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These early studies reveal a deep and inherent connection between emergence and causality.

These early studies reveal a deep and inherent connection between emergence and causality.

==== Causality and its measurement ====

==== Causality and its measurement ====

The causal emergence framework has many similarities with computational mechanics. All historical processes <math>\overleftarrow{s}</math> can be regarded as microscopic states. All <math>R \in \mathcal{R}</math> correspond to macroscopic states. The function <math>\eta</math> can be understood as a possible coarse-graining function. The causal state <math>\epsilon \left ( \overleftarrow{s} \right )</math> is a special state that can at least have the same predictive power as the microscopic state <math>\overleftarrow{s}</math>. Therefore, <math>\epsilon</math> can be understood as an effective [[coarse-graining]] strategy. Causal transfer <math>T</math> corresponds to effective macroscopic dynamics. The characteristic of minimum randomness characterizes the determinism of macroscopic dynamics and can be measured by [[effective information]] in causal emergence.

The causal emergence framework has many similarities with computational mechanics. All historical processes <math>\overleftarrow{s}</math> can be regarded as microscopic states. All <math>R \in \mathcal{R}</math> correspond to macroscopic states. The function <math>\eta</math> can be understood as a possible coarse-graining function. The causal state <math>\epsilon \left ( \overleftarrow{s} \right )</math> is a special state that can at least have the same predictive power as the microscopic state <math>\overleftarrow{s}</math>. Therefore, <math>\epsilon</math> can be understood as an effective [[coarse-graining]] strategy. Causal transfer <math>T</math> corresponds to effective macroscopic dynamics. The characteristic of minimum randomness characterizes the determinism of macroscopic dynamics and can be measured by [[effective information]] in causal emergence.

==== G-emergence ====

==== G-emergence ====

The causal emergence framework also has similarities with the aforementioned G-emergence. The macroscopic states of both methods need to be manually selected. In addition, it should be noted that some of the above methods for quantifying emergence often do not consider true interventionist causality.

The causal emergence framework also has similarities with the aforementioned G-emergence. The macroscopic states of both methods need to be manually selected. In addition, it should be noted that some of the above methods for quantifying emergence often do not consider true interventionist causality.

==== Other theories for quantitatively characterizing emergence ====

==== Other theories for quantitatively characterizing emergence ====

Another approach defines emergence as  "the whole being greater than the sum of its parts" <ref>Teo, Y.M.; Luong, B.L.; Szabo, C. Formalization of emergence in multi-agent systems. In Proceedings of the 1st ACM SIGSIM Conference on Principles of Advanced Discrete Simulation, Montreal, QC, Canada, 19–22 May 2013; pp. 231–240. </ref><ref>Szabo, C.; Teo, Y.M. Formalization of weak emergence in multiagent systems. ACM Trans. Model. Comput. Simul. (TOMACS) 2015, 26, 1–25. [CrossRef] </ref>. This approach focuses on interaction rules and the agent states rather than statistical measures of the system as a whole. Specifically, this measure consists of two subtracting terms. The first term represents the system's collective state, while the second term is the sum of its individual components. This measure emphasizes that emergence arises from the interactions and collective behavior of the system.

Another approach defines emergence as  "the whole being greater than the sum of its parts" <ref>Teo, Y.M.; Luong, B.L.; Szabo, C. Formalization of emergence in multi-agent systems. In Proceedings of the 1st ACM SIGSIM Conference on Principles of Advanced Discrete Simulation, Montreal, QC, Canada, 19–22 May 2013; pp. 231–240. </ref><ref>Szabo, C.; Teo, Y.M. Formalization of weak emergence in multiagent systems. ACM Trans. Model. Comput. Simul. (TOMACS) 2015, 26, 1–25. [CrossRef] </ref>. This approach focuses on interaction rules and the agent states rather than statistical measures of the system as a whole. Specifically, this measure consists of two subtracting terms. The first term represents the system's collective state, while the second term is the sum of its individual components. This measure emphasizes that emergence arises from the interactions and collective behavior of the system.

