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==== Erik Hoel's causal emergence theory ====
==== Erik Hoel's causal emergence theory ====
In 2013, Hoel et al. [1][2] proposed the causal emergence theory. The following figure is an abstract framework for this theory. The horizontal axis represents time and the vertical axis represents scale. This framework can be regarded as a description of the same dynamical system on both microscopic and macroscopic scales. Among them, [math]f_m[/math] represents microscopic dynamics, [math]f_M[/math] represents macroscopic dynamics, and the two are connected by a coarse-graining function [math]\phi[/math]. In a discrete-state Markov dynamical system, both [math]f_m[/math] and [math]f_M[/math] are Markov chains. By performing coarse-graining of the Markov chain on [math]f_m[/math], [math]f_M[/math] can be obtained. [math]\displaystyle{ EI }[/math] is a measure of effective information. Since the microscopic state may have greater randomness, which leads to relatively weak causality of microscopic dynamics, by performing reasonable coarse-graining on the microscopic state at each moment, it is possible to obtain a macroscopic state with stronger causality. The so-called causal emergence refers to the phenomenon that when we perform coarse-graining on the microscopic state, the effective information of macroscopic dynamics will increase, and the difference in effective information between the macroscopic state and the microscopic state is defined as the intensity of causal emergence.
In 2013, Hoel et al. [1][2] proposed the causal emergence theory. The following figure is an abstract framework for this theory. The horizontal axis represents time and the vertical axis represents scale. This framework can be regarded as a description of the same dynamical system on both microscopic and macroscopic scales. Among them, [math]f_m[/math] represents microscopic dynamics, [math]f_M[/math] represents macroscopic dynamics, and the two are connected by a coarse-graining function [math]\phi[/math]. In a discrete-state Markov dynamical system, both [math]f_m[/math] and [math]f_M[/math] are Markov chains. By performing coarse-graining of the Markov chain on [math]f_m[/math], [math]f_M[/math] can be obtained. <math> EI </math> is a measure of effective information. Since the microscopic state may have greater randomness, which leads to relatively weak causality of microscopic dynamics, by performing reasonable coarse-graining on the microscopic state at each moment, it is possible to obtain a macroscopic state with stronger causality. The so-called causal emergence refers to the phenomenon that when we perform coarse-graining on the microscopic state, the effective information of macroscopic dynamics will increase, and the difference in effective information between the macroscopic state and the microscopic state is defined as the intensity of causal emergence.
[[文件:因果涌现理论框架.png|无|缩略图]]
[[文件:因果涌现理论框架.png|无|缩略图]]
Effective Information
Effective Information (<math> EI </math>) was first proposed by Tononi et al. in the study of integrated information theory [41]. In causal emergence research, Erik Hoel and others use this causal effect measure index to quantify the strength of causality of a causal mechanism.
Specifically, the calculation of <math> EI </math> is as follows: use an intervention operation to intervene on the independent variable and examine the mutual information between the cause and effect variables under this intervention. This mutual information is effective information, that is, the causal effect measure of the causal mechanism.
In a Markov chain, the state variable [math]X_t[/math] at any time can be regarded as the cause, and the state variable [math]X_{t + 1}[/math] at the next time can be regarded as the result. Thus, the state transition matrix of the Markov chain is its causal mechanism. Therefore, the calculation formula for <math>EI</math> for a Markov chain is as follows:
<math>
\begin{aligned}
EI(f) \equiv& I(X_t,X_{t+1}|do(X_t)\sim U(\mathcal{X}))\equiv I(\tilde{X}_t,\tilde{X}_{t+1}) \\
&= \frac{1}{N}\sum^N_{i=1}\sum^N_{j=1}p_{ij}\log\frac{N\cdot p_{ij}}{\sum_{k=1}^N p_{kj}}
\end{aligned}
</math>
Here <math>f</math> represents the state transition matrix of a Markov chain, [math]U(\mathcal{X})[/math] represents the uniform distribution on the value space [math]\mathcal{X}[/math] of the state variable [math]X_t[/math]. <math>\tilde{X}_t,\tilde{X}_{t+1}</math> are the states at two consecutive moments after intervening [math]X_t[/math] at time <math>t</math> into a uniform distribution. <math>p_{ij}</math> is the transition probability from the <math>i</math>-th state to the <math>j</math>-th state. From this formula, it is not difficult to see that <math> EI </math> is only a function of the probability transition matrix [math]f[/math]. The intervention operation is performed to make the effective information objectively measure the causal characteristics of the dynamics without being affected by the distribution of the original input data.
