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</math>
</math>
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.