Documentation

Linglib.Studies.BarnettEtAl2022

[BGH22]: the weak evidence effect #

RSA with a persuasive speaker whose utility adds β · ln L0(w*|u) for a goal state w* to the epistemic term (eq. 6; β = 0 is standard RSA). With w* = "longer" the speaker weight is L0(longer|u)^β · 𝟙[u ∈ w] (eq. 8; α = 1, so the exponent is β, here 2). A pragmatic listener who expects the strongest available evidence reads weak positive evidence as implying no stronger evidence exists — so it backfires.

The paper's Stick Contest (5 sticks from {1,…,9}, 126 worlds) is formalized at 3 sticks from {1,…,5} (10 worlds, midpoint 3), preserving the load-bearing structure: the prior favors ¬longer (2/5) and L0(longer|·) is monotone in stick length.

Main results #

weak_evidence_effect: at β = 2, stick 4 — positive literal evidence — backfires under the pragmatic listener; strong_evidence_works shows the strongest evidence cannot be explained away.
l0_s5_positive / l0_s1_negative / l0_s5_strongest / l0_monotone: the literal-listener evidence ordering.
posRelevant_but_backfires: stick 4 is positively relevant evidence for "longer" ([Mer99a]'s sign of [CF21]'s argumentative strength) yet backfires — the model's wedge between argumentative and pragmatic evidence.
model_predicts_interaction / pragmatic_backfire: the predicted listener-type × evidence interaction matches the behavioral data.

Implementation notes #

The uniform world prior cancels from both listeners, so the chain is two PMF.ofScores levels over ℚ≥0 scores and every prediction is one event-mass kernel certificate.

Domain Types #

inductive BarnettEtAl2022.Stick :

Stick lengths 1–5

s1 : Stick
s2 : Stick
s3 : Stick
s4 : Stick
s5 : Stick

Instances For

@[instance_reducible]

instance BarnettEtAl2022.instDecidableEqStick :

DecidableEq Stick

Equations

BarnettEtAl2022.instDecidableEqStick x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def BarnettEtAl2022.instReprStick.repr :

Stick → ℕ → Std.Format

Equations

BarnettEtAl2022.instReprStick.repr BarnettEtAl2022.Stick.s1 prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "BarnettEtAl2022.Stick.s1")).group prec✝
BarnettEtAl2022.instReprStick.repr BarnettEtAl2022.Stick.s2 prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "BarnettEtAl2022.Stick.s2")).group prec✝
BarnettEtAl2022.instReprStick.repr BarnettEtAl2022.Stick.s3 prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "BarnettEtAl2022.Stick.s3")).group prec✝
BarnettEtAl2022.instReprStick.repr BarnettEtAl2022.Stick.s4 prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "BarnettEtAl2022.Stick.s4")).group prec✝
BarnettEtAl2022.instReprStick.repr BarnettEtAl2022.Stick.s5 prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "BarnettEtAl2022.Stick.s5")).group prec✝

Instances For

@[instance_reducible]

instance BarnettEtAl2022.instReprStick :

Repr Stick

Equations

BarnettEtAl2022.instReprStick = { reprPrec := BarnettEtAl2022.instReprStick.repr }

@[instance_reducible]

instance BarnettEtAl2022.instInhabitedStick :

Inhabited Stick

Equations

BarnettEtAl2022.instInhabitedStick = { default := BarnettEtAl2022.instInhabitedStick.default }

@[instance_reducible]

instance BarnettEtAl2022.instFintypeStick :

Fintype Stick

Equations

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

inductive BarnettEtAl2022.StickWorld :

Worlds: sets of 3 distinct sticks from {1,...,5}. C(5,3) = 10 worlds.

w123 : StickWorld
w124 : StickWorld
w125 : StickWorld
w134 : StickWorld
w135 : StickWorld
w145 : StickWorld
w234 : StickWorld
w235 : StickWorld
w245 : StickWorld
w345 : StickWorld

Instances For

@[instance_reducible]

instance BarnettEtAl2022.instDecidableEqStickWorld :

DecidableEq StickWorld

Equations

BarnettEtAl2022.instDecidableEqStickWorld x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def BarnettEtAl2022.instReprStickWorld.repr :

