更改

==Critique==

==Critique==

The theory of causal emergence , first proposed by Erik Hoel and others based on maximizing effective information, has well characterized the phenomenon of causal emergence and has been widely applied in many actual systems. Some scholars, however, have pointed out that this theory still has many flaws and drawbacks. This is mainly reflected in the philosophical level and the technical level of the coarse-graining of Markov dynamical systems.

The theory of causal emergence, first proposed by Erik Hoel and others based on maximizing EI or SVD, has well characterized the phenomenon of emergence and has been widely applied in many actual systems. Some scholars, however, have pointed out that this theory still has many flaws and drawbacks. This is mainly reflected in the philosophical level and the technical level of the coarse-graining of Markov dynamical systems.

Is causal emergence a real phenomenon that exists independently of a specific observer? Here it should be emphasized that for Hoel's theory, different coarse-graining strategies can lead to different macroscopic dynamical mechanisms and different measures of causal effects' results (<math>EI</math>). Essentially, different coarse-graining strategies can represent different observers. Hoel's theory links emergence with causality through intervention and introduces the concept of causal emergence in a quantitative way. Hoel's theory proposes a scheme to eliminate the influence of different coarse-graining methods, that is, maximizing <math>EI</math>. The coarse-graining scheme that maximizes EI is the only objective scheme. Therefore, for a given set of [[Markov dynamics]], only the coarse-graining strategy and corresponding macroscopic dynamics that maximize <math>EI</math> can be considered objective results. If multiple coarse-graining schemes maximize <math>EI</math> is not unique, that is, there are multiple coarse-graining schemes that can maximize <math>EI</math>, it introduces theoretical difficulties and unavoidable subjectivity.

Is causal emergence a real phenomenon that exists independently of a specific observer? Here it should be emphasized that for Hoel's theory, different coarse-graining strategies can lead to different macroscopic dynamical mechanisms and different measures of the strength of causal effects (<math>EI</math>). Essentially, different coarse-graining strategies can represent different observers. Hoel's theory links emergence with causality through intervention and introduces the concept of causal emergence in a quantitative way. Hoel's theory proposes a scheme to eliminate the influence of different coarse-graining strategy, that is, maximizing <math>EI</math>. The coarse-graining strategy that maximizes EI is the only objective scheme. Therefore, for a given set of [[Markov dynamics]], only the coarse-graining strategy and corresponding macroscopic dynamics that maximize <math>EI</math> can be considered objective results. If multiple coarse-graining strategies maximize <math>EI</math> is not unique, that is, there are multiple coarse-graining strategies that can maximize <math>EI</math>, it introduces theoretical difficulties and unavoidable subjectivity.

Dewhurst <ref>Dewhurst, J. (2021). Causal emergence from effective information: Neither causal nor emergent?. Thought: A Journal of Philosophy, 10(3), 158-168.</ref> provides a philosophical clarification of Hoel's theory, arguing that it is epistemological rather than ontological. This indicates that Hoel's macroscopic causality is only a causal explanation based on information theory and does not involve "true causality". This also raises questions about the assumption of [[uniform distribution]] (see the entry for [[effective information]]), as there is no evidence that it should be superior to other distributions.

Dewhurst <ref>Dewhurst, J. (2021). Causal emergence from EI: Neither causal nor emergent?. Thought: A Journal of Philosophy, 10(3), 158-168.</ref> provides a philosophical clarification of Hoel's theory, arguing that it is epistemological rather than ontological. This indicates that Hoel's macroscopic causality is only a causal explanation based on information theory and does not involve "true causality". This also raises questions about the assumption of [[uniform distribution]] (see the entry for EI), as there is no evidence that it should be superior to other distributions.

===From the technical aspect===

===From the technical aspect===

====Non-uniqueness====

====Non-uniqueness====

The result of causal emergence is defined as the difference between the EI of the macroscopic dynamics after coarse-graining and the EI of the original microscopic dynamics. Therefore, this result obviously depends on the choice of the coarse-graining scheme. To address this uncertainty, Hoel and others proposed maximizing EI as the basis for judging and measuring causal emergence. However, there is currently no theoretical guarantee that this way of maximizing the EI of the macroscopic dynamics can ensure the uniqueness of the coarse-graining scheme. In other words, it is entirely possible that multiple coarse-graining schemes will correspond to the same EI of the macroscopic dynamics. In fact, studies on continuous mapping dynamical systems have already shown that there are infinitely many possibilities for the solutions of maximizing EI <ref name="exact">{{cite journal |last1=Liu|first1=K.W.|last2=Yuan|first2=B.|last3=Zhang|first3=J.|title=An Exact Theory of Causal Emergence for Linear Stochastic Iteration Systems|journal=Entropy|volume=26 |issue=8 |year=2024|page=618}}</ref>.

