更改

==Critique==

==Critique==

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.

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.

===From the philosophical aspect===

===From the philosophical aspect===

Throughout history, there has been a long-standing debate on the [[ontological]] and [[epistemological]] aspects of causality and emergence.

Throughout history, there has been a long-standing debate on the [[ontological]] and [[epistemological]] aspects of causality and 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.

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 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.

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 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.

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.

====Ambiguity====

====Ambiguity====

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>.  

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 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.

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 strategy 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].

Then, the commutativity condition of spatial coarse-graining and temporal coarse-graining is the following equation:

Then, the commutativity condition of spatial coarse-graining and temporal coarse-graining is the following equation:

{{NumBlk|:|

{{NumBlk|:|

</math>

</math>

|{{EquationRef|3}}}}

|{{EquationRef|3}}}}

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]]".

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]]".

However, as pointed out in the literature <ref name=":6" />, the above problem can be alleviated by considering the error factor of the model while maximizing EI in the continuous variable space.

However, as pointed out in the literature <ref name=":6" />, the above problem can be alleviated by considering the error factor of the model while maximizing EI in the continuous variable space.

====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 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.

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.

However, although machine learning techniques facilitate the learning of causal relationships, causal mechanisms, and the identification of emergent properties, a key question is whether the results reflect ontological causality and emergence. Alternatively, are they merely an epistemological phenomenon? This is still undecided. Although the introduction of machine learning does not necessarily solve the debate around ontological and epistemological causality and emergence, it can provide a dependence that helps reduce subjectivity. This is because the machine learning agent can be regarded as an "objective" observer who makes judgments about causality and emergence that are independent of human observers. However, the problem of a unique solution still exists in this method. Is the result of machine learning ontological or epistemological? The answer is that the result is epistemological, where the epistemic subject is the machine learning algorithm. However, this does not mean that all results of machine learning are meaningless. If the learning subject is well trained and the defined mathematical objective is effectively optimized, the result can also be considered objective because the algorithm itself is objective and transparent. Combining machine learning methods can help us establish a theoretical framework for observers and study the interaction between observers and the corresponding observed complex systems.

However, although machine learning techniques facilitate the learning of causal relationships, causal mechanisms, and the identification of emergent properties, a key question is whether the results reflect ontological causality and emergence. Alternatively, are they merely an epistemological phenomenon? This is still undecided. Although the introduction of machine learning does not necessarily solve the debate around ontological and epistemological causality and emergence, it can provide a dependence that helps reduce subjectivity. This is because the machine learning agent can be regarded as an "objective" observer who makes judgments about causality and emergence that are independent of human observers. However, the problem of a unique solution still exists in this method. Is the result of machine learning ontological or epistemological? The answer is that the result is epistemological, where the epistemic subject is the machine learning algorithm. However, this does not mean that all results of machine learning are meaningless. If the learning subject is well trained and the defined mathematical objective is effectively optimized, the result can also be considered objective because the algorithm itself is objective and transparent. Combining machine learning methods can help us establish a theoretical framework for observers and study the interaction between observers and the corresponding observed complex systems.