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==Causal Emergence==

==Causal Emergence==

With the metric of Effective Information (EI) in place, we can now discuss causal emergence in Markov chains. For a Markov chain, an observer can adopt a multi-scale perspective to distinguish between micro and macro levels. First, the original Markov transition matrix P defines the micro-level dynamics. Second, after a [[coarse-graining for Markov chain]] that maps microstates into macrostates (typically by grouping microstates together), the observer can obtain a macro-level transition matrix P′, which describes the transition probabilities between macrostates. We can compute EI for both dynamics. If the macro-level EI is greater than the micro-level EI, we say that the system exhibits causal emergence.

With the metric of Effective Information (EI) in place, we can now discuss causal emergence in Markov chains. For a Markov chain, an observer can adopt a multi-scale perspective to distinguish between micro and macro levels. First, the original Markov transition matrix P defines the micro-level dynamics. Second, after [[coarse-graining the Markov chain]] that maps microstates into macrostates (typically by grouping microstates together), the observer can obtain a macro-level transition matrix P′, which describes the transition probabilities between macrostates. We can compute EI for both dynamics. If the macro-level EI is greater than the micro-level EI, we say that the system exhibits causal emergence.

[[文件:CE.png|替代=因果涌现示意图|500x500像素|链接=https://wiki.swarma.org/index.php/%E6%96%87%E4%BB%B6:CE.png]]

[[文件:CE.png|替代=因果涌现示意图|500x500像素|链接=https://wiki.swarma.org/index.php/%E6%96%87%E4%BB%B6:CE.png]]

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Here, [math]P[/math] is the microstate Markov transition matrix with dimensions[math]N\times N[/math], where N is the number of microstates. [math]P'[/math] is the macro-state transition matrix obtained after the coarse-graining of [math]P[/math], with dimensions [math]M\times M[/math], where [math]M<N[/math] represents the number of macrostates.

Here, [math]P[/math] is the microstate Markov transition matrix with dimensions [math]N\times N[/math], where N is the number of microstates. [math]P'[/math] is the macro-state transition matrix obtained after the coarse-graining of [math]P[/math], with dimensions [math]M\times M[/math], where [math]M<N[/math] represents the number of macrostates.

The process of coarse-graining a Markov transition matrix typically involves two steps: 1) grouping N microstates into M macrostates, and 2) reducing the Markov transition matrix accordingly. For more details on the specific methods for coarse-graining a Markov chain, refer to the topic of [[Markov Chain Coarse-graining]].

The process of coarse-graining a Markov transition matrix typically involves two steps: 1) grouping N microstates into M macrostates, and 2) reducing the Markov transition matrix accordingly. For more details on the specific methods for coarse-graining a Markov chain, refer to the topic of [[Markov Chain Coarse-graining]].

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In this example, the microstate transition matrix is a 4*4 matrix, where the first three states transition to each other with a probability of 1/3. This leads to a transition matrix with relatively low determinism, and thus, the EI is not very high, with a value of 0.81. However, when we coarse-grain this matrix—merging the first three states into one macrostate a, and the last state becomes another macrostate b—all transitions between the original three microstates now become internal transitions within macrostate a. Thus, the transition probability matrix becomes [math]P_M[/math], with an EI of 1. In this case, the [[Causal Emergence]] can be measured as:

In this example, the microstate transition matrix [math]P_m[/math] is a 4*4 matrix, where the first three states transition to each other with a probability of 1/3. This leads to a transition matrix with relatively low determinism, and thus, the EI is not very high, with a value of 0.81. However, when we coarse-grain this matrix — merging the first three states into one macrostate a, and the last state becomes another macrostate b — all transitions between the original three microstates now become internal transitions between macrostate a and a. Thus, the transition probability matrix becomes [math]P_M[/math], with an EI of 1. In this case, the [[Causal Emergence]] can be measured as:

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