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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 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.

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

=== Causal emergence theory based on information decomposition ===

Rosas et al. (2020)<ref name=":5" /> proposed a method using [[information decomposition]] to define and quantify causal emergence from an [[information theory|information theory]] perspective, through [[synergistic information]] or [[redundant information]]. [[Information decomposition]] is a new method to analyze the complex interrelationships of various variables in [[complex systems]]. Information decomposition represents each piece of partial information as an [[information atom]]. Partial information is projected onto a information atom using an [[information lattice diagram]]. Both synergistic information and redundant information can be represented by corresponding information atoms. This method is based on the [[non-negative decomposition theory of multivariate information]] proposed by Williams and Beer <ref name=":16">Williams P L, Beer R D. Nonnegative decomposition of multivariate information[J]. arXiv preprint arXiv:10042515, 2010.</ref>. The paper uses [[partial information decomposition]] (PID) to break down the mutual information between microstates and macrostates. However, the PID framework can only decompose the mutual information between multiple source variables and one target variable. Rosas extended this framework and proposed the integrated information decomposition method <math>\Phi ID</math> <ref name=":18">P. A. Mediano, F. Rosas, R. L. Carhart-Harris, A. K. Seth, A. B. Barrett, Beyond integrated information: A taxonomy of information dynamics phenomena, arXiv preprint arXiv:1909.02297 (2019).</ref>.

Rosas et al. (2020)<ref name=":5" /> proposed a method using [[information decomposition]] to define and quantify causal emergence from an [[information theory|information theory]] perspective, through [[synergistic information]] or [[redundant information]]. [[Information decomposition]] is a new method to analyze the complex interrelationships of various variables in [[complex systems]]. Information decomposition represents each piece of partial information as an [[information atom]]. Partial information is projected onto a information atom using an [[information lattice diagram]]. Both synergistic information and redundant information can be represented by corresponding information atoms. This method is based on the [[non-negative decomposition theory of multivariate information]] proposed by Williams and Beer <ref name=":16">Williams P L, Beer R D. Nonnegative decomposition of multivariate information[J]. arXiv preprint arXiv:10042515, 2010.</ref>. The paper uses [[partial information decomposition]] (PID) to break down the mutual information between microstates and macrostates. However, the PID framework can only decompose the mutual information between multiple source variables and one target variable. Rosas extended this framework and proposed the integrated information decomposition method <math>\Phi ID</math> <ref name=":18">P. A. Mediano, F. Rosas, R. L. Carhart-Harris, A. K. Seth, A. B. Barrett, Beyond integrated information: A taxonomy of information dynamics phenomena, arXiv preprint arXiv:1909.02297 (2019).</ref>.

=== Recent Work ===

=== Recent Work ===