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2) The second method bypasses the selection of a specific macroscopic state <math>V_t</math>. It defines causal emergence solely 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>. It defines causal emergence solely 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, 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 [math]V_t[/math] often requires pre-defining 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=====