Documentation

Studies.KehlerRohde2013

Kehler and Rohde (2013): A Probabilistic Reconciliation of Coherence-Driven and Centering-Driven Theories of Pronoun Interpretation #

This file formalizes the Bayesian model of pronoun interpretation of [KR13]: the interpreter's bias P(referent | pronoun) is proportional to the speaker's production bias P(pronoun | referent) times the next-mention prior P(referent) (the paper's (11) and (13)), the prior is the mixture of the relation-conditioned biases weighted by the expected coherence relations ((9)), and coherence expectations are in turn the mixture over the referent mentioned next ((12)). The consequences the paper draws are proved once: a shift in the expected relations with the conditional biases held fixed shifts the prior (mixture_sub), and only then (mixture_const); a referent pronominalized above the overall rate is the pronoun's referent above its prior, the overlaid subject bias (posterior_gt_iff); and raising a referent's next-mention probability raises the expectation of the relations that favor it (relationExpectation_sub).

The paper's passage-completion results (Tables 1–10, Data/Experiments/KehlerRohde2013) then instantiate the model. The instruction manipulation's mixtures computed from Tables 3 and 4 fall on the sides of chance Table 5 observes (instruction_mixtures); the pronoun prompt raises subject mentions and shifts the continuations toward the Source-biased relations (Table 6); the voice manipulation's Bayesian predictions computed from Tables 7 and 9 favor the subject in both voices, more in the active, as Table 10 observes (voice_predictions), though below the predictions Table 10 prints (bayesSubject_lt_predicted); and the pronominalization rates of Table 9 increase with topichood, the passive subject above the active subject above the non-subjects, a gradient the backward-looking center of a grammatical-role Centering cannot see (cb_topichood_dissociation_under_voice).

Implementation notes #

The tables are Data/Experiments/KehlerRohde2013, whose proportions are printed to two places and read here in percent (Decimal.hundredths); computed quantities are compared through their integer numerators. The mixtures (9) range over the five relations the paper codes, whose frequencies fall short of one by the continuations coded otherwise.

References #

The Bayesian model #

def KehlerRohde2013.mixture {C : Type u_1} [Fintype C] (pCR bias : C → ℚ) :

ℚ

(9): the next-mention probability of a referent as the mixture of its relation-conditioned probabilities, weighted by the probabilities of the relations.

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KehlerRohde2013.mixture pCR bias = ∑ c : C, pCR c * bias c

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theorem KehlerRohde2013.mixture_sub {C : Type u_1} [Fintype C] (p q bias : C → ℚ) :

mixture q bias - mixture p bias = ∑ c : C, (q c - p c) * bias c

The mixture is affine in the expected relations: a change of expectations with the conditional biases fixed changes the next-mention probability by the change weighted by the biases.

theorem KehlerRohde2013.mixture_const {C : Type u_1} [Fintype C] (p : C → ℚ) (hp : ∑ c : C, p c = 1) (b : ℚ) :

(mixture p fun (x : C) => b) = b

Without relation-conditioning the mixture over a distribution is the common bias, whatever the expectations: a heuristic account predicts no effect of the instructions.

def KehlerRohde2013.relationExpectation {C : Type u_1} {R : Type u_2} [Fintype R] (pRef : R → ℚ) (crGiven : R → C → ℚ) (c : C) :

ℚ

(12): the probability of a coherence relation as the mixture over the referent mentioned next.

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KehlerRohde2013.relationExpectation pRef crGiven c = ∑ r : R, pRef r * crGiven r c

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theorem KehlerRohde2013.relationExpectation_sub {C : Type u_1} {R : Type u_2} [Fintype R] (p q : R → ℚ) (crGiven : R → C → ℚ) (c : C) :

relationExpectation q crGiven c - relationExpectation p crGiven c = ∑ r : R, (q r - p r) * crGiven r c

A shift in next-mention expectations shifts the expected relations by the shift weighted by each referent's relation profile.

def KehlerRohde2013.posterior {R : Type u_2} [Fintype R] (pRef pPron : R → ℚ) (r : R) :

ℚ

(11), (13): the interpreter's posterior over referents, the production bias times the next-mention prior, normalized by the overall probability of a pronoun.

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KehlerRohde2013.posterior pRef pPron r = pPron r * pRef r / ∑ r' : R, pPron r' * pRef r'

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theorem KehlerRohde2013.posterior_gt_iff {R : Type u_2} [Fintype R] (pRef pPron : R → ℚ) (r : R) (hr : 0 < pRef r) (hZ : 0 < ∑ r' : R, pPron r' * pRef r') :

pRef r < posterior pRef pPron r ↔ ∑ r' : R, pPron r' * pRef r' < pPron r

The overlaid bias: a referent is the pronoun's referent above its prior probability exactly when it is pronominalized above the overall rate of pronominalization.