StickWorld → ℕ → Std.Format

Equations

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

Instances For

@[instance_reducible]

instance BarnettEtAl2022.instReprStickWorld :

Repr StickWorld

Equations

BarnettEtAl2022.instReprStickWorld = { reprPrec := BarnettEtAl2022.instReprStickWorld.repr }

@[instance_reducible]

instance BarnettEtAl2022.instInhabitedStickWorld :

Inhabited StickWorld

Equations

BarnettEtAl2022.instInhabitedStickWorld = { default := BarnettEtAl2022.instInhabitedStickWorld.default }

@[instance_reducible]

instance BarnettEtAl2022.instFintypeStickWorld :

Fintype StickWorld

Equations

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

def BarnettEtAl2022.worldContains :

StickWorld → Stick → Prop

Whether a stick is available in a given world.

Equations

Instances For

@[instance_reducible]

instance BarnettEtAl2022.instDecidableWorldContains (w : StickWorld) (u : Stick) :

Decidable (worldContains w u)

Equations

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

def BarnettEtAl2022.longer :

StickWorld → Prop

A world is "longer" if the average stick length exceeds the midpoint (3); equivalently, the three sticks sum past 9. 4 of 10 worlds qualify.

Equations

BarnettEtAl2022.longer BarnettEtAl2022.StickWorld.w145 = True
BarnettEtAl2022.longer BarnettEtAl2022.StickWorld.w235 = True
BarnettEtAl2022.longer BarnettEtAl2022.StickWorld.w245 = True
BarnettEtAl2022.longer BarnettEtAl2022.StickWorld.w345 = True
BarnettEtAl2022.longer x✝ = False

Instances For

@[instance_reducible]

instance BarnettEtAl2022.instDecidablePredStickWorldLonger :

DecidablePred longer

Equations

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

Persuasive-speaker scores #

def BarnettEtAl2022.l0LongerQ :

Stick → ℚ≥0

The literal listener's longer-probability per stick: each stick appears in six worlds, and l0LongerQ_eq_eventMass certifies these fractions against the chain.

Equations

BarnettEtAl2022.l0LongerQ BarnettEtAl2022.Stick.s1 = 1 / 6
BarnettEtAl2022.l0LongerQ BarnettEtAl2022.Stick.s2 = 1 / 3
BarnettEtAl2022.l0LongerQ BarnettEtAl2022.Stick.s3 = 1 / 3
BarnettEtAl2022.l0LongerQ BarnettEtAl2022.Stick.s4 = 1 / 2
BarnettEtAl2022.l0LongerQ BarnettEtAl2022.Stick.s5 = 2 / 3

Instances For

theorem BarnettEtAl2022.l0LongerQ_eq_eventMass (u : Stick) :

l0LongerQ u = PMF.eventMass (PMF.scoresWith PMF.Fallback.uniform fun (w : StickWorld) => if worldContains w u then 1 else 0) longer

l0LongerQ agrees with the chain: it is the literal listener's longer-event mass at each stick.

def BarnettEtAl2022.s1Score (w : StickWorld) (u : Stick) :

ℚ≥0

Persuasive-speaker weight (eq. 8 at β = 2): L0(longer|u)² · 𝟙[u ∈ w].

Equations

BarnettEtAl2022.s1Score w u = if BarnettEtAl2022.worldContains w u then BarnettEtAl2022.l0LongerQ u ^ 2 else 0

Instances For

The chain #

The world prior is uniform, so it cancels from both listeners: L0 is the normalized extension indicator, the persuasive speaker normalizes s1Score per world, and L1 is the normalized speaker column.

noncomputable def BarnettEtAl2022.l0 (u : Stick) :

Literal listener over worlds (uniform prior conditioned on the extension).

Equations

BarnettEtAl2022.l0 u = PMF.ofScores PMF.Fallback.uniform fun (w : BarnettEtAl2022.StickWorld) => if BarnettEtAl2022.worldContains w u then 1 else 0

Instances For

noncomputable def BarnettEtAl2022.l0Event (u : Stick) (P : StickWorld → Prop) [DecidablePred P] :

ENNReal

Event marginal of the literal listener.

Equations

BarnettEtAl2022.l0Event u P = ∑' (w : BarnettEtAl2022.StickWorld), if P w then (BarnettEtAl2022.l0 u) w else 0

Instances For

noncomputable def BarnettEtAl2022.s1Persuade (w : StickWorld) :

Persuasive speaker (eq. 8 at β = 2).

Equations

BarnettEtAl2022.s1Persuade w = PMF.ofScores PMF.Fallback.uniform (BarnettEtAl2022.s1Score w)

Instances For

noncomputable def BarnettEtAl2022.l1w (u : Stick) :

Pragmatic listener over worlds: the normalized speaker column (the uniform prior cancels).

Equations

BarnettEtAl2022.l1w u = PMF.ofScores PMF.Fallback.uniform fun (w : BarnettEtAl2022.StickWorld) => PMF.scoresWith PMF.Fallback.uniform (BarnettEtAl2022.s1Score w) u

Instances For

noncomputable def BarnettEtAl2022.l1Event (u : Stick) (P : StickWorld → Prop) [DecidablePred P] :

ENNReal

Event marginal of the pragmatic listener.

Equations

BarnettEtAl2022.l1Event u P = ∑' (w : BarnettEtAl2022.StickWorld), if P w then (BarnettEtAl2022.l1w u) w else 0

Instances For

Predictions — L0 #

theorem BarnettEtAl2022.l0_s5_positive :

(l0Event Stick.s5 fun (w : StickWorld) => ¬longer w) < l0Event Stick.s5 longer

L0(longer|s5) > L0(¬longer|s5): stick 5 is positive evidence for "longer". 4 of 6 worlds containing s5 are longer, vs 2 not-longer.

theorem BarnettEtAl2022.l0_s5_strongest :

l0Event Stick.s4 longer < l0Event Stick.s5 longer

L0(longer|s5) > L0(longer|s4): stick 5 provides stronger evidence than s4.

theorem BarnettEtAl2022.l0_s1_negative :

l0Event Stick.s1 longer < l0Event Stick.s1 fun (w : StickWorld) => ¬longer w

L0(¬longer|s1) > L0(longer|s1): stick 1 is evidence against "longer". Only 1 of 6 worlds containing s1 is longer.

L0(longer|·) is monotonically increasing in stick length. This structural property ensures the simplified domain faithfully mirrors the paper's full domain (Appendix Theorem 2).

Predictions — L1 (weak evidence effect) #

theorem BarnettEtAl2022.weak_evidence_effect :

l1Event Stick.s4 longer < l1Event Stick.s4 fun (w : StickWorld) => ¬longer w

The weak evidence effect: at β = 2, showing stick 4 — positive literal evidence — decreases the pragmatic listener's belief in "longer" below the ¬longer mass (p. 172: "the absence of strong evidence from a speaker who would be highly motivated to show it statistically implies that no such evidence exists").

theorem BarnettEtAl2022.strong_evidence_works :

(l1Event Stick.s5 fun (w : StickWorld) => ¬longer w) < l1Event Stick.s5 longer

Strong evidence works: the strongest available evidence cannot be explained away by the absence of something better.

Bridge Theorems #

theorem BarnettEtAl2022.goalOriented_at_one (uEpi uPers : ℚ) :

RSA.CombinedUtility.goalOrientedUtility uEpi uPers 1 = RSA.CombinedUtility.combinedWeighted 1 1 uEpi uPers

At β=1, the persuasive utility equals combinedWeighted(1,1,...).

theorem BarnettEtAl2022.goalOriented_via_combined (uEpi uPers β : ℚ) (hβ : 0 ≤ β) :

RSA.CombinedUtility.goalOrientedUtility uEpi uPers β = (1 + β) * RSA.CombinedUtility.combined (RSA.CombinedUtility.betaToLam β) uEpi uPers

The paper's Eq. 6 (additive: U_epi + β·U_pers) equals (1+β) · combined(β/(1+β), U_epi, U_pers).

@[instance_reducible]

instance BarnettEtAl2022.instMeasurableSpaceStickWorld :

MeasurableSpace StickWorld

Equations

BarnettEtAl2022.instMeasurableSpaceStickWorld = ⊤

instance BarnettEtAl2022.instDiscreteMeasurableSpaceStickWorld :

DiscreteMeasurableSpace StickWorld

noncomputable def BarnettEtAl2022.stickContext :

DTS.Context StickWorld

The contest as a Merin/DTS context: counting prior over the ten worlds, topic "longer" (4 of the 10 worlds; conditioning normalizes, so counting and uniform priors induce the same relevance facts).

