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.

本文讨论了粗粒化对于因果性和 Causal Emergence 的影响，提出了不确定性的重新分配是 Causal Emergence 的原因，并分析了决定 Causal Emergence 的阈值，进一步提供了选择正确粗粒化操作的有价值的建议。