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The authors proposed three new indicators based on [[mutual information]], <math>\mathrm{\Psi}</math>, <math>\mathrm{\Delta}</math> and <math>\mathrm{\Gamma}</math>, which can be used to identify causal emergence, [[causal decoupling]] and [[downward causation]] in the system respectively. The specific calculation formulas of the three indicators are as follows:

The authors proposed three new indicators based on [[mutual information]], <math>\mathrm{\Psi}</math>, <math>\mathrm{\Delta}</math> and <math>\mathrm{\Gamma}</math>, which can be used to identify causal emergence, [[causal decoupling]] and [[downward causation]] in the system respectively. The specific calculation formulas of the three indicators are as follows:

* Indicator for judging causal emergence:

* Indicator for the judgement of causal emergence:

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<math>\Psi_{t, t + 1}(V):=I\left(V_t ; V_{t + 1}\right)-\sum_j I\left(X_t^j ; V_{t + 1}\right)</math>

<math>\Psi_{t, t + 1}(V):=I\left(V_t ; V_{t + 1}\right)-\sum_j I\left(X_t^j ; V_{t + 1}\right)</math>

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Here <math>X_t^j</math> represents the microscopic variable at time t in the j-th dimension, and <math>V_t ; V_{t + 1}</math> respectively represent macroscopic variables at two consecutive times. Rosas et al. defined that when <math>\mathrm{\Psi}>0</math>, emergence occurs in the system; but when <math>\mathrm{\Psi}<0</math>, we cannot determine whether the system has emergence, because this condition is only a sufficient condition for the occurrence of causal emergence.

Here <math>X_t^j</math> represents the microscopic variable at time t in the j-th dimension, and <math>V_t ; V_{t + 1}</math> respectively represent macroscopic variables at two consecutive times. Rosas et al. defined that when <math>\mathrm{\Psi}>0</math>, emergence occurs in the system; but when <math>\mathrm{\Psi}<0</math>, we cannot determine whether emergence occurs, because this condition is only a sufficient condition for the occurrence of causal emergence.

* Indicator for judging [[downward causation]]:

* Indicator for [[downward causation]]:

<math>\Delta_{t, t + 1}(V):=\max _j\left(I\left(V_t ; X_{t + 1}^j\right)-\sum_i I\left(X_t^i ; X_{t + 1}^j\right)\right)</math>

<math>\Delta_{t, t + 1}(V):=\max _j\left(I\left(V_t ; X_{t + 1}^j\right)-\sum_i I\left(X_t^i ; X_{t + 1}^j\right)\right)</math>

When <math>\mathrm{\Delta}>0</math>, there exists [[downward causation]] from macroscopic variable <math>V</math> to microscopic variable <math>X</math>.

When <math>\mathrm{\Delta}>0</math>, there exists [[downward causation]] from macroscopic variable <math>V</math> to microscopic variable <math>X</math>.

* Indicator for judging [[causal decoupling]]:

* Indicator for [[causal decoupling]]:

<math>\Gamma_{t, t + 1}(V):=\max _j I\left(V_t ; X_{t + 1}^j\right)</math>

<math>\Gamma_{t, t + 1}(V):=\max _j I\left(V_t ; X_{t + 1}^j\right)</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>.

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 occurrence 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) The 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.