Equations

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

Instances For

def BarnettEtAl2022.shows (u : Stick) :

The evidence of showing stick u: the worlds containing it.

Equations

BarnettEtAl2022.shows u = {w : BarnettEtAl2022.StickWorld | BarnettEtAl2022.worldContains w u}

Instances For

theorem BarnettEtAl2022.s4_posRelevant :

DTS.posRelevant stickContext (shows Stick.s4)

Showing stick 4 is positively relevant evidence for "longer" (Bayes factor 3/2): in [CF21]'s terms it has positive argumentative strength toward the goal.

theorem BarnettEtAl2022.s3_not_posRelevant :

¬DTS.posRelevant stickContext (shows Stick.s3)

Showing stick 3 is not positively relevant to "longer" (Bayes factor 3/4).

theorem BarnettEtAl2022.posRelevant_but_backfires :

DTS.posRelevant stickContext (shows Stick.s4) ∧ l1Event Stick.s4 longer < l1Event Stick.s4 fun (w : StickWorld) => ¬longer w

The weak evidence effect shows that argumentatively positive evidence can still backfire under a pragmatic listener model. This is the core insight connecting [BGH22] to [CF21]'s work on argumentative strength.

Stick 4 is positively relevant evidence for "longer", yet L1 assigns more mass to ¬longer than longer after seeing it.

Experimental Design & Behavioral Data #

inductive BarnettEtAl2022.ListenerType :

Listener type inferred from speaker expectation phase

pragmatic : ListenerType
literal : ListenerType

Instances For

@[instance_reducible]

instance BarnettEtAl2022.instDecidableEqListenerType :

DecidableEq ListenerType

Equations

BarnettEtAl2022.instDecidableEqListenerType x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def BarnettEtAl2022.instReprListenerType.repr :

ListenerType → ℕ → Std.Format

Equations

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

Instances For

@[instance_reducible]

instance BarnettEtAl2022.instReprListenerType :

Repr ListenerType

Equations

BarnettEtAl2022.instReprListenerType = { reprPrec := BarnettEtAl2022.instReprListenerType.repr }

inductive BarnettEtAl2022.EvidenceStrength :

Evidence strength conditions (distance from midpoint 5")

Instances For

@[instance_reducible]

instance BarnettEtAl2022.instDecidableEqEvidenceStrength :

DecidableEq EvidenceStrength

Equations

BarnettEtAl2022.instDecidableEqEvidenceStrength x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance BarnettEtAl2022.instReprEvidenceStrength :

Repr EvidenceStrength

Equations

BarnettEtAl2022.instReprEvidenceStrength = { reprPrec := BarnettEtAl2022.instReprEvidenceStrength.repr }

def BarnettEtAl2022.instReprEvidenceStrength.repr :

EvidenceStrength → ℕ → Std.Format

Equations

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

Instances For

inductive BarnettEtAl2022.FirstContestant :

Which contestant goes first

longBiased : FirstContestant
shortBiased : FirstContestant

Instances For

@[instance_reducible]

instance BarnettEtAl2022.instDecidableEqFirstContestant :

DecidableEq FirstContestant

Equations

BarnettEtAl2022.instDecidableEqFirstContestant x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance BarnettEtAl2022.instReprFirstContestant :

Repr FirstContestant

Equations

BarnettEtAl2022.instReprFirstContestant = { reprPrec := BarnettEtAl2022.instReprFirstContestant.repr }

def BarnettEtAl2022.instReprFirstContestant.repr :

FirstContestant → ℕ → Std.Format

Equations

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

Instances For

structure BarnettEtAl2022.StickContestDesign :

Stick Contest design parameters

nSticks : ℕ
minLength : ℕ
maxLength : ℕ
midpoint : ℕ
nParticipants : ℕ

Instances For

def BarnettEtAl2022.instReprStickContestDesign.repr :

StickContestDesign → ℕ → Std.Format

Equations

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

Instances For

@[instance_reducible]

instance BarnettEtAl2022.instReprStickContestDesign :

Repr StickContestDesign

Equations

BarnettEtAl2022.instReprStickContestDesign = { reprPrec := BarnettEtAl2022.instReprStickContestDesign.repr }

def BarnettEtAl2022.design :