The result of causal emergence is defined as the difference between the EI of the macroscopic dynamics after coarse-graining and the EI of the original microscopic dynamics. Therefore, this result obviously depends on the choice of the coarse-graining strategy. To address this uncertainty, Hoel and others proposed maximizing EI as the basis for judging and measuring causal emergence. However, there is currently no theoretical guarantee that this way of maximizing the EI of the macroscopic dynamics can ensure the uniqueness of the coarse-graining strategy. In other words, it is entirely possible that multiple coarse-graining strategies will correspond to the same EI of the macroscopic dynamics. In fact, studies on continuous mapping dynamical systems have already shown that there are infinitely many possibilities for the solutions of maximizing EI <ref name="exact">{{cite journal |last1=Liu|first1=K.W.|last2=Yuan|first2=B.|last3=Zhang|first3=J.|title=An Exact Theory of Causal Emergence for Linear Stochastic Iteration Systems|journal=Entropy|volume=26 |issue=8 |year=2024|page=618}}</ref>.

====Ambiguity====

====Ambiguity====

At the same time, it is pointed out that Hoel's theory ignores the constraints on the coarse-graining method, and some coarse-graining methods may lead to ambiguity and irrationality <ref name="Eberhardt">Eberhardt, F., & Lee, L. L. (2022). Causal emergence: When distortions in a map obscure the territory. Philosophies, 7(2), 30.</ref>.  

At the same time, it is pointed out that Hoel's theory ignores the constraints on the coarse-graining strategy, and some coarse-graining strategies may lead to ambiguity and irrationality <ref name="Eberhardt">Eberhardt, F., & Lee, L. L. (2022). Causal emergence: When distortions in a map obscure the territory. Philosophies, 7(2), 30.</ref>.  

First, <ref name="Eberhardt"/> highlights that ambiguity can arise in coarse-graining the TPM if no constraints are imposed on the scheme. For example, when the two row vectors in the TPM corresponding to the two states to be merged are very dissimilar, forcibly merging them (for example, by taking the average) will cause ambiguity. This ambiguity arises in determining the meaning of intervention on the merged macroscopic state. Because the row vectors are dissimilar, the macroscopic state's intervention cannot directly correspond to microscopic state interventions. If the intervention on the macroscopic state is forcibly converted into the intervention on the microscopic states by taking the average, the differences between the microscopic states are ignored. At the same time, new contradictory problems of non-commutativity will also be triggered.

First, <ref name="Eberhardt"/> highlights that ambiguity can arise in coarse-graining the TPM if no constraints are imposed on the strategy. For example, when the two row vectors in the TPM corresponding to the two states to be merged are very dissimilar, forcibly merging them (for example, by taking the average) will cause ambiguity. This ambiguity arises in determining the meaning of intervention on the merged macroscopic state. Because the row vectors are dissimilar, the macroscopic state's intervention cannot directly correspond to microscopic state interventions. If the intervention on the macroscopic state is forcibly converted into the intervention on the microscopic states by taking the average, the differences between the microscopic states are ignored. At the same time, new contradictory problems of non-commutativity will also be triggered.

====Non-commutativity====

====Non-commutativity====

If the two dissimilar row vectors are forcibly averaged, the resulting coarse-grained TPM may break the commutativity between the abstraction operation (i.e., coarse-graining) and marginalization (i.e., the time evolution operator). For example, assume that <math>A_{m\times n}</math> is a state coarse-graining operation (combining n states into m states). In this case, the coarse-graining strategy is defined as the one that maximizes the EI of the macroscopic state transition matrix. <math>(\cdot) \times (\cdot)</math> is a time coarse-graining operation (combining two time steps into one). In this way, [math]A_{m\times n}(TPM_{n\times n})[/math] is to perform coarse-graining on a [math]n\times n[/math] TPM, and the coarse-graining process is simplified as the product of matrix [math]A[/math] and matrix [math]TPM[/math].

If the two dissimilar row vectors are forcibly averaged, the resulting coarse-grained TPM may break the commutativity between the abstraction operation (i.e., coarse-graining) and marginalization (i.e., the time evolution operator). For example, assume that <math>A_{m\times n}</math> is a state coarse-graining operation (combining n states into m states). In this case, the coarse-graining strategy is defined as the one that maximizes the EI of the macroscopic state transition matrix. <math>(\cdot) \times (\cdot)</math> is a time coarse-graining operation (combining two time steps into one). In this way, [math]A_{m\times n}(TPM_{n\times n})[/math] is to perform coarse-graining on a [math]n\times n[/math] TPM, and the coarse-graining strategy is simplified as the product of matrix [math]A[/math] and matrix [math]TPM[/math].

====Other problems and remedies====

====Other problems and remedies====

Hoel's <math>EI</math> calculation and causal emergence quantification rely on two prerequisites: (1) known microscopic dynamics and (2) a known coarse-graining scheme. In practice, both factors are rarely available simultaneously, particularly in observational studies where they may be unknown. Therefore, this limitation hinders the practical applicability of Hoel's theory.

Hoel's <math>EI</math> calculation and causal emergence quantification rely on two prerequisites: (1) known microscopic dynamics and (2) a known coarse-graining strategy. In practice, both factors are rarely available simultaneously, particularly in observational studies where they may be unknown. Therefore, this limitation hinders the practical applicability of Hoel's theory.