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Since <math>Red(V_t, V_{t + 1};X_t)</math> is non-negative, we can thus propose a sufficient but not necessary condition: when <math>\Psi_{t, t + 1}(V)>0</math>.

Since <math>Red(V_t, V_{t + 1};X_t)</math> is non-negative, we can thus propose a sufficient but not necessary condition: when <math>\Psi_{t, t + 1}(V)>0</math>.

In summary, this method is relatively convenient to calculate because it is based on mutual information, and there is no assumption or requirement of Markov property for the dynamics of the system. However, this theory also has many shortcomings: 1) The indicators <math>\mathrm{\Psi}</math>, <math>\mathrm{\Delta}</math> and <math>\mathrm{\Gamma}</math> based only on [[mutual information]] and do not account for causality; 2) The method provides only a sufficient condition for causal emergence; 3) Results depend heavily on the selection of macroscopic variables, with different choices leading to significantly different outcomes; 4) For systems with large amounts of redundant information or many variables, the [[computational complexity]] becomes very high. At the same time, since <math>\Psi</math> is an approximate calculation, there will be a very large error in high-dimensional systems, and it is also very easy to obtain negative values, making it difficult to reliably determine the presence of causal emergence.

In summary, this method is relatively convenient to calculate because it is based on mutual information, and there is no assumption or requirement of Markov property for the dynamics of the system. However, this theory also has many shortcomings: 1) The indicators <math>\mathrm{\Psi}</math>, <math>\mathrm{\Delta}</math> and <math>\mathrm{\Gamma}</math> based only on [[mutual information]] and do not account for causality; 2) The method provides only a sufficient condition for causal emergence; 3) Results depend heavily on the selection of macroscopic variables, with different choices leading to significantly different outcomes; 4) For systems with large amounts of redundant information or many variables, the [[computational complexity]] becomes very high. At the same time, since <math>\Psi</math> is an approximate calculation, there will be a very large error in high-dimensional systems, and it is also very easy to obtain negative values, making it difficult to reliably determine the occurrence of causal emergence.

To verify that the information related to macaque movement is an emergent feature of its cortical activity, Rosas et al. conducted an experiment using the electrocorticogram (ECoG) of macaques as the observational data of microscopic dynamics. To obtain the macroscopic state variable <math>V</math>, the authors chose the time series data of the limb movement trajectory of macaques obtained by motion capture (MoCap). The ECoG consisted of 64 channels, while the MoCap data consisted of 3 channels. Since the original MoCap data does not satisfy the conditional independence assumption of the supervenience feature, they used [[partial least squares]] and [[support vector machine]] algorithms to infer the part of neural activity encoded in the ECoG signal related to predicting macaque behavior. They then speculated that this information is an emergent feature of potential neural activity. Finally, based on the microscopic state and the calculated macroscopic features, the authors verified the existence of causal emergence.

To verify that the information related to macaque movement is an emergent feature of its cortical activity, Rosas et al. conducted an experiment using the electrocorticogram (ECoG) of macaques as the observational data of microscopic dynamics. To obtain the macroscopic state variable <math>V</math>, the authors chose the time series data of the limb movement trajectory of macaques obtained by motion capture (MoCap). The ECoG consisted of 64 channels, while the MoCap data consisted of 3 channels. Since the original MoCap data does not satisfy the conditional independence assumption of the supervenience feature, they used [[partial least squares]] and [[support vector machine]] algorithms to infer the part of neural activity encoded in the ECoG signal related to predicting macaque behavior. They then speculated that this information is an emergent feature of potential neural activity. Finally, based on the microscopic state and the calculated macroscopic features, the authors verified the existence of causal emergence.