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==== Erik Hoel's causal emergence theory ====

==== Erik Hoel's causal emergence theory ====

Hoel et al. (2013) <ref name=":0" /><ref name=":1" />  proposed the theory of causal emergence. The following figure is an  abstract framework for this theory. The horizontal axis represents time  and the vertical axis represents scale. This framework describes the same dynamical system at 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. Causal emergence refers to the phenomenon where coarse-graining a microscopic state increases the [[effective information]] of macroscopic dynamics. The difference in [[effective information]] between the macroscopic and the microscopic states is defined as the intensity of causal emergence.  

Hoel et al. (2013) <ref name=":0" /><ref name=":1" />  proposed the theory of causal emergence. The following figure is an  abstract framework for this theory. The horizontal axis represents time  and the vertical axis represents scale. This framework describes the same dynamical system at 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. Causal emergence refers to the phenomenon where coarse-graining a microscopic state increases the [[effective information]] of macroscopic dynamics. The difference in [[effective information]] between the macroscopic and the microscopic states is defined as the strength of causal emergence.  

=====Causal Emergence in Continuous Variables=====

=====Causal Emergence in Continuous Variables=====

Furthermore, in the paper <ref name="Chvykov_causal_geometry">{{cite journal|author1=Chvykov P|author2=Hoel E.|title=Causal Geometry|journal=Entropy|year=2021|volume=23|issue=1|page=24|url=https://doi.org/10.3390/e2}}</ref>, Hoel et al. proposed the theoretical framework of [[causal geometry]],  with the aim of generalizing 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 this concept  with [[information geometry]]. [[Liu Kaiwei]] et al.<ref name="An_exact_theory_of_causal_emergence">{{cite journal|author1=Liu K|author2=Yuan B|author3=Zhang J|title=An Exact Theory of Causal Emergence for Linear Stochastic Iteration Systems|journal=Entropy|year=2024|volume=26|issue=8|page=618|url=https://arxiv.org/pdf/2405.09207}}</ref> developed an exact analytical framework for causal emergence in [[random iterative dynamical systems]].

Furthermore, in the paper <ref name="Chvykov_causal_geometry">{{cite journal|author1=Chvykov P|author2=Hoel E.|title=Causal Geometry|journal=Entropy|year=2021|volume=23|issue=1|page=24|url=https://doi.org/10.3390/e2}}</ref>, Hoel et al. proposed the theoretical framework of [[causal geometry]],  with the aim of generalizing 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 this concept  with [[information geometry]]. [[Liu Kaiwei]] et al.<ref name="An_exact_theory_of_causal_emergence">{{cite journal|author1=Liu K|author2=Yuan B|author3=Zhang J|title=An Exact Theory of Causal Emergence for Linear Stochastic Iteration Systems|journal=Entropy|year=2024|volume=26|issue=8|page=618|url=https://arxiv.org/pdf/2405.09207}}</ref> developed an exact analytical framework for causal emergence in [[random iterative dynamical systems]].

====Rosas's Causal Emergence Theory====

====Rosas's Causal Emergence Theory====