Qing and Franke (2015): Variations on a Bayesian Theme #
Qing and Franke decompose the rational speech act model of Frank and Goodman's referential game along its design choices. The literal listener conditions a prior on the extension of the word, and the prior is uniform or the measured salience of the objects. The speaker's goal is belief-oriented, the log of the literal listener's probability of the referent, or action-oriented, that probability itself, and colour adjectives carry a cost. The listener inverts a speaker against its own prior and may act on the result by a further soft maximization. The context has a square and a circle of one colour and a circle of the other, and the paper compares the models on the choices of speakers and listeners by their Bayes factors.
Main statements #
speaker_eq_softmax,speaker_belief_uniform: every speaker of the family soft-maximizes its utility over the true words, and the belief-oriented one at the uniform prior and no cost is Frank and Goodman's.prefers_unique_colour_iff,prefers_noun_iff: a speaker who ignores salience prefers the word only the referent satisfies exactly when the adjective costs less than the informativity gap, and at the object with both features shared prefers the noun exactly when it is the cheaper.uniform_listener_narrows_circle,salience_listener_window: a listener with a uniform prior takes both ambiguous words to refer to the object with both features shared, and one with the salience prior does so for green and follows salience for circle, as the paper's listeners did, on a bounded window of rationalities.frankGoodman_listener_green: Frank and Goodman's listener at the measured salience and rationality one takes green to refer to the salient green square.
Implementation notes #
- The lexicon and the model of a context are those of
FrankGoodman2012, and the belief-oriented speaker isRSA.speakeragainst the literal listener. The action-oriented speaker is the softmax of its utility over the true words, the restriction the paper applies in its model comparison. - The salience prior weights each object by its count in the salience condition, without normalizing.
- The action-oriented listener is
RSA.policyfor the game that rewards choosing the referent. It is not restricted to the objects the word is true of, which does not change their order. - The Bayesian model comparison is not formalized. The findings are the directions of preference each model predicts, as conditions on its parameters.
References #
The context, the cost and the salience prior #
Equations
The model M of the sample context gives each object its colour and its shape.
Equations
- QingFranke2015.M QingFranke2015.Object.greenSquare FrankGoodman2012.Attribute.colour = FrankGoodman2012.Colour.green
- QingFranke2015.M QingFranke2015.Object.greenSquare FrankGoodman2012.Attribute.shape = FrankGoodman2012.Shape.square
- QingFranke2015.M QingFranke2015.Object.greenCircle FrankGoodman2012.Attribute.colour = FrankGoodman2012.Colour.green
- QingFranke2015.M QingFranke2015.Object.greenCircle FrankGoodman2012.Attribute.shape = FrankGoodman2012.Shape.circle
- QingFranke2015.M QingFranke2015.Object.blueCircle FrankGoodman2012.Attribute.colour = FrankGoodman2012.Colour.blue
- QingFranke2015.M QingFranke2015.Object.blueCircle FrankGoodman2012.Attribute.shape = FrankGoodman2012.Shape.circle
Instances For
The colour adjectives cost c and the shape nouns nothing (11).
Equations
- QingFranke2015.cost c ⟨FrankGoodman2012.Attribute.colour, snd⟩ = c
- QingFranke2015.cost c ⟨FrankGoodman2012.Attribute.shape, snd⟩ = 0
Instances For
The salience prior S weights each object by the number of listeners who chose it when they
could not understand the speaker.
Equations
Instances For
The literal listener and the speakers #
The literal listener at a prior (1) conditions the prior on the extension of the word.
Equations
Instances For
The speaker's goal is belief-oriented or action-oriented.
Instances For
Equations
- QingFranke2015.instDecidableEqGoal x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
Equations
- QingFranke2015.instReprGoal = { reprPrec := QingFranke2015.instReprGoal.repr }
Equations
- QingFranke2015.instReprGoal.repr QingFranke2015.Goal.belief prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "QingFranke2015.Goal.belief")).group prec✝
- QingFranke2015.instReprGoal.repr QingFranke2015.Goal.action prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "QingFranke2015.Goal.action")).group prec✝
Instances For
The goal's measure of the literal listener's probability of the referent, its logarithm (10) or the probability itself (9).
