nucleation

LIT-1203: Intersection Information Based on Common Randomness

2 min read

Properties6
StatusActive — read in full 2026-10-17 (NOTE-967); worth reading as the measure of redundancy built from the Gács–Körner common random variable, I_∧(X_1, …, X_n : Y) = I(X_1 ∧ … ∧ X_n : Y), defined for any number of sources, the first to satisfy target monotonicity, and by construction a lower bound on any reasonable redundancy. Read it knowing that it is zero whenever the sources' joint distribution has full support, so it is discontinuous and fails the identity property, that its synergy goes negative (−0.99 bit in ImperfectRdn, −0.252 in Subtle), and that its zero-error variant, which satisfies identity, goes negative too. It is the formal version of the view that only correlation between the sources makes redundancy. Filed as THEORY-715.
Tagsinformation-theory, complex-systems
ExtendsLIT-1172 — Nonnegative Decomposition of Multivariate Information
Compared againstLIT-1172 — Nonnegative Decomposition of Multivariate Information, LIT-1168 — A Bivariate Measure of Redundant Information
Cited as “paper” byNOTE-967 — Griffith, Chong, James, Ellison and Crutchfield — Intersection information based on common randomness
Cited as “source” byTHEORY-715 — The information about a target carried by the Gács–Körner common random variable of the sources is a redundancy defined for any number of sources that is nonnegative, target-monotone and no larger than I_min, and is a lower bound on any reasonable redundancy; but it counts only deterministic shared structure, so it is zero whenever the sources' joint distribution has full support, fails the identity property and continuity, and gives negative synergy already for two sources

Graph View

Virgil Griffith, Edwin K. P. Chong, Ryan G. James, Christopher J. Ellison and James P. Crutchfield (2014), “Intersection Information Based on Common Randomness”, Entropy 16(4): 1985–2000 — ARXIV-1310.1538

Key takeaways

From the reading, NOTE-967.

  • The measure (Eqs. 4–5, Lemma 8). The largest I(Q : Y) over Q that is a function of every source is attained by the common random variable X_1 ∧ … ∧ X_n, so I_∧ = I(X_1 ∧ … ∧ X_n : Y). It is defined for any number of sources.
  • Properties (Lemmas 4–6). It satisfies global positivity, equivalence-class invariance, target monotonicity, weak monotonicity, weak symmetry, the lower bound and self-redundancy. It fails identity, weak and strong local positivity, strong monotonicity and strong symmetry. And 0 ≤ I_∧ ≤ I_min (Lemma 9).
  • Its failure (§5.3). It sees only deterministic shared structure. With any noise between the sources, so that their joint distribution has full support, it is zero, and the synergy goes negative: ImperfectRdn gives −0.99 bit.
  • The zero-error variant (§3). H(X_1 ∧ … ∧ X_n ∧ Y) satisfies nine of eleven properties, identity among them in its zero-error form, but not local positivity. Subtle gives −0.252 bit.
  • Diagnosis (§§6–7). Negative synergy means the intersection is too small. Identity and local positivity are incompatible within zero-error information, as Bertschinger et al. show more generally.

Standing in the record

Filed on 2026-10-17 with Griffith and Koch, and read the same day. Filed as THEORY-715. It is the one redundancy measure the record holds that is defined for any number of sources. It does not close the gap the record named, an identity-property redundancy for ΦID beyond two sources: it fails identity and goes negative already for two. It formalises “source redundancy only”. In AND it gives 0 redundant, 0.311 unique each and 0.189 synergy, which is exactly Griffith and Koch’s v1 answer, and the opposite of Harder, Salge and Polani’s mechanistic redundancy. That correspondence is the reading’s. It carries no instruction for machine-learning practice.

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