=== 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 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.

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 ===

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 ===

[[Zhang Jiang]] et al. (2024) <ref name=":2">Zhang J, Tao R, Yuan B. Dynamical Reversibility and A New Theory of Causal Emergence. arXiv preprint arXiv:2402.15054. 2024 Feb 23.</ref> introduced a new causal emergence theory using [[singular value decomposition]].  The core idea equates causal emergence with dynamical reversibility. The [[reversibility measure]] of Markov dynamics (<math>\Gamma_{\alpha}\equiv \sum_{i=1}^N\sigma_i^{\alpha}</math>) is defined as the sum of <math>\alpha</math> power of its singular values. Here, [math]\sigma_i[/math] represents the singular value. This index is highly correlated with [[effective information]] and can also be used to characterize the causal effect strength of  dynamics. This method defines '''clear emergence''' and '''vague emergence''' directly from the spectrum of singular values, without requiring a coarse-graining scheme.

[[Zhang Jiang]] et al. (2024) <ref name=":2">Zhang J, Tao R, Yuan B. Dynamical Reversibility and A New Theory of Causal Emergence. arXiv preprint arXiv:2402.15054. 2024 Feb 23.</ref> introduced a new causal emergence theory using [[singular value decomposition]].  The core idea equates causal emergence with dynamical reversibility. The [[reversibility measure]] of Markov dynamics (<math>\Gamma_{\alpha}\equiv \sum_{i=1}^N\sigma_i^{\alpha}</math>) is defined as the sum of <math>\alpha</math> power of its singular values. Here, [math]\sigma_i[/math] represents the singular value. This index is highly correlated with [[effective information]] and can also be used to characterize the causal effect strength of  dynamics. This method defines '''clear emergence''' and '''vague emergence''' directly from the spectrum of singular values, without requiring a coarse-graining scheme.

== Quantification of causal emergence ==

== Quantification of causal emergence ==

|Dynamic independence <ref name=":6"/>||Granger causality||Requires specifying a coarse-graining method||Arbitrary dynamics||Dynamic independence: transfer entropy

|Dynamic independence <ref name=":6"/>||Granger causality||Requires specifying a coarse-graining method||Arbitrary dynamics||Dynamic independence: transfer entropy

|}

|}

==Identification of Causal Emergence==

==Identification of Causal Emergence==

The results show that in the simple example shown in Figure (b), by maximizing <math>\mathrm{\Psi}</math> through the model constructed in Figure a, the experiment finds that the learned <math>\mathrm{\Psi}</math> is approximately equal to the true groundtruth <math>\mathrm{\Psi}</math>. This outcome verifies the effectiveness of model learning. This system is effective in identifying the occurrence of causal emergence. However, this method also has the problem of being difficult to deal with complex multivariate situations. This is because the number of neural networks on the right side of the figure is proportional to the number of macroscopic and microscopic variable pairs. Therefore, the more the number of microscopic variables (dimensions), the more the number of neural networks will increase proportionally, which will lead to an increase in computational complexity. In addition, this method is only tested on very few cases, so it cannot be scaled up yet. Moreover, since the network calculates an approximate index of causal emergence, it inherits the limitations of the earlier approximate algorithms.

The results show that in the simple example shown in Figure (b), by maximizing <math>\mathrm{\Psi}</math> through the model constructed in Figure a, the experiment finds that the learned <math>\mathrm{\Psi}</math> is approximately equal to the true groundtruth <math>\mathrm{\Psi}</math>. This outcome verifies the effectiveness of model learning. This system is effective in identifying the occurrence of causal emergence. However, this method also has the problem of being difficult to deal with complex multivariate situations. This is because the number of neural networks on the right side of the figure is proportional to the number of macroscopic and microscopic variable pairs. Therefore, the more the number of microscopic variables (dimensions), the more the number of neural networks will increase proportionally, which will lead to an increase in computational complexity. In addition, this method is only tested on very few cases, so it cannot be scaled up yet. Moreover, since the network calculates an approximate index of causal emergence, it inherits the limitations of the earlier approximate algorithms.