Equations
- QingFranke2015.Goal.belief.value = Real.log
- QingFranke2015.Goal.action.value = id
Instances For
The utility of a word for a referent (4) is the goal's measure of the literal listener's probability of the referent, less the cost of the word.
Equations
- QingFranke2015.utility g μ c t u = g.value (((QingFranke2015.literal μ) u).real {t}) - QingFranke2015.cost c u
Instances For
The speaker models σ_xy with goal x, belief y about the literal listener's prior,
rationality lam and adjective cost c. The belief-oriented speaker is the rational speech act
speaker against the literal listener, and the action-oriented speaker soft-maximizes its utility
over the true words.
Equations
- One or more equations did not get rendered due to their size.
- QingFranke2015.speaker QingFranke2015.Goal.belief x✝² x✝¹ x✝ = RSA.speaker x✝¹ (QingFranke2015.cost x✝) (⇑(QingFranke2015.literal x✝²)) MeasureTheory.Measure.dirac
Instances For
At the uniform prior the literal listener is Frank and Goodman's.
At the uniform prior and no cost the belief-oriented speaker is Frank and Goodman's.
Every speaker of the family soft-maximizes its utility over the true words, (9) and (10).
Of two words true of an object the speaker prefers the one of greater utility.
An object has two true words, the one for its colour and the one for its shape, and the share of one is the logistic function of its utility over the other's, scaled by the rationality.
Speakers who ignore salience #
At the uniform prior a word is worth the goal's measure of the reciprocal of its extension's size.
Every word of the context is true of one object or of two, and the difference in their worth is
log 2 to the belief-oriented speaker and 1/2 to the action-oriented one.
At the uniform prior the literal listener gives each object in the extension of a word the reciprocal of the extension's size.
The gap exceeds every cost in the paper's support (-0.4, 0.4).
At the blue circle the speaker prefers blue, which only it satisfies, to circle exactly when the adjective costs less than the informativity gap.
At the green square it prefers square, which only it satisfies, to green exactly when the adjective is not cheaper than the noun by the informativity gap.
At the green circle, whose colour and shape are each shared, the two words are equally informative and the speaker prefers the noun exactly when it is the cheaper.
The belief-oriented speaker who heeds salience #
At the salience prior the literal listener gives each object in the extension of a word its share of the extension's salience.
The belief-oriented speaker who takes the literal listener to hold the salience prior (7)
prefers blue to circle at the blue circle exactly when the adjective costs less than
log (169 / 139), the salient blue circle being the likelier circle.
At the green circle it prefers the adjective green, which sets the green circle against the
less salient of its two rivals, to circle exactly when the adjective costs less than
log (169 / 101).
The first threshold lies inside the paper's cost support (-0.4, 0.4) and the second beyond
it, so at every cost in the support this speaker prefers the adjective at the green circle.
Listeners #
The belief-oriented listener (15) is RSA.listener at its prior and a speaker, and the paper's
listeners take the speaker to ignore salience. The action-oriented listener (14) soft-maximizes
that belief.
A listener with a uniform prior who hears circle prefers the green circle, since for the blue circle the speaker had blue.
Hearing green it prefers the green circle, since for the green square the speaker had square.
A listener with the salience prior who hears circle prefers the salient blue circle exactly when its salience outweighs the speaker's preference for blue there.
Hearing green it prefers the green circle to the more salient green square exactly when the speaker's preference for square there outweighs the salience.
Without cost a listener with the salience prior follows salience on circle and narrows on green, as the paper's listeners did, exactly on a window of rationalities of the speaker.
Frank and Goodman's listener at the measured salience and rationality one takes green to refer to the salient green square.
The action-oriented listener (14) soft-maximizes its belief and so orders the objects as the belief does, at every rationality of its own. With the salience prior and the action-oriented speaker, the model the paper's comparison favours, it follows salience on circle and narrows on green on the window of its belief.