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Furthermore, in the paper <ref name="Chvykov_causal_geometry">{{cite journal|author1=Chvykov P|author2=Hoel E.|title=Causal Geometry|journal=Entropy|year=2021|volume=23|issue=1|page=24|url=https://doi.org/10.3390/e2}}</ref>, Hoel et al. proposed the theoretical framework of [[causal geometry]],  with the aim of generalizing causal emergence theory to function mappings  and dynamical systems with continuous states. This article defines <math>EI</math> for [[random function mapping]], and also introduces the concepts of intervention noise and [[causal geometry]], and compares this concept  with [[information geometry]]. [[Liu Kaiwei]] et al.<ref name="An_exact_theory_of_causal_emergence">{{cite journal|author1=Liu K|author2=Yuan B|author3=Zhang J|title=An Exact Theory of Causal Emergence for Linear Stochastic Iteration Systems|journal=Entropy|year=2024|volume=26|issue=8|page=618|url=https://arxiv.org/pdf/2405.09207}}</ref> developed an exact analytical framework for causal emergence in [[random iterative dynamical systems]].

Furthermore, in the paper <ref name="Chvykov_causal_geometry">{{cite journal|author1=Chvykov P|author2=Hoel E.|title=Causal Geometry|journal=Entropy|year=2021|volume=23|issue=1|page=24|url=https://doi.org/10.3390/e2}}</ref>, Hoel et al. proposed the theoretical framework of [[causal geometry]],  with the aim of generalizing causal emergence theory to function mappings  and dynamical systems with continuous states. This article defines <math>EI</math> for [[random function mapping]], and also introduces the concepts of intervention noise and [[causal geometry]], and compares this concept  with [[information geometry]]. [[Liu Kaiwei]] et al.<ref name="An_exact_theory_of_causal_emergence">{{cite journal|author1=Liu K|author2=Yuan B|author3=Zhang J|title=An Exact Theory of Causal Emergence for Linear Stochastic Iteration Systems|journal=Entropy|year=2024|volume=26|issue=8|page=618|url=https://arxiv.org/pdf/2405.09207}}</ref> developed an exact analytical framework for causal emergence in [[random iterative dynamical systems]].

====Rosas's Causal Emergence Theory====

====Rosas's causal emergence theory====

Rosas et al. <ref name=":5" /> propose from the perspective of [[information decomposition]] theory a method for defining causal emergence based on [[integrated information decomposition]], and further divide causal emergence into two parts: [[causal decoupling]] (Causal Decoupling) and [[downward causation]] (Downward Causation). Causal decoupling represents the causal effect of the macroscopic state at the current moment on the  macroscopic state at the next moment. Downward causation represents the causal effect of the macroscopic state at the previous moment on the microscopic state at the next moment. The schematic diagrams of causal  decoupling and downward causation are shown in the following figure. The microscopic state input is <math>X_t\ (X_t^1,X_t^2,…,X_t^n ) </math>, while the macroscopic state, represented as <math>V_t </math>, is derived through the coarse-graining of the microscopic state variable <math>X_t </math>. Consequently, it serves as a supervenient feature of <math>X_t </math>. Additionally, <math>X_{t + 1} </math> and <math>V_{t + 1} </math> denote the microscopic and macroscopic states at the subsequent moment, respectively.  

Rosas et al. <ref name=":5" /> propose from the perspective of [[information decomposition]] theory a method for defining causal emergence based on [[integrated information decomposition]], and further divide causal emergence into two parts: [[causal decoupling]] (Causal Decoupling) and [[downward causation]] (Downward Causation). Causal decoupling represents the causal effect of the macroscopic state at the current moment on the  macroscopic state at the next moment. Downward causation represents the causal effect of the macroscopic state at the previous moment on the microscopic state at the next moment. The schematic diagrams of causal  decoupling and downward causation are shown in the following figure. The microscopic state input is <math>X_t\ (X_t^1,X_t^2,…,X_t^n ) </math>, while the macroscopic state, represented as <math>V_t </math>, is derived through the coarse-graining of the microscopic state variable <math>X_t </math>. Consequently, it serves as a supervenient feature of <math>X_t </math>. Additionally, <math>X_{t + 1} </math> and <math>V_{t + 1} </math> denote the microscopic and macroscopic states at the subsequent moment, respectively.  

=====Partial Information Decomposition=====

=====Partial information decomposition=====

This method is based on the nonnegative decomposition of multivariate information theory proposed by Williams and Beer <ref name=":16" />. This paper uses [[partial information decomposition]] (PID) to decompose the [[mutual information]] between microstates and macrostates.

This method is based on the nonnegative decomposition of multivariate information theory proposed by Williams and Beer <ref name=":16" />. This paper uses [[partial information decomposition]] (PID) to decompose the [[mutual information]] between microstates and macrostates.