====Neural Information Compression Method====

====Neural Information Compression Method====

These experiments demonstrates that [[NIS+]] identifies causal emergence, discovers macroscopic dynamics, and develops coarse-graining strategies. Additionally, other experiments show that EI maximization enhances the model's e out-of-distribution generalization ability.

These experiments demonstrates that [[NIS+]] identifies causal emergence, discovers macroscopic dynamics, and develops coarse-graining strategies. Additionally, other experiments show that EI maximization enhances the model's e out-of-distribution generalization ability.

==Applications==

==Applications==

For example, in Eastern philosophy, the [[five elements]] can be interpreted as  macroscopic states, with their mutual generation and restraint representing a macroscopic causal mechanism. The process of identifying these five states from all phenomena is a coarse-graining process dependent on the observer's ability to analogize. Therefore, the theory of five elements can be regarded as an abstract causal emergence theory for everything. Similarly, the concept of causal emergence can be extended to fields like traditional Chinese medicine, divination, and feng shui. The common point of these applications will be that its causal mechanism is simpler and possibly has stronger causality compared to Western science. However, the process of obtaining such an abstract coarse-graining is more complex and more dependent on experienced abstractors. This explains why Eastern philosophies all emphasize the self-cultivation of practitioners. This is because Eastern philosophical theories place significant complexity and computational demands on '''analogical thinking'''.

For example, in Eastern philosophy, the [[five elements]] can be interpreted as  macroscopic states, with their mutual generation and restraint representing a macroscopic causal mechanism. The process of identifying these five states from all phenomena is a coarse-graining process dependent on the observer's ability to analogize. Therefore, the theory of five elements can be regarded as an abstract causal emergence theory for everything. Similarly, the concept of causal emergence can be extended to fields like traditional Chinese medicine, divination, and feng shui. The common point of these applications will be that its causal mechanism is simpler and possibly has stronger causality compared to Western science. However, the process of obtaining such an abstract coarse-graining is more complex and more dependent on experienced abstractors. This explains why Eastern philosophies all emphasize the self-cultivation of practitioners. This is because Eastern philosophical theories place significant complexity and computational demands on '''analogical thinking'''.

==Critique==

==Critique==

For specific methods of coarse-graining Markov chains, please refer to [[coarse-graining of Markov chains]].

For specific methods of coarse-graining Markov chains, please refer to [[coarse-graining of Markov chains]].

==References==

==References==

*[https://pattern.swarma.org/study_group/28 Causal Emergence Reading Club Season 3]

*[https://pattern.swarma.org/study_group/28 Causal Emergence Reading Club Season 3]

In the third season reading club of causal emergence, we will further conduct in-depth learning and discussion around the core research questions of causal emergence, namely "the definition of causal emergence" and "the identification of causal emergence". We will conduct in-depth discussions and analyses on the core theories of causal emergence such as Causal Emergence and Causal Geometry proposed by Erik Hoel, and carefully sort out the methodologies involved, including learning and drawing on relevant research ideas from other research fields such as dynamic reduction and latent space dynamic learning. Finally, we will discuss the applications of causal emergence, including expanding on issues such as biological networks, brain networks, or emergence detection to explore more practical application scenarios.

In the third season reading club of causal emergence, we will further conduct in-depth learning and discussion around the core research questions of causal emergence, namely "the definition of causal emergence" and "the identification of causal emergence". We will conduct in-depth discussions and analyses on the core theories of causal emergence such as Causal Emergence and Causal Geometry proposed by Erik Hoel, and carefully sort out the methodologies involved, including learning and drawing on relevant research ideas from other research fields such as dynamic reduction and latent space dynamic learning. Finally, we will discuss the applications of causal emergence, including expanding on issues such as biological networks, brain networks, or emergence detection to explore more practical application scenarios.

===Path Recommendation===

===Path Recommendation===