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The coarse-graining of this matrix is as follows: First, merge the first 7 states into a macroscopic state, which may be called A. And sum up the probability values in the first 7 columns of the first 7 rows in [math]f_m[/math] to obtain the probability of state transition from macroscopic state A to state A, and keep other values of the [math]f_m[/math] matrix unchanged. The new probability transition matrix after merging is shown in the right figure, denoted as [math]f_M[/math]. This is a definite macroscopic Markov transition matrix, that is, the future state of the system can be completely determined by the current state. At this time <math>EI(f_M)>EI(f_m)</math>, and causal emergence occurs in the system.

The coarse-graining of this matrix is as follows: First, merge the first 7 states into a macroscopic state, which may be called A. And sum up the probability values in the first 7 columns of the first 7 rows in [math]f_m[/math] to obtain the probability of state transition from macroscopic state A to state A, and keep other values of the [math]f_m[/math] matrix unchanged. The new probability transition matrix after merging is shown in the right figure, denoted as [math]f_M[/math]. This is a definite macroscopic Markov transition matrix, that is, the future state of the system can be completely determined by the current state. At this time <math>EI(f_M)>EI(f_m)</math>, and causal emergence occurs in the system.

[[文件:Causal Emergence in the State Space1.png|无|缩略图]]

[[文件:Causal emergence in the state space2.png|无|缩略图|600x600像素]]