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==Critique==

==Critique==

Although the theory of causal emergence based on maximizing effective information, first proposed by Erik Hoel and others, has well characterized the phenomenon of causal emergence and has been widely applied in many actual systems, some scholars 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 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.

For example, Yurchenko pointed out in the literature <ref>Yurchenko, S. B. (2023). Can there be a synergistic core emerging in the brain hierarchy to control neural activity by downward causation?. Authorea Preprints.</ref> that the concept of "causation" is often ambiguous and should be distinguished into two different concepts of '''"cause"''' and '''"reason"''', which respectively conform to ontological and epistemological causality. Among them, cause refers to the real cause that fully leads to the result, while reason is only the observer's explanation of the result. Reason may not be as strict as a real cause, but it does provide a certain degree of [[predictability]]. Similarly, there is also a debate about the nature of causal emergence.

For example, Yurchenko <ref>Yurchenko, S. B. (2023). Can there be a synergistic core emerging in the brain hierarchy to control neural activity by downward causation?. Authorea Preprints.</ref> highlights the ambiguity in the concept of 'causation,' distinguishing it into 'cause,' which aligns with ontological causality, and 'reason,' which aligns with epistemological causality. Among them, cause refers to the real cause that fully leads to the result, while reason is only the observer's explanation of the result. Reason may not be as strict as a real cause, but it does provide a certain degree of [[predictability]]. Similarly, there is also a debate about the nature of causal emergence.

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 causal effect measurement 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. However, when the solution that maximizes <math>EI</math> is not unique, that is, there are multiple coarse-graining schemes that can maximize <math>EI</math>, it will lead to difficulties in this theory and a certain degree of subjectivity cannot be avoided.

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 causal effect measurement 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.

===From the technical aspect===

===From the technical aspect===

====Non-uniqueness====

====Non-uniqueness====

The result of causal emergence is defined as the difference between the Effective Information (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 eliminate this uncertainty, Hoel and others introduced the concept of maximizing Effective Information, that is, the judgment and measurement of causal emergence in a system are based on maximizing the EI of the macroscopic dynamics. 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 Effective Information (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 Effective Information 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>.

====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>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 method, and some coarse-graining methods may lead to ambiguity and irrationality <ref>Eberhardt, F., & Lee, L. L. (2022). Causal emergence: When distortions in a map obscure the territory. Philosophies, 7(2), 30.</ref>.  

First of all, <ref name="Eberhardt"/> points out that if there are no constraints on the coarse-graining scheme, ambiguity may arise when coarse-graining the transition probability matrix (TPM) (merging states and adding up probabilities). 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 is mainly manifested in the question of what exactly the intervention on the merged macroscopic state means? Since the row vectors are dissimilar, the intervention on the merged macroscopic state cannot be simply reduced to the intervention on the microscopic states. 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 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.

 

====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). Here, the coarse-graining strategy is the strategy that maximizes the effective information 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 effective information 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].

The left side represents first performing coarse-graining on the states of two consecutive time steps, and then multiplying the dynamics TPM of the two time steps together to obtain a transfer matrix for two-step evolution; the right side of the equation represents first multiplying the TPMs of two time steps together to obtain the two-step evolution of the microscopic state, and then using A for coarse-graining to obtain the macroscopic TPM. The non-satisfaction of this equation indicates that some coarse-graining operations will cause differences between the evolution of macroscopic states and the coarse-grained states of the microscopic system after evolution. This implies that certain consistency constraints need to be imposed on the coarse-graining strategy, such as the lumpable conditions of Markov chains. See the entry of "[[Coarse-graining of Markov Chains]]".

The left side represents first performing coarse-graining on the states of two consecutive time steps, and then multiplying the dynamics TPM of the two time steps together to obtain a transfer matrix for two-step evolution; the right side of the equation represents first multiplying the TPMs of two time steps together to obtain the two-step evolution of the microscopic state, and then using A for coarse-graining to obtain the macroscopic TPM. If this equation is not satisfied, it suggests that certain coarse-graining operations cause discrepancies between macroscopic and microscopic state evolutions. This implies that certain consistency constraints need to be imposed on the coarse-graining strategy, such as the lumpable conditions of Markov chains. See the entry of "[[Coarse-graining of Markov Chains]]".

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

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

In addition, Hoel's <math>EI</math> calculation and the quantification of causal emergence depend on two known prerequisite factors: (1) known microscopic dynamics; (2) known coarse-graining scheme. However, in practice, people rarely can obtain both of these factors at the same time, especially in observational studies, these two factors 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 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.

 

==Related research fields==

==Related research fields==