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→Causal Emergence Measurement
Here <math>CE</math> is the causal emergence intensity. If the effective information of macroscopic dynamics is greater than that of microscopic dynamics (that is, <math>CE>0</math>), then we consider that macroscopic dynamics has causal emergence characteristics on the basis of this coarse-graining.
Here <math>CE</math> is the causal emergence intensity. If the effective information of macroscopic dynamics is greater than that of microscopic dynamics (that is, <math>CE>0</math>), then we consider that macroscopic dynamics has causal emergence characteristics on the basis of this coarse-graining.
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. Since the macroscopic state after coarse-graining is reduced, <math>\Delta I_{Size}</math> must be less than 0. In order for causal emergence to occur, the increase in <math>\Delta I_{Eff}</math> must be greater than the decrease in <math>\Delta I_{Size}</math>.
=====Markov Chain Example=====
=====Markov Chain Example=====