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Here <math>z\sim\mathcal{Ν}\left (0,I_{p - q}\right )[/math] is a <math>p - q</math>-dimensional random vector that obeys the standard normal distribution.

Here <math>z\sim\mathcal{Ν}\left (0,I_{p - q}\right )[/math] is a <math>p - q</math>-dimensional random vector that obeys the standard normal distribution.

However, if we directly optimize the dimension-averaged effective information, there will be certain difficulties. The article [46] does not directly optimize Equation {{EquationNote|1}}, but adopts a clever method. To solve this problem, the author divides the optimization process into two stages. The first stage is to minimize the microscopic state prediction error under the condition of a given macroscopic scale <math>q</math>, that is, <math>\min _{\phi, f_q, \phi^{\dagger}}\left\|\phi^{\dagger}(Y(t + 1)) - X_{t + 1}\right\|<\epsilon</math> and obtain the optimal macroscopic state dynamics <math>f_q^\ast[/math>; the second stage is to search for the hyperparameter <math>q</math> to maximize the effective information <math>\mathcal{J}[/math>, that is, <math>\max_{q}\mathcal{J}(f_{q}^\ast)</math>. Practice has proved that this method can effectively find macroscopic dynamics and coarse-graining functions, but it cannot truly maximize EI in advance.

However, if we directly optimize the dimension-averaged effective information, there will be certain difficulties. The article [46] does not directly optimize Equation {{EquationNote|1}}, but adopts a clever method. To solve this problem, the author divides the optimization process into two stages. The first stage is to minimize the microscopic state prediction error under the condition of a given macroscopic scale <math>q</math>, that is, <math>\min _{\phi, f_q, \phi^{\dagger}}\left\|\phi^{\dagger}(Y(t + 1)) - X_{t + 1}\right\|<\epsilon</math> and obtain the optimal macroscopic state dynamics [math]f_q^\ast[/math]; the second stage is to search for the hyperparameter <math>q</math> to maximize the effective information [math]\mathcal{J}[/math], that is, <math>\max_{q}\mathcal{J}(f_{q}^\ast)</math>. Practice has proved that this method can effectively find macroscopic dynamics and coarse-graining functions, but it cannot truly maximize EI in advance.

In addition to being able to automatically identify causal emergence based on time series data, this framework also has good theoretical properties. There are two important theorems:

In addition to being able to automatically identify causal emergence based on time series data, this framework also has good theoretical properties. There are two important theorems: