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However, if we directly optimize the dimension-averaged [[effective information]], there will be certain difficulties. The article <ref name="NIS" /> 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 EI, there will be certain difficulties. The article <ref name="NIS" /> 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 EI [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.

======Comparison with Classical Theories======

======Comparison with Classical Theories======

The [[NIS]] framework has many similarities with the [[computational mechanics]] framework mentioned in the previous sections. NIS can be regarded as an <math>\epsilon</math>-machine. The set of all historical processes <math>\overleftarrow{S}</math> in [[computational mechanics]] can be regarded as a set of microscopic states. All <math>R \in \mathcal{R}</math> represent macroscopic states. The function <math>\eta</math> acts as a coarse-graining function, while <math>\epsilon</math> represents an effective coarse-graining strategy. <math>T</math> corresponds to a effective macroscopic dynamic. The characteristic of minimum randomness index characterizes the determinism of macroscopic dynamics and can be replaced by [[effective information]] in causal emergence. At this point, the encoded macroscopic state converges to the effective state, which can be regarded as the [[causal state]] in computational mechanics.

The [[NIS]] framework has many similarities with the [[computational mechanics]] framework mentioned in the previous sections. NIS can be regarded as an <math>\epsilon</math>-machine. The set of all historical processes <math>\overleftarrow{S}</math> in [[computational mechanics]] can be regarded as a set of microscopic states. All <math>R \in \mathcal{R}</math> represent macroscopic states. The function <math>\eta</math> acts as a coarse-graining function, while <math>\epsilon</math> represents an effective coarse-graining strategy. <math>T</math> corresponds to a effective macroscopic dynamic. The characteristic of minimum randomness index characterizes the determinism of macroscopic dynamics and can be replaced by EI in causal emergence. At this point, the encoded macroscopic state converges to the effective state, which can be regarded as the [[causal state]] in computational mechanics.