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=== Causal emergence theory based on effective information ===

=== Causal emergence theory based on effective information ===

In history, the first relatively complete and explicit quantitative theory that uses causality to define emergence is the causal emergence theory proposed by Erik Hoel, Larissa Albantakis, and Giulio Tononi [1][2]. This theory defines so-called causal emergence for Markov chains as the phenomenon that the coarsened Markov chain has a greater causal effect strength than the original Markov chain. Here, the causal effect strength is measured by effective information. This indicator is a modification of the mutual information indicator. The main difference is that the state variable at time $t$ is intervened by do-intervention and transformed into a uniform distribution (or maximum entropy distribution). The effective information indicator was proposed by Giulio Tononi as early as 2003 when studying integrated information theory. As Giulio Tononi's student, Erik Hoel applied effective information to Markov chains and proposed the causal emergence theory based on effective information.

In history, the first relatively complete and explicit quantitative theory that uses causality to define emergence is the causal emergence theory proposed by Erik Hoel, Larissa Albantakis, and Giulio Tononi <ref name=":0" /><ref name=":1" />. This theory defines so-called causal emergence for Markov chains as the phenomenon that the coarsened Markov chain has a greater causal effect strength than the original Markov chain. Here, the causal effect strength is measured by effective information. This indicator is a modification of the mutual information indicator. The main difference is that the state variable at time $t$ is intervened by do-intervention and transformed into a uniform distribution (or maximum entropy distribution). The effective information indicator was proposed by Giulio Tononi as early as 2003 when studying integrated information theory. As Giulio Tononi's student, Erik Hoel applied effective information to Markov chains and proposed the causal emergence theory based on effective information.

=== Causal Emergence Theory Based on Information Decomposition ===

=== Causal Emergence Theory Based on Information Decomposition ===