The data #

def KehlerRohde2013.Voice.toUD :

Voice → UD.Voice

The voice as UD codes it.

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def KehlerRohde2013.Position.select {α : Type u_1} (s n : α) :

Position → α

A function on positions by its two values.

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Coherence-conditioned biases (Tables 1–5) #

def KehlerRohde2013.aspectSource (a : Aspect) :

ℕ

Table 1: the Source interpretation rate by aspect.

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KehlerRohde2013.aspectSource a = (KehlerRohde2013.sourceByAspect a).sourceInterpretation.hundredths.toNat

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theorem KehlerRohde2013.imperfective_more_source :

aspectSource Aspect.perfective < aspectSource Aspect.imperfective

The event-structure hypothesis: the imperfective keeps the Source central, the perfective focuses the end state, so the imperfective draws more Source interpretations.

def KehlerRohde2013.perfectiveFrequency (c : Relation) :

ℕ

Table 2: the frequency of a relation in the perfective continuations.

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KehlerRohde2013.perfectiveFrequency c = (KehlerRohde2013.relationsPerfective c).frequency.hundredths.toNat

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def KehlerRohde2013.perfectiveSourceGiven (c : Relation) :

ℕ

Table 2: the Source bias of a relation in the perfective continuations.

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KehlerRohde2013.perfectiveSourceGiven c = (KehlerRohde2013.relationsPerfective c).biasToSource.hundredths.toNat

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theorem KehlerRohde2013.perfective_mixture_masks_biases :

(2 / 5 < mixture (fun (c : Relation) => ↑(perfectiveFrequency c) / 100) fun (c : Relation) => ↑(perfectiveSourceGiven c) / 100) ∧ (mixture (fun (c : Relation) => ↑(perfectiveFrequency c) / 100) fun (c : Relation) => ↑(perfectiveSourceGiven c) / 100) < 3 / 5 ∧ perfectiveSourceGiven Relation.occasion < 20 ∧ 90 < perfectiveSourceGiven Relation.elaboration ∧ ∀ (c : Relation), perfectiveFrequency c ≤ perfectiveFrequency Relation.occasion

The near-chance overall bias of the perfective continuations is the mixture (9), over the five coded relations, of strongly opposed relation-conditioned biases: Occasion, the most common relation, favors the Goal and Elaboration the Source.

def KehlerRohde2013.frequency (i : Instruction) (c : Relation) :

ℕ

Table 3: the frequency of a relation under an instruction.

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KehlerRohde2013.frequency i c = (KehlerRohde2013.relationsByInstruction c i).frequency.hundredths.toNat

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def KehlerRohde2013.sourceGiven (c : Relation) :

ℕ

Table 4: the Source bias of a relation in the instruction experiment.

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KehlerRohde2013.sourceGiven c = (KehlerRohde2013.biasesByRelation c).instructionManipulation.hundredths.toNat

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theorem KehlerRohde2013.biasesByRelation_original (c : Relation) :

(biasesByRelation c).original = (relationsPerfective c).biasToSource

Table 4 repeats the biases of Table 2 as those of the original experiment.

def KehlerRohde2013.observedSource (i : Instruction) :

ℕ

Table 5: the observed Source interpretation rate under an instruction.

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KehlerRohde2013.observedSource i = (KehlerRohde2013.sourceByInstruction i).sourceInterpretation.hundredths.toNat

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def KehlerRohde2013.predictedSource (i : Instruction) :

ℚ

(9) on Tables 3 and 4: the next-mention probability of the Source under an instruction.

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One or more equations did not get rendered due to their size.

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theorem KehlerRohde2013.instruction_mixtures :

(1 / 2 < predictedSource Instruction.why ∧ 50 < observedSource Instruction.why) ∧ predictedSource Instruction.whatNext < 1 / 2 ∧ observedSource Instruction.whatNext < 50

The instruction manipulation: the expectations of Table 3, with the conditional biases of Table 4 held fixed, put the Why? mixture on the Source side of chance and the What happened next? mixture on the Goal side, where Table 5 finds the interpretations of identical stimuli.

Bidirectionality (Table 6) #

def KehlerRohde2013.promptFrequency (p : Prompt) (c : Relation) :

ℕ

Table 6: the frequency of a relation by prompt type.

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KehlerRohde2013.promptFrequency p c = (KehlerRohde2013.relationsByPrompt c p).frequency.hundredths.toNat

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def KehlerRohde2013.goalMention (p : Prompt) :

ℕ

The share of first mentions to the Goal by prompt type, in percent.

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KehlerRohde2013.goalMention p = (KehlerRohde2013.goalFirstMentions p).percent

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An ambiguous pronoun is not inert: the prompt draws first mentions from the Goal to the Source, the overlaid subject bias, and with them the continuations move from the Goal-biased Occasion and Result toward the Source-biased Elaboration and Explanation, as (12) predicts.

