Source
Bertschinger, Rauh, Olbrich, Jost & Ay (2014), “Quantifying unique information”, arXiv 1311.2852v2 (Entropy 16(4): 2161–2183), LIT-1162, read in full in NOTE-937. The Entropy version was not compared.
What was actually shown
The definitions. For target X and sources Y, Z, Δ_P is the set of joints with P’s (X,Y) and (X,Z) marginals. UĨ(X : Y \ Z) = min_{Q∈Δ_P} MI_Q(X : Y | Z), SĨ = max CoI_Q, CĨ = MI(X : (Y,Z)) − min MI_Q(X : (Y,Z)). One convex programme gives all three (Lemma 4).
The bounds (Lemma 3). If a non-negative, continuous decomposition has unique information depending only on p(x), p(y|x), p(z|x), then its unique information is at most UĨ, and its shared and synergistic information at least SĨ and CĨ. Equality holds exactly when some joint in Δ_P has no synergy.
Operational meaning (Lemma 6, Corollary 7). UĨ(X : Y \ Z) = 0 if and only if Y is a garbling of Z with respect to X, that is, Z dominates Y in Blackwell’s order.
Properties. Non-negativity (Lemma 5); symmetry, consistency and the bivariate PI axioms (Lemmas 11–13); the identity property, SĨ((Y,Z) : Y; Z) = MI(Y : Z) (Proposition 18); additivity (Lemma 19); unique information that cannot grow as sources are added (Lemma 25).
The two-term split (Remark 14). For any binary decomposition, MI(X : Z | Y) = UI(X : Z \ Y) + CI(X : Y; Z).
Checked (NOTE-937): AND gives SĨ = 0.311 and CĨ = 0.5, as in Example 30. XOR, RDN and Copy reproduce Table 1, except its Unq row (CĨ printed as 1; Proposition 18 and the computation give 0) and its Copy entry (printed as MI(X : Y); Proposition 18 and the computation give MI(Y : Z)).
What could have come out otherwise: the bound need not have been attained by any decomposition. Lemma 3 shows it is attained exactly when Δ_P holds a synergy-free joint, and Lemma 5’s construction shows the shared part is non-negative.
What this does not say
- That property (∗) is required of unique information. It is argued from decision problems. A measure without it is not shown wrong.
- Anything for three or more sources. By §6, citing earlier work, the identity property is incompatible with a non-negative decomposition on the PI lattice for three sources.
- That the optimum is easy to find. Convex, but ill-conditioned (Example 31). An unparametrised solver run got AND wrong in this reading.
- That transfer entropy’s split depends on I_min in the record’s examples. It does not (NOTE-937): where the target’s past carries no information about its next value, any such decomposition gives the same split.
Connections
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For two sources its synergy is Griffith and Koch’s S_VK: the constraint set is the same, and H(Y) is fixed on it (THEORY-716; the record’s derivation). Rauh et al. show its unique information is monotone and its shared and synergistic parts are not, and that no identity-respecting extension to three sources keeps atoms nonnegative (THEORY-713).
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It answers the flaw in Williams and Beer’s I_min (LIT-1172) that Harder, Salge and Polani identified (LIT-1168), and is compared with their I_red.
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Remark 14 is the two-term split behind THEORY-712, THEORY-707 and CLAIM-335.
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THEORY-568’s Shannon invariants hold for this decomposition, as for any that satisfies the consistency equation.
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Lemma 6 is Blackwell’s theorem as THEORY-156 states it (Proposed): being at least as informative for every decision problem is being able to simulate the other by a garbling. Here it is relative to the target X.
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THEORY-497 (ΦID) needs a redundancy function for more than two variables. This one, as it stands, is not one.