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→Markov Chain Example
The coarse-graining of this matrix is as follows: First, merge the first 7 states into a macroscopic state, which may be called A. And sum up the probability values in the first 7 columns of the first 7 rows in [math]f_m[/math] to obtain the probability of state transition from macroscopic state A to state A, and keep other values of the [math]f_m[/math] matrix unchanged. The new probability transition matrix after merging is shown in the right figure, denoted as [math]f_M[/math]. This is a definite macroscopic Markov transition matrix, that is, the future state of the system can be completely determined by the current state. At this time <math>EI(f_M)>EI(f_m)</math>, and causal emergence occurs in the system.
The coarse-graining of this matrix is as follows: First, merge the first 7 states into a macroscopic state, which may be called A. And sum up the probability values in the first 7 columns of the first 7 rows in [math]f_m[/math] to obtain the probability of state transition from macroscopic state A to state A, and keep other values of the [math]f_m[/math] matrix unchanged. The new probability transition matrix after merging is shown in the right figure, denoted as [math]f_M[/math]. This is a definite macroscopic Markov transition matrix, that is, the future state of the system can be completely determined by the current state. At this time <math>EI(f_M)>EI(f_m)</math>, and causal emergence occurs in the system.
[[文件:状态空间中的因果涌现1.png|无|缩略图]]
=====Boolean Network Example=====
=====Boolean Network Example=====
Through comparison, we find that the effective information of macroscopic dynamics is greater than that of microscopic dynamics <math>EI(f_M\ )>EI(f_m\ ) </math>. Causal emergence occurs in this system.
Through comparison, we find that the effective information of macroscopic dynamics is greater than that of microscopic dynamics <math>EI(f_M\ )>EI(f_m\ ) </math>. Causal emergence occurs in this system.
[[文件:含有4个节点的布尔网络.png|无|缩略图]]
=====Causal Emergence in Continuous Variables=====
=====Causal Emergence in Continuous Variables=====
Furthermore, in the paper "xxx", Hoel et al. proposed the theoretical framework of causal geometry, trying to generalize the causal emergence theory to function mappings and dynamical systems with continuous states. This article defines <math>EI</math> for random function mapping, and also introduces the concepts of intervention noise and causal geometry, and compares and analogizes this concept with information geometry. Liu Kaiwei et al. further gave an exact analytical causal emergence theory for random iterative dynamical systems.
Furthermore, in the paper "xxx", Hoel et al. proposed the theoretical framework of causal geometry, trying to generalize the causal emergence theory to function mappings and dynamical systems with continuous states. This article defines <math>EI</math> for random function mapping, and also introduces the concepts of intervention noise and causal geometry, and compares and analogizes this concept with information geometry. Liu Kaiwei et al. further gave an exact analytical causal emergence theory for random iterative dynamical systems.
====Rosas's Causal Emergence Theory====
Rosas et al. [5] From the perspective of [[information decomposition]] theory, propose a method for defining causal emergence based on [[integrated information decomposition]], and further divide causal emergence into two parts: [[causal decoupling]] (Causal Decoupling) and [[downward causation]] (Downward Causation). Among them, causal decoupling represents the causal effect of the macroscopic state at the current moment on the macroscopic state at the next moment, and downward causation represents the causal effect of the macroscopic state at the previous moment on the microscopic state at the next moment. The schematic diagrams of causal decoupling and downward causation are shown in the following figure. The microscopic state input is <math>X_t\ (X_t^1,X_t^2,…,X_t^n ) </math>, and the macroscopic state is <math>V_t </math>, which is obtained by coarse-graining the microscopic state variable <math>X_t </math>, so it is a supervenient feature of <math>X_t </math>, <math>X_{t + 1} </math> and <math>V_{t + 1} </math> represent the microscopic and macroscopic states at the next moment respectively.
[[文件:向下因果与因果解耦2.png|居左|300x300像素|因果解耦与向下因果]]