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'''Theorem 2''': For a trained model, <math>I\left(X_t;{\hat{X}}_{t + 1}\right)\approx I\left(X_t;X_{t + 1}\right)</math>. Therefore, combining Theorem 1 and Theorem 2, we can obtain for a trained model:

'''Theorem 2''': For a trained model, <math>I\left(X_t;{\hat{X}}_{t + 1}\right)\approx I\left(X_t;X_{t + 1}\right)</math>.  

 

Therefore, combining Theorem 1 and Theorem 2, we can obtain for a trained model:

======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 microscopic state. 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> 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.