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==Identification of causal emergence==

==Identification of causal emergence==

Some works on quantifying emergence through measures of causal effects and other information-theoretic indicators have been introduced previously. However, in practical applications, we can often only collect observational data and cannot obtain the true dynamics of the system. Therefore, identifying whether causal emergence has occurred in a system from observable data is a more important problem. The following describes two identification methods of causal emergence. The first is the approximate method based on [[Rosas's causal emergence theory]] (the method based on mutual information approximation and the method based on [[machine learning]]). The second is the [[neural information squeezer]] (NIS, NIS+) method proposed by Chinese scholars.

Some works on quantifying emergence through measures of causal effects and other information-theoretic indicators have been introduced previously. However, in practical applications, we often can only collect observational data and cannot obtain the true dynamics of the system. Therefore, identifying whether causal emergence has occurred in a system from observable data is a more important problem. The following section describes two identification methods of causal emergence. The first is the approximate method based on [[Rosas's causal emergence theory]] (the method based on mutual information approximation and the method based on [[machine learning]]). The second is the [[neural information squeezer]] (NIS, NIS+) method proposed by J.Zhang's group.

====Approximate method based on Rosas's causal emergence theory====

====Approximate method based on Rosas's causal emergence theory====

[[Rosas's causal emergence theory]] employs quantification method based on both [[synergistic information]] and [[unique information]]. The second method can bypass the combinatorial explosion problem of multivariate variables, but it depends on the coarse-graining method and the selection of macroscopic variable <math>V</math>. To solve this problem, the author gives two solutions. One is to specify a macroscopic state <math>V</math> by the researcher, and the other is a machine learning-based method that allows the system to automatically learn the macroscopic variable <math>V</math> by maximizing <math>\mathrm{\Psi}</math>.  The following sections introduce these two methods in detail:

[[Rosas's causal emergence theory]] employs quantification method based on both [[synergistic information]] and [[unique information]]. The second method can bypass the combinatorial explosion problem of multivariate variables, but it depends on the selection of macroscopic variable <math>V</math> or coarse-graining method. To solve this problem, the authors give two solutions. One is to specify a macroscopic variable <math>V</math>, and the other is a machine learning-based method that allows the system to automatically learn the macroscopic variable <math>V</math> by maximizing <math>\mathrm{\Psi}</math>.  The following sections introduce these two methods in detail:

=====Method based on mutual information approximation=====

=====Method based on mutual information approximation=====