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→Markov Chain Example
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 space3.png|无|缩略图|600x600像素]]
[[文件:Causal emergence in the state space3.png|居左|缩略图|600x600像素]]
However, for more general Markov chains and more general state groupings, this simple operation of averaging probabilities is not always feasible. This is because the merged probability transition matrix may not satisfy the conditions of a Markov chain (such as the rows of the matrix not satisfying the normalization condition, or the element values exceeding the range of [0, 1]). For what kind of Markov chains and state groupings can a feasible macroscopic Markov chain be obtained, please refer to the section “Reduction of Markov Chains” later in this entry, or see the entry [[Coarse-graining of Markov Chains]].
However, for more general Markov chains and more general state groupings, this simple operation of averaging probabilities is not always feasible. This is because the merged probability transition matrix may not satisfy the conditions of a Markov chain (such as the rows of the matrix not satisfying the normalization condition, or the element values exceeding the range of [0, 1]). For what kind of Markov chains and state groupings can a feasible macroscopic Markov chain be obtained, please refer to the section “Reduction of Markov Chains” later in this entry, or see the entry [[Coarse-graining of Markov Chains]].
=====Boolean Network Example=====
=====Boolean Network Example=====