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Develop an α‑synergy analogue of Integrated Information Decomposition (ΦID)

Develop an analogue of Integrated Information Decomposition (ΦID) within the α‑synergy framework to decompose the information shared between two sets of random variables X and Y (i.e., I(X; Y)), given that ΦID is not expressible as a Kullback–Leibler divergence and therefore cannot currently be reconstructed using the α‑synergy methods based on Kullback–Leibler decompositions.

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Background

The α‑synergy framework introduced in this paper provides a synergy-first decomposition for entropy and for quantities expressible via Kullback–Leibler (KL) divergences, including total correlation and single-target mutual information. However, Integrated Information Decomposition (ΦID) extends beyond single-target settings to decompose I(X; Y) between two sets of variables using a product-lattice, and is not currently definable via KL divergence.

Because ΦID is not KL-based, the α‑synergy approach cannot directly reconstruct ΦID. Establishing an α‑synergy analogue for ΦID would extend the framework’s applicability from single-target to two-set (bidirectional) information-sharing, aligning it with the broader taxonomy of information decomposition methods.

References

Currently, the ΦID is not describable as a Kullback-Leibler divergence, instead being based on a product-lattice structure, and cannot currently be reconstructed from the framework presented here. Future work on the α-synergy decomposition should focus one finding a way to produce an analogue of the ΦID in the same way that the single-target decomposition above is analogous to the PID.

— A scalable, synergy-first backbone decomposition of higher-order structures in complex systems (2402.08135 - Varley, 13 Feb 2024) in Section “Single-target mutual information”