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Furthermore, <math>CE</math> can be decomposed into the sum of two terms, <math>CE = \Delta I_{Eff} + \Delta I_{Size}</math>, where <math>\Delta I_{Eff} = (Eff(f_M) - Eff(f_m))\cdot \log_2(M)</math>, <math>\Delta I_{Size} = Eff(f_m)\cdot(\log_2(M) - \log_2(m))</math>. Here, <math>M</math> and <math>m</math> represent the sizes of the macroscopic and microscopic states respectively. Because coarse-graining reduces the macroscopic state, <math>\Delta I_{Size}</math> is necessarily less than 0. Causal emergence occurs if the increase in <math>\Delta I_{Eff}</math> exceeds the decrease in <math>\Delta I_{Size}</math>.

Furthermore, <math>CE</math> can be decomposed into the sum of two terms, <math>CE = \Delta I_{Eff} + \Delta I_{Size}</math>, where <math>\Delta I_{Eff} = (Eff(f_M) - Eff(f_m))\cdot \log_2(M)</math>, <math>\Delta I_{Size} = Eff(f_m)\cdot(\log_2(M) - \log_2(m))</math>. Here, <math>M</math> and <math>m</math> represent the sizes of the macroscopic and microscopic states respectively. Because coarse-graining reduces the macroscopic state, <math>\Delta I_{Size}</math> is necessarily less than 0. Causal emergence occurs if the increase in <math>\Delta I_{Eff}</math> exceeds the decrease in <math>\Delta I_{Size}</math>.

=====Markov Chain Example=====

=====Markov chain example=====

In the literature <ref name=":0"/>, Hoel gives an example of a state transition matrix ([math]f_m[/math]) of a Markov chain with 8 states, as shown in the left figure below. Among them, the first 7 states transfer with equal probability, and the last state is independent and can only transition to its own state.

In the literature <ref name=":0"/>, Hoel gives an example of a state transition matrix ([math]f_m[/math]) of a Markov chain with 8 states, as shown in the left figure below. Among them, the first 7 states transfer with equal probability, and the last state is independent and can only transition to its own state.