更改

=== Several causal emergence theories  ===

=== Several causal emergence theories  ===

Defining causal emergence is a key issue. Several representative works include the method based on [[effective information]] 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 Jiang 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 Jiang 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 causal emergence theory ====

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 [[effective information]]. 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 [[effective information]] of macroscopic dynamics. The difference in [[effective information]] 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 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.  

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 [[Effective Information]], quantifying the causal effect of the 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.  

<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 effective information 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 ensures that EI measures the causal effects of the dynamics objectively. This approach avoids influence from the original data distribution.  

Effective information 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 Effective Information, 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 effective information. The formula is shown as follows:  

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

=====Causal Emergence Measurement=====

=====Causal Emergence Measurement=====

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

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

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

A comparison reveals that the [[effective information]] 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.

=====Example=====

=====Example=====

The paper gives an example of a linear dynamical system. Its dynamics is a vector autoregressive model. By using genetic algorithms to iteratively evolve different initial conditions, the degree of dynamic independence of the system can also gradually increase. At the same time, it is found that different coarse-graining scales will affect the degree of optimization to dynamic independence. The experiment finds that [[dynamic independence]] can only be achieved at certain scales, but not at other scales. Therefore, the choice of scale is also very important.

The paper gives an example of a linear dynamical system. Its dynamics is a vector autoregressive model. By using genetic algorithms to iteratively evolve different initial conditions, the degree of dynamic independence of the system can also gradually increase. At the same time, it is found that different coarse-graining scales will affect the degree of optimization to dynamic independence. The experiment finds that [[dynamic independence]] can only be achieved at certain scales, but not at other scales. Therefore, the choice of scale is also very important.

===Comparison of Several Causal Emergence Theories===

===Comparison of Several Causal Emergence Theories===