Contribution
Before this paper, the multivariate redundancy measures on the Williams–Beer lattice either overstated redundancy (I_min, two-bit copy) or were bivariate (I_red, I_broja). I_ccs is defined for any number of sources and gets the two-bit copy right. It pays for that, as Rauh et al. prove some measure must, with negative atoms, and it argues they should be read as “misinformation” and “mechanistic redundancy” rather than as defects.
Key insight
Mutual information is an average of local terms that can be positive (information) or negative (misinformation). A local co-information term is an overlap only if every quantity it is built from moves the same way. Count those and drop the rest, and redundancy becomes a set intersection at the level of single events. Monotonicity then fails for a plain reason: a source can carry local misinformation the other source does not.
Assumptions
- Finite or Gaussian variables. Discrete in §§4–6; univariate and multivariate Gaussians in §7, by numerical integration.
- The joint used is P̂, not P (Def. 4.2). The maximum-entropy distribution with Q(Ai, S) = P(Ai, S) for every i and Q(A1…An) = P(A1…An). So redundancy depends only on those marginals. Def. 4.3, without the source–source constraint, is offered for readers who keep Bertschinger et al.’s operational view.
- Unique information may depend on P(X1, X2) (§4.3). This is the paper’s operational premise: a simultaneous, one-shot, non-cooperative game with equal but asymmetric utility (one player sets the stake) operationalises unique information.
- A fundamental distinction between local information and local misinformation. Terms of opposite sign are never treated as overlapping.
- Continuity is conjectured, not proved (§4.4). The code checks the local condition (Eq. 42) in each example.
Key results
- Def. 4.1 — I_ccs(S; A1…An) = Σ p̂ · Δ_s h_com, with Δ_s h_com = c_p̂(a1…an, s) when sgn Δ_s h(a1) = … = sgn Δ_s h(an) = sgn Δ_s h(a1…an) = sgn c, and 0 otherwise.
- Properties (§4.4) — symmetry; self-redundancy, since for n = 1 the local co-information is the local mutual information; subset equality, proved by cancelling terms; the independent identity property (Eq. 22). Not Harder et al.’s identity: “any distribution with negative local information terms serves as a counter example”. Not monotonicity.
- Table 7 — x1, x2, s with p(0,0,0) = 0.4, p(0,1,0) = 0.1, p(1,1,1) = 0.5: I(S; X1) = 1, I(S; X2) = 0.61, I_ccs({1}{2}) = 0.77. The PID is 0.77, 0.23, −0.16, 0.16. Checked: 0.7685, 0.2315, −0.1585, 0.1585.
- REDUCEDOR (Table 6) — I_ccs gives redundancy 0, unique 0.31 each, synergy 0.38; I_min and I_broja give 0.31 redundancy and 0.69 synergy. Checked: 0, 0.3113, 0.3113, 0.3774.
- SUM (Table 5) — redundancy 0, unique 0.5 each, synergy 0.5, where I_broja gives 0.5 redundancy. Checked: 0.0001 (iterative-fitting error), 0.4999, 0.4999, 0.5001.
- AND (Table 12) — one redundant event, (0,0,0). Computed: redundancy 0.1038, unique 0.2075 each, synergy 0.2925. The extracted Table 11 lost its numbers, so these are mine, consistent with Table 12’s single pointwise term (¼ × 0.415).
- XOR, RDN, two-bit copy — 1 synergistic; 1 redundant; 0 redundant and 1 unique each. Checked.
- Three sources (§6) — XORCOPY: {i}{jk} = 1, {12}{13}{23} = −1. DBLXOR: {12} = {13} = {23} = 1, {123} = −1. Giant bit: {1}{2}{3} = 1. Parity: {123} = 1. XORUNQ: {3} = {12} = 1. All checked with a full Möbius inversion on the 18-node lattice.
- Gaussians (§7, Fig. 10) — the minimum-MI redundancy is constant in the predictor–predictor correlation b, while I_ccs moves from unique at negative b to redundant at strong positive b. Read from the prose only.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | I_ccs is a redundancy measure for any number of sources satisfying symmetry, self-redundancy, subset equality and independent identity | strong for three; false for subset equality (note of 2026-10-20) | §4.4, proved for these four |
| C2 | I_ccs is not monotone, so its PID has negative atoms | strong | Table 7, checked |
| C3 | Negative atoms are a necessary consequence of measuring overlapping content | interpretive | §§4.4, 8; supported by DBLXOR and by Rauh et al.’s theorem, not proved for every such measure |
| C4 | Bertschinger et al.’s operational definition is not necessary for unique information | moderate | REDUCEDOR game (§4.3.1); a counterexample to necessity of their criterion, given the paper’s game-theoretic premise |
| C5 | So I_broja is “invalid”, since redundancy is not fixed on Δ_P | weak as stated | follows only if the game-theoretic premise is accepted; Bertschinger et al. would reject it |
| C6 | I_broja conflates synergy into redundancy | moderate | Fig. 5A example: one local term of −0.138 bit enters its co-information |
| C7 | I_ccs gives intuitive PIDs across the literature’s examples | interpretive | §§5–6; judged against intuition, reproduced here for the discrete ones |
| C8 | I_ccs is continuous | conjecture | §4.4 |
Method
- Build P̂ by maximum entropy under the marginal constraints (the paper uses dit; iterative proportional fitting gives the same here).
