Source
Griffith, Chong, James, Ellison & Crutchfield (2014), “Intersection Information Based on Common Randomness”, arXiv 1310.1538v3 (Entropy 16, 1985–2000), LIT-1203, read in full in NOTE-967.
What was actually shown
The measure (Lemma 8). I_∧(X_1, …, X_n : Y) = I(X_1 ∧ … ∧ X_n : Y), the most information about Y in any variable that is a function of every source.
Properties (Lemmas 4–6, 9). Global positivity, equivalence-class invariance, target monotonicity, weak monotonicity, weak symmetry, lower bound, self-redundancy; I_∧ ≤ I_min. Not identity, local positivity, strong monotonicity or strong symmetry.
The failure. Any noise between the sources makes their common random variable trivial. ImperfectRdn: I_∧ = 0 and synergy −0.99 bit (recomputed −0.9896). On Harder, Salge and Polani’s copy family I_∧ is 0 for every correlation short of perfect, against I(X : Y) up to 0.24 bit, so synergy is −I(X : Y).
The zero-error variant. H(∧X_i ∧ Y) satisfies nine of eleven properties, identity for zero-error information among them, and still gives Subtle −0.252 bit (recomputed −0.2516).
More than two sources (recomputed). For a target copying three equally correlated bits, I_∧ is 1 at full correlation and 0 at λ = 0.5, where each pairwise mutual information is 0.046.
What could have come out otherwise: the common random variable could have captured ImperfectRdn’s 0.99 bit of near-copy. It captures none.
What this does not say
- That redundancy is only shared structure. That is the measure’s premise, the source-redundancy pole of the dispute THEORY-708 takes the other side of.
- That a multivariate identity-property redundancy exists or does not. This measure is multivariate and lacks identity. Whether any measure can have both with local positivity is a separate question (Rauh et al., LIT-1202).
- Anything for noisy data. It is zero there.
Connections
- It is the “only correlation makes redundancy” view of Griffith and Koch’s first version (THEORY-716), made exact: in AND it gives their v1 answer, 0 redundant and 0.189 synergistic.
- It lies below THEORY-706’s I_min, and disagrees with THEORY-708 and THEORY-705 wherever the sources are noisily correlated.
- THEORY-568’s Shannon invariants do not depend on it, since its atoms satisfy the consistency equations.