Emergence and causality are two fundamental concepts for understanding complex systems. They are interconnected. On one hand, emergence refers to the phenomenon where macroscopic properties cannot be solely attributed to the cause of individual properties. On the other hand, causality can exhibit emergence, meaning that new causal laws may arise as we increase the level of abstraction. Causal emergence theory aims to bridge these two concepts and even employs measures of causality to quantify emergence. This paper provides a comprehensive review of recent advancements in quantitative theories and applications of causal emergence. Two key problems are addressed: quantifying causal emergence and identifying it in data. Addressing the latter requires the use of machine learning techniques, thus establishing a connection between causal emergence and artificial intelligence. We highlighted that the architectures used for identifying causal emergence are shared by causal representation learning, causal model abstraction, and world model-based reinforcement learning. Consequently, progress in any of these areas can benefit the others. Potential applications and future perspectives are also discussed in the final section of the review.

Quantifying emergence and modeling emergent dynamics in a data-driven manner for complex dynamical systems is challenging due to the lack of direct observations at the micro-level. Thus, it's crucial to develop a framework to identify emergent phenomena and capture emergent dynamics at the macro-level using available data. Inspired by the theory of causal emergence (CE), this paper introduces a machine learning framework to learn macro-dynamics in an emergent latent space and quantify the degree of CE. The framework maximizes effective information, resulting in a macro-dynamics model with enhanced causal effects. Experimental results on simulated and real data demonstrate the effectiveness of the proposed framework. It quantifies degrees of CE effectively under various conditions and reveals distinct influences of different noise types. It can learn a one-dimensional coarse-grained macro-state from fMRI data, to represent complex neural activities during movie clip viewing. Furthermore, improved generalization to different test environments is observed across all simulation data.

The classic studies of causal emergence have revealed that in some Markovian dynamical systems, far stronger causal connections can be found on the higher-level descriptions than the lower-level of the same systems if we coarse-grain the system states in an appropriate way. However, identifying this emergent causality from the data is still a hard problem that has not been solved because the correct coarse-graining strategy can not be found easily. This paper proposes a general machine learning framework called Neural Information Squeezer to automatically extract the effective coarse-graining strategy and the macro-state dynamics, as well as identify causal emergence directly from the time series data. By decomposing a coarse-graining operation into two processes: information conversion and information dropping out, we can not only exactly control the width of the information channel, but also can derive some important properties analytically including the exact expression of the effective information of a macro-dynamics. We also show how our framework can extract the dynamics on different levels and identify causal emergence from the data on several exampled systems.

Using the mathematical language of information theory, Hoel and his collaborators claim to show that new causes--things that produce effects--can emerge at macroscopic scales. They say coarse-grained macroscopic states of a physical system (such as the psychological state of a brain) can have more causal power over the system's future than a more detailed, fine-grained description of the system possibly could. Just as codes reduce noise (and thus uncertainty) in transmitted data--Claude Shannon's 1948 insight that formed the bedrock of information theory--Hoel claims that macro states also reduce noise and uncertainty in a system's causal structure, strengthening causal relationships and making the system's behavior more deterministic. With Albantakis and Tononi, Hoel formalized a measure of causal power called "effective information," which indicates how effectively a particular state influences the future state of a system.