Source
Harder, Salge & Polani (2013), “A Bivariate Measure of Redundant Information”, arXiv 1207.2080v3 (Phys. Rev. E 87, 012130), LIT-1168, read in full in NOTE-944. The PRE version was not compared.
What was actually shown
The counterexample. For Z = (X, Y), X and Y independent uniform bits, I_min(Z; X, Y) = 1 bit. Recomputed: 1.0000, and I_min stays 1 as the correlation between X and Y varies, while I(X; Y) goes from 1 to 0.
The measure. Project each p(z|x) by reverse KL onto conv{p(z|y)}; the projected information is Σ_x p(x)[D_KL(p(z|x) ‖ p(z)) − D_KL(p(z|x) ‖ projection)]. I_red is the smaller of the two directions.
Its properties. Symmetry, self-redundancy, monotonicity, identity (I_red((X,Y); X, Y) = I(X; Y)), and nonnegative synergy (Lemma 7), all proved.
Comparison. Same as I_min on XOR (0), AND (0.311), RdnXor (1) and XorAnd (0.5); different on copying (I(X; Y) against 1) and RdnUnqXor (1 against 2). All recomputed.
Transfer entropy. Williams and Beer’s coupled pair: same decomposition under both measures, at every d (recomputed at five values). A process with an independent bystander: at d = 1, all state-independent under I_red, all state-dependent under I_min (recomputed at d = 0.75 and 1).
What could have come out otherwise: a measure satisfying the identity axiom could have broken monotonicity or produced negative synergy. This one does neither, for two sources.
What this does not say
- That I_red is the right redundancy. It is one measure that satisfies four axioms. The identity axiom is argued for, not derived, and other measures with it exist.
- That independent sources share nothing. The paper says they can, by the mechanism (AND keeps 0.311 bit). That is a position.
- Anything for three or more sources. No multivariate I_red is given, so it cannot replace I_min in ΦID or in the three-source lattice.
- That I_min always overstates. Fig. 3’s upper-bound pattern has unexplained exceptions.
Connections
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The view this paper opposes is now held. Griffith and Koch’s later version concedes mechanistic redundancy (THEORY-716), so AND’s redundancy is 0.311 under I_min, I_red and Bertschinger et al.’s measure alike. Griffith et al.’s I_∧ is the opposite pole, with 0 (THEORY-715). Neither I_red nor any identity-respecting measure extends to three sources with nonnegative atoms (THEORY-713).
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It corrects THEORY-706’s I_min on shared information, and keeps its lattice and inversion.
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It leaves THEORY-712’s two-term split and THEORY-707’s examples unchanged in value. It confirms that the SITE/SDTE amounts depend on the redundancy measure.
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THEORY-581’s synergy bias is computed with I_min, the measure this corrects.
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THEORY-568’s invariants hold for I_red too, since its atoms satisfy the consistency equations.