The voice manipulation (Tables 7–10) #

def KehlerRohde2013.causalMention (v : Voice) (p : Prompt) :

ℕ

Table 7: the rate of next mention of the causally implicated referent by voice and prompt.

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KehlerRohde2013.causalMention v p = (KehlerRohde2013.causalMentions v p).proportion.hundredths.toNat

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def KehlerRohde2013.explanationRate (v : Voice) (p : Prompt) :

ℕ

Table 8: the rate of Explanation continuations by voice and prompt.

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KehlerRohde2013.explanationRate v p = (KehlerRohde2013.explanations v p).proportion.hundredths.toNat

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def KehlerRohde2013.pronominalized (v : Voice) (pos : Position) :

ℕ

Table 9: the pronominalization rate of a position by voice, without a pronoun prompt.

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KehlerRohde2013.pronominalized v pos = (KehlerRohde2013.pronominalizations v pos).proportion.hundredths.toNat

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def KehlerRohde2013.actualSubject (v : Voice) :

ℕ

Table 10: the observed bias of the pronoun toward the subject by voice.

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KehlerRohde2013.actualSubject v = (KehlerRohde2013.subjectBiases v).actual.hundredths.toNat

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def KehlerRohde2013.subjectMention :

Voice → ℕ

Table 7 without a pronoun prompt as the next-mention rate of the subject: the causally implicated referent is the subject of the active and the by-phrase of the passive.

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theorem KehlerRohde2013.voice_shifts_interpretation_and_coherence :

causalMention Voice.passive Prompt.pronoun < causalMention Voice.active Prompt.pronoun ∧ explanationRate Voice.passive Prompt.pronoun < explanationRate Voice.active Prompt.pronoun

The voice manipulation with the propositions constant: the pronoun refers to the causally implicated referent less once the passive moves it out of subject position (Table 7), and the continuations are Explanations less often (Table 8), the shift in coherence mediated by the shift in reference.

def KehlerRohde2013.bayesSubject (v : Voice) :

ℚ

(13) from Tables 7 and 9: the predicted bias of an ambiguous pronoun toward the subject.

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One or more equations did not get rendered due to their size.

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theorem KehlerRohde2013.voice_predictions :

(1 / 2 < bayesSubject Voice.active ∧ 1 / 2 < bayesSubject Voice.passive ∧ bayesSubject Voice.passive < bayesSubject Voice.active) ∧ 50 < actualSubject Voice.active ∧ 50 < actualSubject Voice.passive ∧ actualSubject Voice.passive < actualSubject Voice.active

The Bayesian predictions of Table 10: from the next-mention rates without a pronoun (Table 7) and the pronominalization rates (Table 9), the pronoun refers to the subject in both voices and more strongly in the active, as the interpretations measured with the pronoun prompt show.

theorem KehlerRohde2013.bayesSubject_lt_predicted (v : Voice) :

bayesSubject v < (subjectBiases v).predicted.toRat

Table 10's predictions are not the posterior of the aggregate rates of Tables 7 and 9, which fall below them in both voices (0.79 against 0.81 in the active, 0.54 against 0.59 in the passive); the paper does not say how it estimated them.

theorem KehlerRohde2013.overlaid_subject_bias :

(↑(subjectMention Voice.active) / 100 < bayesSubject Voice.active ∧ ↑(subjectMention Voice.passive) / 100 < bayesSubject Voice.passive) ∧ causalMention Voice.active Prompt.noPronoun < causalMention Voice.active Prompt.pronoun

The overlaid subject bias in the voice experiment: in both voices the subject is pronominalized above the overall rate, so (13) puts the pronoun's bias toward it above its next-mention prior, as the pronoun prompt's rise in subject mentions over the no-pronoun condition shows for the active (Table 7).

Production tracks topichood, not expectancy #

inductive KehlerRohde2013.TopichoodLevel :

How strongly a referent is signalled as the sentence topic: the subject of a passive, promoted by a marked construction, more strongly than the subject of an active ([Dav84]), and non-subjects not at all.

low : TopichoodLevel
default_ : TopichoodLevel
strong : TopichoodLevel

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@[instance_reducible]

instance KehlerRohde2013.instDecidableEqTopichoodLevel :

DecidableEq TopichoodLevel

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KehlerRohde2013.instDecidableEqTopichoodLevel x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance KehlerRohde2013.instReprTopichoodLevel :

Repr TopichoodLevel

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KehlerRohde2013.instReprTopichoodLevel = { reprPrec := KehlerRohde2013.instReprTopichoodLevel.repr }

def KehlerRohde2013.instReprTopichoodLevel.repr :

TopichoodLevel → ℕ → Std.Format

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One or more equations did not get rendered due to their size.

