Can the whole be more than the sum of its parts? Most definitely.
Rigorous theory of causal emegence based on information decomposition.
Information decomposition based on an operational definition of synergy.
Covering Integrated Information Decomposition (ΦID), with a few extra details for a PID audience.
Tracking the information decomposition of neural networks as they learn.
The love baby of IIT and PID: ΦID.
Shannon-style measure of high-order effects for high-dimensional data.