Among them, <math>Red(X^1,X^2;V)</math> represents [[redundant information]], which refers to the information repeatedly provided by two microstates <math>X^1</math> and <math>X^2</math> to the macrostate <math>V</math>; <math>Un(X^1;V│X^2)</math> and <math>Un(X^2;V│X^1)</math> represent [[unique information]], which refers to the information provided by each microstate variable alone to the macrostate; <math>Syn(X^1,X^2;V)</math> represents [[synergistic information]], which refers to the information provided by all microstates <math>X</math> jointly to the macrostate <math>V</math>.

Among them, <math>Red(X^1,X^2;V)</math> represents [[redundant information]], which refers to the information repeatedly provided by two microstates <math>X^1</math> and <math>X^2</math> to the macrostate <math>V</math>; <math>Un(X^1;V│X^2)</math> and <math>Un(X^2;V│X^1)</math> represent [[unique information]], which refers to the information provided by each microstate variable alone to the macrostate; <math>Syn(X^1,X^2;V)</math> represents [[synergistic information]], which refers to the information provided by all microstates <math>X</math> jointly to the macrostate <math>V</math>.

=====Definition of Causal Emergence=====

=====Definition of causal emergence=====

However, the PID framework is limited to decomposing the mutual information between multiple source variables and one target variable. Rosas extended this framework and proposed the integrated information decomposition method <math>\Phi ID</math><ref name=":18" /> to handle the mutual information between multiple source variables and multiple target variables. It can also be used to decompose the mutual information between different moments. Based on the decomposed information, the author proposed two methods for defining causal emergence:

However, the PID framework is limited to decomposing the mutual information between multiple source variables and one target variable. Rosas extended this framework and proposed the integrated information decomposition method <math>\Phi ID</math><ref name=":18" /> to handle the mutual information between multiple source variables and multiple target variables. It can also be used to decompose the mutual information between different moments. Based on the decomposed information, the author proposed two methods for defining causal emergence:

It should be noted that for the first method to judge the occurrence of causal emergence, its accuracy depends on the selection of the macroscopic state <math>V_t</math>. The first method is the lower bound of the second method. This is because <math>Syn(X_t;X_{t+1}\ ) ≥ Un(V_t;X_{t+1}| X_t\ )</math> always holds. So, if <math>Un(V_t;X_{t + 1}|X_t)</math> is greater than 0, then causal emergence occurs in the system. However, the selection of Vt often requires predefining a coarse-graining function, so the limitations of the [[Erik Hoel causal emergence theory]] cannot be avoided. Another natural idea is to use the second method to judge the occurrence of causal emergence with the help of synergistic information. However, the calculation of synergistic information is very difficult and there is a combinatorial explosion problem. Therefore, the calculation based on synergistic information in the second method is often infeasible. In short, both quantitative characterization methods of causal emergence have some weaknesses, so a more reasonable quantification method needs to be proposed.

It should be noted that for the first method to judge the occurrence of causal emergence, its accuracy depends on the selection of the macroscopic state <math>V_t</math>. The first method is the lower bound of the second method. This is because <math>Syn(X_t;X_{t+1}\ ) ≥ Un(V_t;X_{t+1}| X_t\ )</math> always holds. So, if <math>Un(V_t;X_{t + 1}|X_t)</math> is greater than 0, then causal emergence occurs in the system. However, the selection of Vt often requires predefining a coarse-graining function, so the limitations of the [[Erik Hoel causal emergence theory]] cannot be avoided. Another natural idea is to use the second method to judge the occurrence of causal emergence with the help of synergistic information. However, the calculation of synergistic information is very difficult and there is a combinatorial explosion problem. Therefore, the calculation based on synergistic information in the second method is often infeasible. In short, both quantitative characterization methods of causal emergence have some weaknesses, so a more reasonable quantification method needs to be proposed.

=====Specific Example=====

=====Specific example=====

[[文件:Examples of causal decoupling and downward causation.png|居左]]

[[文件:Examples of causal decoupling and downward causation.png|居左]]

Therefore, the macroscopic state of this process can be regarded as the parity of the overall parity of the sequence, and the probability distribution of this parity is the result of the XOR calculation of the microstate. <math>x_{t + 1}^1</math> is a special microstate that always remains consistent with the macroscopic state of the sequence at the previous moment. Therefore, when only the first item in the second judgment condition is satisfied, the downward causation of the system occurs. When only the second item is satisfied, the causal decoupling of the system occurs. When both items are satisfied simultaneously, it is said that causal emergence occurs in the system.

Therefore, the macroscopic state of this process can be regarded as the parity of the overall parity of the sequence, and the probability distribution of this parity is the result of the XOR calculation of the microstate. <math>x_{t + 1}^1</math> is a special microstate that always remains consistent with the macroscopic state of the sequence at the previous moment. Therefore, when only the first item in the second judgment condition is satisfied, the downward causation of the system occurs. When only the second item is satisfied, the causal decoupling of the system occurs. When both items are satisfied simultaneously, it is said that causal emergence occurs in the system.

====Causal Emergence Theory Based on Singular Value Decomposition====

====Causal Emergence Theory Based on Singular Value Decomposition====