Contribution
It gives the Gricean maxim of quantity a formal, quantitative form for simple reference: a speaker’s utility is the negative KL divergence from her intended meaning to the literal meaning of the word she chooses. Inverting that speaker gives a learner who can infer what a novel word means from the context it was used in. Two experiments test the two sides: learners’ graded judgements (well fit) and speakers’ free productions (weakly fit).
Key insight
A word is informative to the degree that it narrows the context to the intended object. If speakers choose words that way, the choice of word is itself evidence. A learner who sees a speaker point at the one red circle among blue shapes and say “lipfy” should lean towards “red”, because “red” singles the object out and “circular” does not. The size principle, which had been justified as a rule about generalisation, here becomes a consequence of the speaker’s goal.
Assumptions
- A fixed context C of objects. Meanings are distributions over C.
- Words have Boolean, truth-functional meanings. A word’s literal meaning w̃_C is uniform over its extension and 0 elsewhere (Eq. 4).
- The speaker soft-maximises U(w; M_S, C) = −D_KL(M_S ‖ w̃_C) + F with decision noise α (Eqs. 5–6). α = 1 and F = 0 throughout the model predictions.
- The intended meaning is a point mass on the referent in both experiments. The framework allows vague meanings, but they are not tested.
- The learner knows the referent (ostensive learning), and in Eq. 12 there are exactly two candidate lexicons with a uniform prior.
Key results
- Eq. 8–10. For M_S = δ_{o_S}, D_KL(M_S ‖ w̃_C) = −log w̃_C(o_S) = log |w|. So P(w | M_S, C) = |w|^(−α) / Σ_{w′ true of o_S} |w′|^(−α) over the true words, and 0 for words false of the referent.
- Eq. 12. For two features and two candidate lexicons, P(L₁ | w₁, M_S, C) = |f₁|⁻¹ / (|f₁|⁻¹ + |f₂|⁻¹). In the Figure 1 display (one red object, two circles) this gives 2/3.
- Experiment 1. 700 participants from MIT mailing lists, three trials each, six objects varying on two binary dimensions. They split 50 on the diagonal (equal counts), and the Figure 1 case gave 67.
- Experiment 2. 44 participants saw all 304 superball photos, then wrote descriptions of 50 so that someone could pick each out, about ten descriptions per ball. Each description was treated as a bag of words. Tag-words never used for a ball were excluded. The probability of using a tag-word against Eq. 10’s prediction: r = .19 (p < .0001), r = .02 for basic colour terms (p = .63), r = .51 for the rest. A word-frequency estimate of F from Google Images moves these to .22 and .52.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Minimising KL from intended to literal meaning yields, for a single referent, word choice ∝ |extension|^(−α) | strong (derivation) | Eqs. 7–10 |
| C2 | Learners inferring a novel word’s meaning from a known referent behave as if inverting such a speaker | moderate | Experiment 1: r = .93 on condition means, one-shot web survey, no competing model fitted |
| C3 | Speakers’ word choice in free description is related to informativeness | weak | Experiment 2: r = .19 overall, .51 excluding colour terms; tags not chosen for this task |
| C4 | Colour-term use fails because of how well a colour applies to the object, not production cost | weak (conjecture) | Experiment 2 discussion; adding a frequency proxy for F barely changes the fit |
| C5 | The framework can handle scalar implicature, anaphora and other phenomena | not supported here | general discussion; no such case is modelled |
Method
The model is a one-step speaker, P(w | M_S, C) ∝ exp(−α D_KL(M_S ‖ w̃_C)), inverted once by Bayes’ rule for a learner who knows M_S and is uncertain about the lexicon. The predictions in Experiment 1 have no free parameters (α = 1, uniform prior). Experiment 2 applies Eq. 10 to the ball’s tags, read as its features, with extensions counted over the whole 304-photo set.
Concepts
- meaning: a probability distribution over the objects in the context.
- literal meaning of a word (w̃_C): uniform over the word’s extension in C.
- informativeness: the negative KL divergence from the intended meaning to the word’s literal meaning, i.e. the information about the intended meaning that the word fails to carry.
- F: a residual term in the utility for other factors, such as utterance complexity. It is set to 0 in the model and estimated from word frequency in one post-hoc analysis.
- ostensive learning: learning where the referent is known and only the word’s meaning is inferred.
Connections
It builds on Grice’s maxim of quantity, Sperber and Wilson’s relevance and Clark’s common ground, and gives them a formal analogue. The size principle it re-derives is Shepard’s (1987), as used by Tenenbaum and Griffiths (2002) and Xu and Tenenbaum (2007). It cites Shafto and Goodman’s pedagogical sampling (2008) as the teacher’s side of the same idea. The authors place it next to game-theoretic pragmatics (Benz, Jäger and van Rooij 2005) and distinguish it by having no recursion. Their companion word-learning model (cited “in press”, now LIT-815) handles the case where the referent is also unknown.
Bearing on the record
- THEORY-172. That account (listeners invert an informative
speaker, so a fixed form’s interpretation depends on the alternatives)
is sourced from Frank and Goodman 2012, Bergen, Goodman and Levy 2012
and Goodman and Frank 2016. This paper is its precursor and supports
it in a neighbouring form: the inference is about the lexicon, given
the referent, not about the referent. Experiment 1’s r = .93 is again a
correlation of condition means with a parameter-free model, which is
exactly the kind of evidence that THEORY’s
promote_whensays cannot settle it. It adds support, not promotion. - On the speaker side. THEORY-172 explicitly does not say speakers are rational. Experiment 2 is the record’s only direct test of production, and it shows a weak relation that fails for colour terms. It does not warrant a THEORY of its own: one corpus of tags, chosen by a photo owner, a bag-of-words scoring and a post-hoc account of the failure.
- On the LIT for the Psychological Science paper. LIT-815’s status note says that RSA work cites this paper, not that one, for the informative speaker. This reading confirms it: the informative speaker is here (Eqs. 5–10), and the Psychological Science paper has none.
- No instruction for machine-learning practice; nothing for the anthology.
Limitations
- Single-referent intended meanings only, with Boolean word meanings and uniform literal distributions. The framework’s claimed generality is not exercised.
- Experiment 1 tests condition means against one model with no alternative model fitted. The histograms in Figure 2 show wide spread within conditions, which the mean fit does not address.
- Experiment 2’s “features” are an album owner’s tags. Extensions are counted over all 304 photos, though participants described each ball alone. The bag-of-words scoring ignores what a description was about.
- The fit in Experiment 2 is weak, and the explanation for colour terms is not tested.
- α is fixed at 1 rather than estimated.
Open questions
- Does a model with estimated α and a cost term, fitted on part of the data and tested on the rest, predict production better than frequency alone?
- Does the learner’s inference survive when the referent is uncertain? The companion model treats that case, but without an informative speaker.
- Can the KL formulation with non-uniform literal meanings, or vague intended meanings, separate this account from a plain size-principle learner? Here the two coincide by construction.