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.

翻译：为了构建一个高效且信息丰富的复杂系统表征，选择合适的尺度至关重要。科学家们精心选择实验尺度，以提取描述系统中因果关系的变量。他们发现，粗尺度（宏观）有时比多参数观测（微观）更具因果性和信息量。这种通过粗粒化而使因果关系涌现的现象被称为因果涌现（Causal Emergence, CE）。基于信息论，近年来的多项工作定量表明，在对微观模型进行粗粒化到宏观时，因果涌现确实会发生。然而，现有工作尚未讨论因果涌现为何及何时发生的问题。我们定量分析了粗粒化过程中不确定性的重新分布，并提出不确定性的重新分布是因果涌现的成因。我们进一步分析了决定因果涌现是否发生的阈值。基于离散系统的转移概率矩阵（Transition Probability Matrix, TPM）的规律性，推导出了模型性质的数学表达式，并计算了不同操作对应的阈值取值。这些结果为选择恰当的粗粒化操作提供了有助于因果涌现的临界与具体条件，同时也为更好地理解因果关系和因果涌现的本质提供了新的途径。