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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]] <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 <math>t</math> 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.

The first relatively comprehensive quantitative theory using causality to define emergence is the causal emergence theory by [[Erik Hoel]], [[Larissa Albantakis]] and [[Giulio Tononi]] <ref name=":0" /><ref name=":1" />. This theory defines causal emergence of 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 key distinction is the use of [[do-intervention]] to transform the state variable at time <math>t</math> 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]]. Building on  [[Giulio Tononi]]'s work, [[Erik Hoe]]l applied effective information to Markov chains, leading to the development of the causal emergence theory.

 

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

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