Quantifying causal emergence

Causal interactions within complex systems can be analyzed at multiple spatial and temporal scales.

Supervenience - once the micro level is fixed, macro levels are fixed too. Then: causality is causally complete on the macro level.

A convincing case for emergence must demonstrate that higher levels can be causal above and beyond lower levels [“causal emergence” (CE)].

Paper proposes a measure, Effective Information (EI), that "depends on both the effectiveness of a system’s mechanisms and the size of its state space: EI is higher the more the mechanisms constrain the system’s possible past and future states."

EI is maximal for systems that are deterministic and not degenerate, and decreases with noise (causal divergence) and/or degeneracy (causal convergence).

Coarse grained macro-mechanisms can be "more effective (more deterministic and/or less degenerate)", ie has a higher EI.