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</math>

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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 gaussian random vector.

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.

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}} directly, but adopts a trick. To solve this problem, the authors separate 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>. It has been 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:

'''Theorem 1''': The [[information bottleneck]] of the neural information squeezer. That is, for any bijection <math>\mathrm{\psi}</math>, projection <math>\chi</math>, macroscopic dynamics <math>f</math> and Gaussian noise <math>z_{p - q}\sim\mathcal{Ν}\left (0,I_{p - q}\right )</math>,

'''Theorem 1''': The [[information bottleneck]] of the neural information squeezer, that is, for any bijection <math>\mathrm{\psi}</math>, projection <math>\chi</math>, macroscopic dynamics <math>f</math> and Gaussian noise <math>z_{p - q}\sim\mathcal{Ν}\left (0,I_{p - q}\right )</math>,

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======Computational examples======

======Computational examples======

The author of NIS conducted experiments in the [[spring oscillator model]], and the results are shown in the following figure. Figure a demostrates that the results for the next moment align linearly with the iterative macroscopic dynamics, validating the model's effectiveness. Figure b illustrates that the learned and real dynamics align, further validating the model. Figure c illustrates the model's accuracy in multi-step predictions, as the predicted and real curves are closely aligned. Figure d shows the degree of causal emergence at different scales. Causal emergence is most significant at scale is 2, consistent with the real spring oscillator model, where only two states (position and velocity) are needed to describe the system.

The authors of NIS conducted experiments in the [[spring oscillator model]], and the results are shown in the following figure. Figure a demostrates that the results for the next moment align linearly with the iterative macroscopic dynamics, validating the model's effectiveness. Figure b illustrates that the learned and real dynamics align, further validating the model. Figure c illustrates the model's accuracy in multi-step predictions, as the predicted and real curves are closely aligned. Figure d shows the degree of causal emergence at different scales. Causal emergence is most significant at scale is 2, consistent with the real spring oscillator model, where only two states (position and velocity) are needed to describe the system.

[[文件:弹簧振子模型1.png|居左|600x600像素|替代=弹簧振子模型1|弹簧振子模型]]

[[文件:弹簧振子模型1.png|居左|600x600像素|替代=弹簧振子模型1|弹簧振子模型]]