Source
- Zaslavsky, Kemp, Regier & Tishby (2018), LIT-810, read in NOTE-631: Eqs. 1–7, Fig. 3, Table 1, the rotation control, Fig. 5.
- Tishby, Pereira & Bialek (1999), LIT-338, read in NOTE-300: the objective and its self-consistent solution.
The claim
Let a language’s colour lexicon be a naming policy q(w|m) from meanings m (distributions over colours, one per World Color Survey chip) to words, with a Bayesian listener reconstructing m̂_w. Score it by complexity I(M;W) and accuracy I(W;U). Then the attested systems of the WCS languages and of English lie close to the information-bottleneck curve, the set of minimizers of I(M;W) − β I(W;U), each at a β only slightly above 1, and closer to it than systematically rotated versions of themselves. Because optima at finite β are stochastic, graded category membership and regions named inconsistently across speakers are properties of efficient systems.
What was actually shown
Each language was fitted to the β at which it is closest to optimal, and its distance measured (ε_l = 0.18 with the LI source, held-out languages). This could have failed in three ways, and did not: the languages could have been far from the curve at every β; their full naming distributions could have matched the IB encoders no better than a deterministic efficiency model (gNID 0.18 for IB against 0.47 for RKK+); and rotated variants, which keep each system’s complexity profile but misplace it in colour space, could have done as well (93% of languages beat all 39 of their own). The yellow category IB predicts at the lowest complexities is not in the data, a stated failure.
What this does not say
- Not that a drive to efficiency produced these systems. Near the bound is a property of the end states; no transmission or learning process was modelled, and the “evolution” is the path of optima through β, compared with Berlin and Kay’s stages by eye. How a population would reach the bound is open; iterated learning (THEORY-155) is a theory of where transmission goes, and the two have not been joined.
- Not that the bound is fitted without help. The need distribution that gives the best fit is built from the naming data, averaged and cross-validated over languages; with a uniform source the fit is worse (gNID 0.39).
- Not that any lexicon in any domain is IB-efficient. The result is for colour, with a fixed CIELAB perceptual model; other domains are the authors’ future work.
- Not that I(W;U) is the right notion of accuracy for communication in general. It follows from taking KL as the distortion between meanings, which is one choice of loss.
Note of 2026-10-20: the wider claim filed separately
When this THEORY was narrowed to what its source showed, the wider claim it had stated was left with nowhere to go but this body and a QUESTION. Under ADR-047 v4 it is now its own Proposed THEORY, THEORY-791, which extends this one: the general claim about the world’s languages. This entry is that THEORY’s evidence and is not changed by it.