- For each event, compute each source’s local information about s, the joint source’s, and the local co-information (Eq. 29).
- Keep the co-information where all signs agree; average under P̂.
- Möbius-invert on the lattice to get atoms, which may be negative.
Concepts
- Local misinformation — a negative local mutual information, the observation raising the surprisal of the target value.
- Independent identity property — I(A1; A2) = 0 ⇒ I∩((A1, A2); A1, A2) = 0 (Eq. 22), introduced here.
- Subset equality — split off from monotonicity, and in Ince’s view the axiom the lattice actually needs.
- Mechanistic redundancy — redundancy between sources that are independent, created by the mechanism; the paper’s reading of the negative atoms in DBLXOR and XORCOPY.
Connections
- LIT-1202 (Rauh et al.). Their XORCOPY is Ince’s §6.2.2, and Ince cites them for it. Theorem 2 bounds the atom {12}{13}{23} at −1 for any measure with identity and non-negative atoms. I_ccs meets the bound exactly while keeping only independent identity. So it is an instance of the theorem’s escape route, not a refutation.
- LIT-1162 (Bertschinger et al.). The disagreement is over what unique information is, operationally. Under Def. 4.3, without the source–source constraint, I_ccs can be computed in their framework. The two agree on every example in the record (NOTE-937) and differ on REDUCEDOR, SUM and AND.
- LIT-1168 (Harder et al.). Ince keeps their two-bit-copy verdict but rejects their identity in general, because mutual information can contain synergistic entropy.
- ΦID in this record. NOTE-799 (Luppi et al.) uses the minimum-mutual-information redundancy for Gaussian BOLD signals. No held ΦID work uses I_ccs. §7 shows the two differ for Gaussians whenever the predictors are correlated, so the choice is not cosmetic for those analyses. My connection.
Bearing on the record
- Filed as THEORY-714.
- CLAIM-335’s examples are unchanged under I_ccs. Y_t = X_{t−1} ⊕ Y_{t−1}: redundancy 0, unique 0 and 0, synergy 1. Z_t = X_{t−1} ⊕ Y_{t−1}, pairwise sources (Z_{t−1}, X_{t−1}): every atom 0; joint sources (X_{t−1}, Y_{t−1}): synergy 1. Computed here.
- Williams and Beer’s coupled pair (THEORY-712) is unchanged: SITE/SDTE at d = 0.25, 0.5, 0.75 are 0.4564/0.1379, 0.1887/0.3113, 0.0456/0.5488, as under I_min, I_red and I_broja.
- Harder et al.’s bystander example is not: at d = 0.75 I_ccs gives SITE 0.392, SDTE 0.157 and redundancy 0.065, against 0/0.549 (I_min) and 0.411/0.138 (I_red); at d = 0.25 it gives a negative SITE, −0.019. A third measure confirming that the split depends on the measure off the record’s examples. Computed here.
- NOTE-010’s permutation (X¹⊕X², X¹): redundancy 0, unique 1 each, synergy 0, as under I_red. Independent identity is enough here.
- For the Anthology of the SOTA: no instruction for ML practice. The method is offered for neuroimaging analyses, but as a measure, not as practice.
Limitations
- The case against I_broja rests on a premise, the game-theoretic operationalisation, that the other side need not grant. REDUCEDOR shows the two premises disagree, not which is right.
- “Correct” and “intuitive” PIDs are judged by the author’s intuition about the systems.
- Continuity is a conjecture; the measure is not differentiable, by its thresholding.
- Bias and inference from finite samples are left open (§8).
- Four-variable lattices (166 nodes) are discussed but not computed.
Open questions
- Separate lattices for local information and local misinformation, each conjectured monotone (§8).
- Whether I_ccs satisfies Rauh’s secret-sharing property in general (§8).
- A partial entropy decomposition compatible with it (Ince 2017, not held).
Note of 2026-10-20: subset equality does not hold
The promotion audit (batch 056) re-derived §4.4. The subset-equality proof (Eqs. 39–41) shows that the local co-information of a collection is unchanged when a source contained in another is added. That part is right. But Def. 4.1 keeps an event only if every source’s local information has the same sign, and a source and a source containing it can disagree. Take A1 = X1 and A2 = (X1, X2) with p(x1, x2, s) ∝ (0,0,0) 0.2, (0,0,1) 0.05, (0,1,0) 0.05, (0,1,1) 0.2, (1,0,0) 0.3, (1,0,1) 0.2. Then I_ccs({1}{12}) = 0.0272 bit against I(S; X1) = 0.0073. The other kept properties, and every value this reading recomputed, stand. C1 is corrected above, and the summary’s “keeps … subset equality” should be read with this note.