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======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> cts as a coarse-graining function, while <math>\epsilon</math> represents an effective coarse-graining strategy. <math>T</math> corresponds to effective macroscopic dynamics. 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 effective macroscopic dynamics. 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.

At the same time, the [[NIS]] framework also has similarities with the G-emergence theory mentioned earlier. For example, [[NIS]] also adopts the idea of [[Granger causality]]: optimizing the effective macroscopic state by predicting the microscopic state at the next time step. However, there are several obvious differences between these two frameworks: a) In the G-emergence theory, the macroscopic state needs to be manually selected, while in [[NIS]], the macroscopic state is obtained by automatically optimizing the coarse-graining strategy; b) NIS uses neural networks to predict future states, whereas G-emergence relies on autoregressive techniques to fit the data.

At the same time, the [[NIS]] framework also has similarities with the G-emergence theory mentioned earlier. For example, [[NIS]] also adopts the idea of [[Granger causality]]: optimizing the effective macroscopic state by predicting the microscopic state at the next time step. However, there are several obvious differences between these two frameworks: a) In the G-emergence theory, the macroscopic state needs to be manually selected, while in [[NIS]], the macroscopic state is obtained by automatically optimizing the coarse-graining strategy; b) NIS uses neural networks to predict future states, whereas G-emergence relies on autoregressive techniques to fit the data.

======Computational Examples======

======Computational Examples======