Effective information can be decomposed into two parts: determinism and degeneracy. Normalization can also be introduced to eliminate the influence of the size of the state space. For more detailed information about effective information, please refer to the entry: Effective Information.
=====Causal Emergence Measurement=====
We can judge the occurrence of causal emergence by comparing the magnitudes of effective information of macroscopic and microscopic dynamics in the system:
<math>
CE = EI\left ( f_M \right ) - EI\left (f_m \right )
</math>
Here <math>CE</math> is the causal emergence intensity. If the effective information of macroscopic dynamics is greater than that of microscopic dynamics (that is, <math>CE>0</math>), then we consider that macroscopic dynamics has causal emergence characteristics on the basis of this coarse-graining.
=====Markov Chain Example=====
In the literature, Hoel gives an example of a state transition matrix ([math]f_m[/math]) of a Markov chain with 8 states, as shown in the left figure below. Among them, the first 7 states transfer with equal probability, and the last state is independent and can only transition to its own state.
The coarse-graining of this matrix is as follows: First, merge the first 7 states into a macroscopic state, which may be called A. And sum up the probability values in the first 7 columns of the first 7 rows in [math]f_m[/math] to obtain the probability of state transition from macroscopic state A to state A, and keep other values of the [math]f_m[/math] matrix unchanged. The new probability transition matrix after merging is shown in the right figure, denoted as [math]f_M[/math]. This is a definite macroscopic Markov transition matrix, that is, the future state of the system can be completely determined by the current state. At this time <math>EI(f_M\)>EI(f_m\)</math>, and causal emergence occurs in the system.
=====Boolean Network Example=====
Another example in the literature is an example of causal emergence in a Boolean network. As shown in the figure, this is a Boolean network with 4 nodes. Each node has two states, 0 and 1. Each node is connected to two other nodes and follows the same microscopic dynamics mechanism (figure a). Therefore, this system contains a total of sixteen microscopic states, and its dynamics can be represented by a state transition matrix (figure c).
The coarse-graining operation of this system is divided into two steps. The first step is to cluster the nodes in the Boolean network. As shown in figure b below, merge A and B to obtain the macroscopic node [math]\alpha[/math], and merge C and D to obtain the macroscopic node [math]\beta[/math]. The second step is to map the microscopic node states in each group to the merged macroscopic node states. This mapping function is shown in figure d below. All microscopic node states containing 0 are transformed into the off state of the macroscopic node, while the microscopic 11 state is transformed into the on state of the macroscopic. In this way, we can obtain a new macroscopic Boolean network, and obtain the dynamic mechanism of the macroscopic Boolean network according to the dynamic mechanism of the microscopic nodes. According to this mechanism, the state transition matrix of the macroscopic network can be obtained (as shown in figure e).
Through comparison, we find that the effective information of macroscopic dynamics is greater than that of microscopic dynamics (<math>EI(f_M\)>EI(f_m\)</math>). Causal emergence occurs in this system.
=====Causal Emergence in Continuous Variables=====
Furthermore, in the paper "xxx", Hoel et al. proposed the theoretical framework of causal geometry, trying to generalize the causal emergence theory to function mappings and dynamical systems with continuous states. This article defines <math>EI</math> for random function mapping, and also introduces the concepts of intervention noise and causal geometry, and compares and analogizes this concept with information geometry. Liu Kaiwei et al. further gave an exact analytical causal emergence theory for random iterative dynamical systems.