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

=== 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 the strength of causal emergence than the original Markov chain. Here, the strength of causal emergence is measured by EI. 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 EI 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 EI to Markov chains, leading to the development of the causal emergence theory.

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 stronger causal effect than the original Markov chain. Here, the strength of causal effect is measured by EI. 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 EI 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 EI 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 ===