Cause of Causal Emergence: Redistribution of Uncertainty and Its Critical Condition

It is crucial to choose the appropriate scale in order to build an effective and informative representation of a complex system. Scientists carefully choose the scales for their experiments to extract the variables that describe the causalities in the system. They have found that the coarse scale(macro) is sometimes more causal and informative than the numerous-parameter observations(micro). The phenomenon that the causality emerges by coarse-graining is called Causal Emergence(CE). Based on information theory, a number of recent works have quantitatively shown that CE indeed occurs while coarse-graining a micro model to the macro. However, the existing works have not discussed the question of why and when the CE occurs. We quantitatively analyze the redistribution of uncertainties for coarse-graining and suggest that the redistribution of uncertainties is the cause of causal emergence. We further analyze the thresholds that determine if CE occurs or not. From the regularity of the transition probability matrix(TPM) of discrete systems, the mathematical expressions of the model properties are derived. The values of thresholds for different operations are computed. The results provide the critical and specific conditions of CE as helpful suggestions for choosing the proper coarse-graining operation. The results also provide a new way to better understand the nature of causality and causal emergence.