[Boolean network]]. The figure illustrates a Boolean network with 4 nodes. Each node can exist in one of two states, 0 or 1. Each node is connected to two other nodes and operates according to identical microscopic dynamics (figure a).  Therefore, this system contains a total of sixteen microscopic states, and its dynamics can be represented by a state transition matrix (figure c).  

Another instance of causal emergence in the literature <ref name=":0"/> involves a [[Boolean network]]. The figure illustrates a Boolean network with 4 nodes. Each node can exist in one of two states, 0 or 1. Each node is connected to two other nodes and operates according to identical microscopic node mechanism (figure a).  Therefore, this system contains a total of sixteen microscopic states, and its dynamics can be represented by a TPM (figure c).  

The coarse-graining operation of this system is divided into two steps. The first step is to cluster the nodes in the Boolean network. As shown in figure b below, node A and B are merged to form the macroscopic node [math]\alpha[/math], and node C and D are merged to obtain the macroscopic node [math]\beta[/math]. The second step is to map the microscopic node states in each group to the merged macroscopic node states. This mapping function is shown in figure d below. All microscopic node states containing 0 are transformed into the "off" state of the macroscopic node, while the microscopic 11  state is transformed into the on state of the macroscopic. In this way, we can obtain a new macroscopic Boolean network, and derive the dynamic  mechanism of the macroscopic Boolean network according to the dynamic  mechanism of the microscopic nodes. According to this mechanism, the state transition matrix of the macroscopic network can be obtained (as shown in figure e).  

The coarse-graining operation of this system is divided into two steps. The first step is to cluster the nodes in the Boolean network. As shown in figure b below, node A and B are merged to form the macroscopic node [math]\alpha[/math], and node C and D are merged to obtain the macroscopic node [math]\beta[/math]. The second step is to map the microscopic node states in each group to the merged macroscopic node states. This mapping function is shown in figure d below. All microscopic node states containing 0 are transformed into the "off" state of the macroscopic node, while the microscopic 11  state is transformed into the "on" state of the macroscopic. In this way, we can obtain a new macroscopic Boolean network, and derive the node mechanism of the macroscopic Boolean network according to the mechanism of the microscopic nodes. According to this mechanism, the state transition matrix of the macroscopic network can be obtained (as shown in figure e).  

A comparison reveals that the EI of macroscopic dynamics exceeds that of microscopic dynamics <math>EI(f_M\ )>EI(f_m\ ) </math>. This demonstrates the occurrence of causal emergence in this system.

A comparison reveals that the EI of macroscopic dynamics exceeds that of microscopic dynamics <math>EI(f_M\ )>EI(f_m\ ) </math>. This demonstrates the occurrence of causal emergence in this system.