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→Erik Hoel's causal emergence theory
[[文件:因果涌现理论.png|因果涌现理论框架|alt=因果涌现理论抽象框架|居左|400x400像素]]
[[文件:CE.png|替代=因果涌现示意图|500x500像素]]
=====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]], 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.<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> further gave an exact analytical causal emergence theory for [[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]], 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.<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> further gave an exact analytical causal emergence theory for [[random iterative dynamical systems]].
====Rosas's Causal Emergence Theory====
====Rosas's Causal Emergence Theory====