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=== 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]] theory to define and quantify causal emergence from an [[information theory|information theory]] perspective. [[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 the corresponding information atoms. This method is based on the [[partial information decomposition]](PID) theory 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> and redefine causal emergence by synergistic information. And this information can further be decomposed into [[causal decoupling]] and [[downward causation]] .

=== Recent work ===

=== Recent work ===