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However, the PID framework can only decompose 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>[45] 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 definition methods of causal emergence:

However, the PID framework can only decompose 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>[45] 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 definition methods of causal emergence:

1) When the unique information <math>Un(V_t;X_{t + 1}|X_t^1,\ldots,X_t^n\)>0</math>, it means that the macroscopic state <math>V_t</math> at the current moment can provide more information to the overall system <math>X_{t + 1}</math> at the next moment than the microscopic state <math>X_t</math> at the current moment. At this time, there is causal emergence in the system;

1) When the unique information <math>Un(V_t;X_{t+1}| X_t^1,\ldots,X_t^n\ )>0 </math>, it means that the macroscopic state <math>V_t</math> at the current moment can provide more information to the overall system <math>X_{t + 1}</math> at the next moment than the microscopic state <math>X_t</math> at the current moment. At this time, there is causal emergence in the system;

2) The second method bypasses the selection of a specific macroscopic state <math>V_t</math>, and defines causal emergence only based on the synergistic information between the microscopic state <math>X_t</math> and the microscopic state <math>X_{t + 1}</math> at the next moment of the system. When the synergistic information <math>Syn(X_t^1,…,X_t^n;X_{t + 1}^1,…,X_{t + 1}^n)>0</math>, causal emergence occurs in the system.

2) The second method bypasses the selection of a specific macroscopic state <math>V_t</math>, and defines causal emergence only based on the synergistic information between the microscopic state <math>X_t</math> and the microscopic state <math>X_{t + 1}</math> at the next moment of the system. When the synergistic information <math>Syn(X_t^1,…,X_t^n;X_{t + 1}^1,…,X_{t + 1}^n)>0</math>, causal emergence occurs in the system.

It should be noted that for the first method to judge the occurrence of causal emergence, it 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 <math>V_t</math> 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, it 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 <math>V_t</math> 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=====