Zaslavsky et al. (2019): Color Naming Reflects Both Perceptual Structure and Communicative Need #
This file formalizes [ZKTR19]'s information-theoretic link between communicative
need and communicative precision in color naming. A language's color naming distribution
p(w | c) is a channel κ : Kernel C W from colors to words and a need distribution is a prior
μ over colors. The expected surprisal S(c) of a color (eq. 1) is the surprisal of the color
to a listener who hears its name and inverts the lexicon by Bayes' rule (eq. 2), the measure of
communicative imprecision with which [GFJE+17] found warm colors named more precisely
than cool ones. The capacity of the lexicon (eqs. 3 and 4) is channelCapacity κ, and the
paper's central observation is that the need distributions attaining it are those with
p(c) ∝ exp(−S(c)) (eq. 5, capacityAchieving_iff), so that −log p(c) is linear in S(c)
with slope one (eq. 6, neg_log_eq_add_channelCapacity). The intercept log Z of eq. 6 is the
capacity itself.
The divergence of the row κ c from the word marginal is −S(c) − log p(c)
(toReal_klDiv_eq), so eq. 5 is the divergence condition of InformationTheory's capacity
theorems: sufficiency (measureMutualInfo_compProd_eq_channelCapacity) and necessity at a prior
of positive mass everywhere (toReal_klDiv_eq_channelCapacity).
The artificial naming systems obtained by clustering the chips in perceptual space are
deterministic channels (ofPartition). Under a uniform need the surprisal of a color is the log
size of its cluster (expectedSurprisal_ofPartition_uniformOn), so that their warm–cool
asymmetry is an asymmetry in cluster size (warmCoolAsymmetry_ofPartition_uniformOn_iff). The
prior spreading mass equally over clusters (clusterPrior) is capacity-achieving
(capacityAchieving_clusterPrior) and the capacity of a k-term clustering system is log k
(channelCapacity_ofPartition). The universal need distribution inferred from the World Color
Survey averages the per-language capacity-achieving priors (eq. 7), which are the least
informative priors of [ZKRT18] (isLeastInformative_iff).
Implementation notes #
- The paper states eq. 5 as necessary and sufficient for a prior to be capacity-achieving.
capacityAchieving_iffstates it with the positivity that eq. 5 itself entails: a need distribution satisfies eq. 5 exactly when it is capacity-achieving and gives every color positive mass. A capacity-achieving prior may leave a color unused, and eq. 5 then fails. - Every prior giving the clusters of a clustering system equal mass satisfies eq. 5, so the
capacity-achieving prior of such a system is not unique;
clusterPrioris the one uniform within each cluster. - The perceptual coordinates of the chips and the
k-means procedure are not represented; a clustering enters only through its assignment of chips to terms. The survey data that instantiateWarmCoolAsymmetryare not in the library.
References #
- N. Zaslavsky, C. Kemp, N. Tishby and T. Regier, Color naming reflects both perceptual structure and communicative need (2019)
- E. Gibson, R. Futrell, J. Jara-Ettinger, K. Mahowald, L. Bergen, S. Ratnasingam, M. Gibson, S. T. Piantadosi and B. R. Conway, Color naming across languages reflects color use (2017)
- T. M. Cover and J. A. Thomas, Elements of Information Theory (2006)
- C. E. Shannon, A Mathematical Theory of Communication (1948)
- N. Zaslavsky, C. Kemp, T. Regier and N. Tishby, Efficient compression in color naming and its evolution (2018)
Communicative precision #
The expected surprisal S(c) of a color (eq. 1): the surprisal of the color to a listener
who hears its name and recovers a color by Bayes' rule (eq. 2), averaged over its names. Lower
values mean more precise communication.
Equations
- ZaslavskyEtAl2019.expectedSurprisal κ μ c = ∫ (w : W), InformationTheory.surprisal ((ProbabilityTheory.posterior κ μ) w) c ∂κ c
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Equations
- ZaslavskyEtAl2019.instDecidableEqTemperature x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
Equations
- One or more equations did not get rendered due to their size.
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Equations
The warm–cool asymmetry of [GFJE+17]: under the need distribution μ, the warm
colors have lower mean expected surprisal than the cool ones.
Equations
- One or more equations did not get rendered due to their size.
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Need and precision at capacity #
The expected surprisal is the negative expected log posterior.
The identity behind eq. 5: the divergence of a color's naming distribution from the word marginal is its negative expected surprisal less its log need.
Eq. 6: at a capacity-achieving need distribution of positive mass everywhere, −log p(c) is
linear in the expected surprisal with slope one, and the intercept is the capacity of the
lexicon.
Eq. 5: a need distribution satisfies p(c) ∝ exp(−S(c)) exactly when it is
capacity-achieving and gives every color positive mass. The condition says that the need
distribution is a fixed point of the Blahut–Arimoto update (eq_blahutArimoto_iff).
Naming systems from a hard clustering #
The naming system of a hard clustering: each color is named by the term of its cluster.
Equations
- ZaslavskyEtAl2019.ofPartition f = ProbabilityTheory.Kernel.deterministic f ⋯
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Under a clustering system, the surprisal of a color is the log need of its cluster less its own log need.
Under a uniform need, the surprisal of a color in a clustering system is the log size of its cluster (Fig. 3C).
For a clustering system, the warm–cool asymmetry under a uniform need is an asymmetry in mean log cluster size (Fig. 3C).
The capacity-achieving prior of a clustering system #
The need distribution that spreads mass equally over the clusters of f and uniformly
within each: a capacity-achieving prior of one clustering system, of the kind averaged into
KM-CAP (Fig. 4B).
Equations
- ZaslavskyEtAl2019.clusterPrior f = (↑(Finset.image f Finset.univ).card)⁻¹ • ∑ w ∈ Finset.image f Finset.univ, ProbabilityTheory.uniformOn (f ⁻¹' {w})
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Under the cluster prior, the surprisal of a color is the log size of its cluster.
The cluster prior satisfies eq. 5 with normalizer the number of terms, so it is a capacity-achieving prior of the clustering system (Fig. 4B).
The capacity of a k-term clustering system is log k: the intercept of eq. 6 at its
capacity-achieving prior.
The universal need distribution #
Eq. 7 averages the per-language capacity-achieving priors into a universal need distribution,
following [ZKRT18]: it is that paper's least informative source
(ZaslavskyKempRegierTishby2018.averagePrior), whose per-language priors are the least
informative priors of the languages' naming distributions.
The per-language priors averaged in eq. 7 are the least informative priors of [ZKRT18]; those of positive mass everywhere are exactly the need distributions satisfying eq. 5.