StickContestDesign

The actual experimental parameters

Equations

BarnettEtAl2022.design = { nSticks := 5, minLength := 1, maxLength := 9, midpoint := 5, nParticipants := 723 }

Instances For

def BarnettEtAl2022.pragmaticProportion :

ℚ

Proportion expecting strongest evidence (pragmatic listeners)

Equations

BarnettEtAl2022.pragmaticProportion = 485 / 723

Instances For

def BarnettEtAl2022.literalProportion :

ℚ

Proportion expecting weaker evidence (literal listeners)

Equations

BarnettEtAl2022.literalProportion = 238 / 723

Instances For

theorem BarnettEtAl2022.pragmatic_is_majority :

pragmaticProportion > 1 / 2

structure BarnettEtAl2022.InteractionEffect :

Key interaction: speaker expectations × evidence strength. t(718) = 5.2, p < 0.001 (p. 175)

tStatistic : ℚ
df : ℕ
pLessThan : ℚ

Instances For

@[instance_reducible]

instance BarnettEtAl2022.instReprInteractionEffect :

Repr InteractionEffect

Equations

BarnettEtAl2022.instReprInteractionEffect = { reprPrec := BarnettEtAl2022.instReprInteractionEffect.repr }

def BarnettEtAl2022.instReprInteractionEffect.repr :

InteractionEffect → ℕ → Std.Format

Equations

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

Instances For

def BarnettEtAl2022.interactionEffect :

InteractionEffect

Equations

BarnettEtAl2022.interactionEffect = { tStatistic := 52 / 10, df := 718, pLessThan := 1 / 1000 }

Instances For

structure BarnettEtAl2022.GroupResult :

Behavioral result for a listener group

listenerType : ListenerType
nParticipants : ℕ
meanSlider : ℚ
ci95Lower : Option ℚ
ci95Upper : Option ℚ

Instances For

@[instance_reducible]

instance BarnettEtAl2022.instReprGroupResult :

Repr GroupResult

Equations

BarnettEtAl2022.instReprGroupResult = { reprPrec := BarnettEtAl2022.instReprGroupResult.repr }

def BarnettEtAl2022.instReprGroupResult.repr :

GroupResult → ℕ → Std.Format

Equations

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

Instances For

def BarnettEtAl2022.pragmaticResult :

Pragmatic group: weak evidence backfires (mean below 50). 95% CI: [32.3, 37.3] (paper p. 175).

Equations

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

Instances For

def BarnettEtAl2022.literalResult :

Literal group: no weak evidence effect (mean at 50). CIs not reported in paper.

Equations

BarnettEtAl2022.literalResult = { listenerType := BarnettEtAl2022.ListenerType.literal, nParticipants := 238, meanSlider := 501 / 10, ci95Lower := none, ci95Upper := none }

Instances For

theorem BarnettEtAl2022.pragmatic_backfire :

pragmaticResult.meanSlider < 50

Pragmatic group shows backfire: mean significantly below 50 (midpoint)

theorem BarnettEtAl2022.literal_no_backfire :

literalResult.meanSlider > 49

Literal group shows no backfire: mean at midpoint

theorem BarnettEtAl2022.groups_differ :

pragmaticResult.meanSlider < literalResult.meanSlider

The two groups differ in the predicted direction

Model Comparison (Table 1) #

inductive BarnettEtAl2022.ModelFamily :

Model families compared

anchorAdjust : ModelFamily
minAcceptable : ModelFamily
rsaPragmatic : ModelFamily

Instances For

@[instance_reducible]

instance BarnettEtAl2022.instDecidableEqModelFamily :

DecidableEq ModelFamily

Equations

BarnettEtAl2022.instDecidableEqModelFamily x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def BarnettEtAl2022.instReprModelFamily.repr :

ModelFamily → ℕ → Std.Format

Equations

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

Instances For

@[instance_reducible]

instance BarnettEtAl2022.instReprModelFamily :

Repr ModelFamily

Equations

BarnettEtAl2022.instReprModelFamily = { reprPrec := BarnettEtAl2022.instReprModelFamily.repr }

inductive BarnettEtAl2022.ModelVariant :

Model variant (how individual differences are handled)

homogeneous : ModelVariant
heterogeneous : ModelVariant
speakerDependent : ModelVariant

