Source
Griffith & Koch (2014), “Quantifying synergistic mutual information”, arXiv 1205.4265v6 (Guided Self-Organization: Inception, Springer), LIT-1204, read in full in NOTE-966. v1 and v2 were read for the passage v6 withdraws. The book chapter was not compared.
What was actually shown
The measure. I_VK = min I*(X_1…n : Y) over Pr* with every Pr*(X_i, Y) fixed; S_VK = I(X_1…n : Y) − I_VK (Eqs. 13–14). The product Pr(y)ΠPr(x_i|y) is feasible and bounds I_VK above.
Against the prior measures (Table 1, recomputed). S_max gives Unq 1 bit of synergy, where there are two unique bits. WholeMinusSum gives RdnXor 0, one bit of synergy minus one of redundancy. ΔI gives AND 0.104, below WholeMinusSum’s 0.189, and AndDuplicate 0.038. S_VK gives the intended answer wherever one is stipulated.
Properties. Proved: global positivity, self-redundancy, weak symmetry, weak local positivity for two sources, strong identity I_VK(X : X_1…n) = H(X_1…n). Claimed without proof: monotonicity, target monotonicity. Disproved: strong symmetry.
Two sources. The constraint set is Bertschinger et al.’s Δ_P, and H(Y) is fixed on it, so minimising I*(X_1X_2 : Y) is their maximisation of H(Y | X_1X_2). S_VK is their CI. For AND that gives 0.5 bit of synergy and 0.311 of redundancy, the top of the paper’s open interval [0.270, 0.5]. The record’s derivation, checked numerically.
The reversal. v1–v2 give AND zero redundancy “because X1 and X2 are independent”. v6 calls that an error and bounds AND’s redundancy below by 0.082.
What could have come out otherwise: I_VK’s bound could have failed to separate AND’s synergy from WholeMinusSum’s 0.189. It gives at least 0.270.
What this does not say
- That S_VK is the right synergy. The test is stipulated intuition on circuits, and AND’s answer was left open.
- Anything settled for three or more predictors. I_VK is defined for any n and keeps strong identity, but nonnegativity of the induced atoms and monotonicity are not proved there, and the authors say they were blocked.
- That independent sources must share information. v6 says they can under this measure; whether they should is the dispute between THEORY-708 and THEORY-715.
Connections
- For two sources it coincides with THEORY-705, which supplies the proofs of nonnegativity and identity this paper lacks there.
- It corrects the S_max built from Williams and Beer’s I_max (LIT-1172) on unique information, as THEORY-708 corrects THEORY-706’s I_min on shared information.
- THEORY-715 formalises the v1 position this version withdraws.