A model of hyperbole understanding as pragmatic reasoning:

The speaker chooses an utterance conditioned on the listener inferring information that is correct and relevant to the speaker’s communicative goal (or QUD). The goal can either be to communicate the state of the world, the speaker’s attitude towards the state of the world (affect), or both. The listener chooses an interpretation conditioned on the speaker selecting the given utterance when intending to communicate this meaning. In this example the state of the world is how much an electric kettle cost.

``````;; Define list of kettle prices under consideration (possible price states)
(define states
(list 50 51 500 501 1000 1001 5000 5001 10000 10001))

;; Prior probability of kettle prices (taken from human experiments)
(define (state-prior)
(multinomial states
'(0.4205 0.3865 0.0533 0.0538 0.0223 0.0211 0.0112 0.0111 0.0083 0.0120)))

;; Probability that given a price state, the speaker thinks it's too
;; expensive (taken from human experiments)
(define (valence-prior state)
(if (flip (second (assoc state
(list (list 50 0.3173)
(list 51 0.3173)
(list 500 0.7920)
(list 501 0.7920)
(list 1000 0.8933)
(list 1001 0.8933)
(list 5000 0.9524)
(list 5001 0.9524)
(list 10000 0.9864)
(list 10001 0.9864)))))
1
0))

;; Uniform prior over QUDs
;; (de-refernce through qud name since mem doesn't play nice with function values)
;; (define (qud-prior)
;;   (uniform-draw (list 's 'v 'sv 'as 'asv)))

(define (qud-prior)
(multinomial (list 's 'v 'sv 'as 'asv) '(0.17 0.32 0.17 0.17 0.17)))

(define (qud-fn qud)
(second
(assoc qud
(list
(list 's (lambda (state valence) state))
(list 'v (lambda (state valence) valence))
(list 'sv (lambda (state valence) (list state valence)))
(list 'as (lambda (state valence) (approx state 10)))
(list 'asv (lambda (state valence) (list (approx state 10) valence)))))))

;; Round x to nearest multiple of b (used for approximate interpretation):
(define (approx x b) (* b (round (/ x b))))

;; Define list of possible utterances (same as price states)
(define utterances states)

;; Sharp numbers are costlier
(define (utterance-prior)
(multinomial utterances
'(0.18 0.1 0.18 0.1 0.18 0.1 0.18 0.1 0.18 0.1)))

;; Literal interpretation "meaning" function, just check if uttered number reflects price state
(define (literal-interpretation utterance state)
(equal? utterance state))

;; Pragmatic listener, jointly infers the price state, speaker valence, and QUD
(define prag-listener
(mem
(lambda (utterance)
(enumeration-query
(define state (state-prior))
(define valence (valence-prior state))
(define qud (qud-prior))
(define val ((qud-fn qud) state valence))
(list state valence)
(equal? utterance
(apply multinomial (speaker val qud)))))))

;; Speaker, chooses an utterance to convey a particular value of the qud
(define speaker
(mem
(lambda (val qud)
(enumeration-query
(define utterance (utterance-prior))
utterance
(equal? val (apply multinomial (lit-listener utterance qud)))))))

;; Literal listener, infers the qud value assuming the utterance is true of the state
(define lit-listener
(mem
(lambda (utterance qud)
(enumeration-query
(define state (state-prior))
(define valence (valence-prior state))
((qud-fn qud) state valence)
(literal-interpretation utterance state)))))

(barplot (prag-listener 10000))

``````

## Another version

The same as above, written slightly differently (maintained for posterity):

``````;; Prior probability of kettle prices (taken from human experiments)
(define (state-prior)
(multinomial states
'(0.4205 0.3865 0.0533 0.0538 0.0223 0.0211 0.0112 0.0111 0.0083 0.0120)))

;; Probability that given a price state, the speaker thinks it's too
;; expensive (taken from human experiments)
(define valence-prior
(list (list 50 0.3173)
(list 51 0.3173)
(list 500 0.7920)
(list 501 0.7920)
(list 1000 0.8933)
(list 1001 0.8933)
(list 5000 0.9524)
(list 5001 0.9524)
(list 10000 0.9864)
(list 10001 0.9864)))

;; Parameters
(define depth 1)
(define hardness 1)

;; Define communicative goals and goal priors
(define goals
(list 'state-and-valence-precise
'state-and-valence-imprecise
'just-state-precise
'just-state-imprecise
'just-valence))

(define (goal-prior)
(multinomial goals '(0.17 0.17 0.17 0.17 0.32)))

;; Define list of prices under consideration (possible price states)
(define states
(list 50 51 500 501 1000 1001 5000 5001 10000 10001))

;; Define list of possible utterances (same as price states)
(define utterances states)

;; Sharp numbers are costlier
(define (utterance-prior)
(multinomial utterances
'(0.18 0.1 0.18 0.1 0.18 0.1 0.18 0.1 0.18 0.1)))

;; Define valences. 0 is no valence; 1 is with valence
(define valences (list 0 1))

(define (sample-valence state prior)
(let ((current-state-valence-pair (first prior)))
(if (equal? state (first current-state-valence-pair))
(if (flip (second current-state-valence-pair))
1
0)
(sample-valence state (rest prior)))))

(define (literal-interpretation utterance state)
(equal? utterance state))

(define (evaluate-state-goal listener-state speaker-state precision)
(if (equal? precision 'precise)
(equal? listener-state speaker-state)
(<= (abs (- listener-state speaker-state)) 1)))

(define (goal-satisfied? goal listener-state-valence-pair speaker-state speaker-valence)
(case goal
(('state-and-valence-precise)
(and (evaluate-state-goal (first listener-state-valence-pair) speaker-state 'precise)
(equal? (second listener-state-valence-pair) speaker-valence)))
(('state-and-valence-imprecise)
(and (evaluate-state-goal (first listener-state-valence-pair) speaker-state 'imprecise)
(equal? (second listener-state-valence-pair) speaker-valence)))
(('just-state-precise)
(evaluate-state-goal (first listener-state-valence-pair) speaker-state 'precise))
(('just-state-imprecise)
(evaluate-state-goal (first listener-state-valence-pair) speaker-state 'imprecise))
(('just-valence)
(equal? (second listener-state-valence-pair) speaker-valence))))

;; Speaker model
(define speaker
(mem
(lambda (state valence goal depth)
(enumeration-query
(define utterance (utterance-prior))
utterance
(goal-satisfied? goal (apply multinomial (listener utterance depth)) state valence)))))

;; Listener model
(define listener
(mem
(lambda (utterance depth)
(enumeration-query
(define state (state-prior))
(define valence (sample-valence state valence-prior))
(define speaker-goal (goal-prior))
(list state valence)
(if (equal? depth 0)
(literal-interpretation utterance state)
(equal? utterance
(apply multinomial (speaker state valence speaker-goal (- depth 1)))))))))

(define (sample-one utterance)
(listener utterance depth))

(barplot (sample-one 1000))
``````

References:

• Cite:Kao2014hyp