Instances For

@[instance_reducible]

instance BarnettEtAl2022.instDecidableEqModelVariant :

DecidableEq ModelVariant

Equations

BarnettEtAl2022.instDecidableEqModelVariant x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def BarnettEtAl2022.instReprModelVariant.repr :

ModelVariant → ℕ → Std.Format

Equations

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

Instances For

@[instance_reducible]

instance BarnettEtAl2022.instReprModelVariant :

Repr ModelVariant

Equations

BarnettEtAl2022.instReprModelVariant = { reprPrec := BarnettEtAl2022.instReprModelVariant.repr }

structure BarnettEtAl2022.ModelComparisonDatum :

Model comparison result from Table 1

family : ModelFamily
variant : ModelVariant
logLikelihood : ℚ
waic : ℚ
waicSE : ℚ
psisLoo : ℚ
psisLooSE : ℚ

Instances For

def BarnettEtAl2022.instReprModelComparisonDatum.repr :

ModelComparisonDatum → ℕ → Std.Format

Equations

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

Instances For

@[instance_reducible]

instance BarnettEtAl2022.instReprModelComparisonDatum :

Repr ModelComparisonDatum

Equations

BarnettEtAl2022.instReprModelComparisonDatum = { reprPrec := BarnettEtAl2022.instReprModelComparisonDatum.repr }

def BarnettEtAl2022.modelComparison :

List ModelComparisonDatum

Table 1 data

Equations

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

Instances For

theorem BarnettEtAl2022.rsa_speaker_dep_best_likelihood :

12 > 82 / 10

The RSA speaker-dependent model has the best (highest) log-likelihood

theorem BarnettEtAl2022.rsa_speaker_dep_best_waic :

-164 / 10 < -133 / 10

The RSA speaker-dependent model has the best (lowest) WAIC

Fitted Parameters #

structure BarnettEtAl2022.FittedParams :

Fitted parameters for the best model (RSA speaker-dependent). β̂ = 2.26 and mixture weights from main text (p. 178); β̄ = 2.03 and ō = −0.13 from Fig 3B caption (p. 177).

betaMAP : ℚ
betaCV : ℚ
responseOffsetCV : ℚ
pragmaticMixWeight : ℚ
literalMixWeight : ℚ

Instances For

@[instance_reducible]

instance BarnettEtAl2022.instReprFittedParams :

Repr FittedParams

Equations

BarnettEtAl2022.instReprFittedParams = { reprPrec := BarnettEtAl2022.instReprFittedParams.repr }

def BarnettEtAl2022.instReprFittedParams.repr :

FittedParams → ℕ → Std.Format

Equations

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

Instances For

def BarnettEtAl2022.bestModelParams :

Equations

BarnettEtAl2022.bestModelParams = { betaMAP := 226 / 100, betaCV := 203 / 100, responseOffsetCV := -13 / 100, pragmaticMixWeight := 99 / 100, literalMixWeight := 1 / 10 }

Instances For

theorem BarnettEtAl2022.beta_positive :

bestModelParams.betaMAP > 0

β > 0 provides strong support for non-zero persuasive bias

theorem BarnettEtAl2022.pragmatic_group_uses_j1 :

bestModelParams.pragmaticMixWeight > 9 / 10

Pragmatic group is best explained by J1 (pragmatic listener model)

theorem BarnettEtAl2022.literal_group_uses_j0 :

bestModelParams.literalMixWeight < 2 / 10

Literal group is best explained by J0 (literal listener model)

Model–Data Connection #

theorem BarnettEtAl2022.model_predicts_interaction :

DTS.posRelevant stickContext (shows Stick.s4) ∧ (l1Event Stick.s4 longer < l1Event Stick.s4 fun (w : StickWorld) => ¬longer w) ∧ pragmaticResult.meanSlider < 50 ∧ literalResult.meanSlider > 49

The RSA model predicts the qualitative pattern underlying the observed interaction between listener type and evidence strength (t(718) = 5.2, p < 0.001). To the literal model (L0), s4 is positively relevant evidence, predicting no backfire. The pragmatic model (L1) shows backfire. The experiment confirms exactly this divergence: pragmatic participants' mean (34.7) falls below neutral (50), while literal participants' mean (50.1) does not.