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@[instance_reducible]

instance KehlerRohde2013.instFintypeTopichoodLevel :

Fintype TopichoodLevel

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One or more equations did not get rendered due to their size.

def KehlerRohde2013.TopichoodLevel.rank :

TopichoodLevel → Fin 3

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@[instance_reducible]

instance KehlerRohde2013.instLinearOrderTopichoodLevel :

LinearOrder TopichoodLevel

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KehlerRohde2013.instLinearOrderTopichoodLevel = LinearOrder.lift' KehlerRohde2013.TopichoodLevel.rank KehlerRohde2013.instLinearOrderTopichoodLevel._proof_1

def KehlerRohde2013.topichood :

UD.Voice → Discourse.Centering.GrammaticalRole → TopichoodLevel

The topichood of a grammatical role by voice.

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def KehlerRohde2013.pronounRate (v : Voice) :

Discourse.Centering.GrammaticalRole → ℕ

Table 9 by grammatical role: the object of the active and the by-phrase of the passive are the non-subjects.

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theorem KehlerRohde2013.pronounRate_strictMono (v v' : Voice) (r r' : Discourse.Centering.GrammaticalRole) :

topichood v.toUD r < topichood v'.toUD r' → pronounRate v r < pronounRate v' r'

Production tracks topichood: across the voices and roles of Table 9 a higher topichood level is pronominalized at a higher rate, the passive subject above the active subject above the non-subjects, so not all grammatical subjects are equal.

theorem KehlerRohde2013.production_not_expectancy :

subjectMention Voice.passive < 100 - subjectMention Voice.passive ∧ pronominalized Voice.passive Position.nonSubject < pronominalized Voice.passive Position.subject

Production is not expectancy: without a pronoun prompt the passive's by-phrase referent is mentioned next far more often than its subject, yet it is pronominalized far less: the rate of pronominalization follows topichood, not the next-mention bias.

The design premise and the Centering term #

The stimuli pair same-gender characters, so the prompt pronoun φ-agrees with both and the model must decide between them; and under the grammatical-role ranking of forward-looking centers the backward-looking center is the same in both voices, so the voice-induced gradient of Table 9 lives in the topichood term rather than in Centering's center.

def KehlerRohde2013.amanda :

Morphology.Word

The two characters of (20), both third-person singular feminine.

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KehlerRohde2013.amanda = { form := "Amanda", cat := UD.UPOS.PROPN, features := Morphology.Features.of (some Person.third) (some Number.singular) (some Gender.feminine) }

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def KehlerRohde2013.brittany :

Morphology.Word

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KehlerRohde2013.brittany = { form := "Brittany", cat := UD.UPOS.PROPN, features := Morphology.Features.of (some Person.third) (some Number.singular) (some Gender.feminine) }

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theorem KehlerRohde2013.she_ambiguous_over_stimuli :

English.Pronouns.she.CandidateAntecedent amanda ∧ English.Pronouns.she.CandidateAntecedent brittany

Both characters are candidate antecedents of the prompt She.

def KehlerRohde2013.prevAmandaActive :

Discourse.Centering.Utterance Morphology.Word Discourse.Centering.GrammaticalRole

The context sentence of (20a): Amanda the subject, Brittany the object.

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One or more equations did not get rendered due to their size.

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def KehlerRohde2013.curActive :

Discourse.Centering.Utterance Morphology.Word Discourse.Centering.GrammaticalRole

An active continuation pronominalizing both.

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One or more equations did not get rendered due to their size.

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def KehlerRohde2013.curPassive :

Discourse.Centering.Utterance Morphology.Word Discourse.Centering.GrammaticalRole

The passive continuation: Amanda promoted to subject, Brittany in the by-phrase.

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One or more equations did not get rendered due to their size.

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theorem KehlerRohde2013.cb_invariant_under_voice :

Discourse.Centering.cb prevAmandaActive curActive = Discourse.Centering.cb prevAmandaActive curPassive

The backward-looking center is the same in both voices: the grammatical-role ranking does not see voice.

theorem KehlerRohde2013.topichood_distinguishes_voice :

topichood UD.Voice.Act Discourse.Centering.GrammaticalRole.subject < topichood UD.Voice.Pass Discourse.Centering.GrammaticalRole.subject

Topichood does see it.

theorem KehlerRohde2013.cb_topichood_dissociation_under_voice :

Discourse.Centering.cb prevAmandaActive curActive = Discourse.Centering.cb prevAmandaActive curPassive ∧ topichood UD.Voice.Act Discourse.Centering.GrammaticalRole.subject < topichood UD.Voice.Pass Discourse.Centering.GrammaticalRole.subject

The dissociation: Centering's center is voice-blind where the topichood term is voice-sensitive, which is where the 87% against 62% of Table 9 lives.