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How$do$cognitive$agents$handle$the$tradeoff$between$   speed$and$accuracy?                         Tatsuji(Takahashi(高橋(達二(...
Tatsuji(Takahashi(高橋達二★ Studied(philosophy(and(history(of(science(with(  KomachiGsan.(★ Got(a(Ph.D.(in(science(of(complex(...
Purpose(and(metaGtheory★ Purpose: ★ To(analytically(and(constructively(understand(the(   flexibility(and(creativity(of(huma...
The(problem(★ How$do$cognitive$agents$like$us$handle$the$  speed–accuracy$tradeoff$that$is$inevitable$in$  this$uncertain$w...
Illogical(biases(in(cognition★ In(our(classroom(experience:( ★ we(have(difficulty(in(understanding(material(   implication((...
Illogical(biases(in(cognition★ We(dont(follow(P(if(p(then(q)(=(P(notGp(or(q)((material(  implication)  ★ Generally(P(p)(is...
Illogical(biases(can(be(rational(         and(even(logical★ The(illogical(biases(in(human(cognition(can(be(  rationalized(...
Two(topics(of(this(talk:(★ (pARIs(part)(Study(of(how(we(reason,(with(  emphasis(on(conditionals((sentences(of(the(  form(R...
pARIs(part
Reasoning(and(conditional★ Three(forms(of(reasoning:(deduction,(induction,(abduction  ★ Deduction(uses(conditionals    ★ p...
Causality(and(conditional★ Causal(relationship(is(usually(expressed(by(conditional.  ★ If(global(warming(continues((W)(the...
Material(implication★ Modeling(conditional(by(material.implication ★ Rif(p,(then(qR(⇔Rnot(p,(or(qR                        ...
Material(implication                                     A(⊃(C                                                         A=T...
Defective(conditional★ For(half(a(century((since(1966),(it(  has(been(known(that(humans(  follow(the(Rdefective(truth(tabl...
Defective(biconditional★ There(is(our(tendency(of(         If and only if                                                 ...
From(defective(conditional(to(      conditional(event★ P(if(p(then(q)(=(P(q|p) ★ Not(P(if(p(then(q)(=(P(p(⊃(q)(=(P(¬p(or(q...
Overview★ New(paradigm(psychology(of(reasoning★ De(FineSis(conditional$and(biconditional$event★ biconditional(event(in(cau...
toc★ New$paradigm$psychology$of$reasoning★ Reasoning$and$conditional★ Conditional$and$biconditional$event★ Biconditional$e...
New(paradigm(psychology(of(         reasoning★ Very(naively(expressed...  ★ Old(paradigm:    ★ The(normative(theory(is(the...
toc★ New$paradigm$psychology$of$reasoning★ Reasoning$and$conditional★ Conditional$and$biconditional$event★ Biconditional$e...
defective conditional and     defective biconditional★ Defective truth table in the older paradigms ★ (Wason, 1966; Johnso...
de(FineSis(conditional$event★ Conditional(event,(formerly(called(defective(conditional,(is(a(   core(notion(in(the(new(par...
toc★ New$paradigm$psychology$of$reasoning★ Reasoning$and$conditional★ Conditional$and$biconditional$event★ Biconditional$e...
Causal(induction★ Example:(We(want(to(know(the(cause(of(a(health(problem,(  right(now(just(from(pure(observation,(no(inter...
Causal(induction(experiment(Stimulus$presentation:(a(     showing b-cell type joint event   pair(of(two(kinds(of( pictures...
Causal((intensity)(induction★ Two(phases(of(causal(induction((HaSori(&(Oaksford(  2007) ★ Phase$1:$observational((statisti...
Causal(Induction★ Here(we(study(the(causal(intensity.★ Recent(studies(emphasize(the(structure((the(  topology(of(Bayes(net...
∆P = P (E|C) − P (E|¬C) = (a + b)(c + d)                           (a + b)(c + d)  Framework(and(models(of(causal(        ...
The(pARIs(rule★ The(frequency(information(of(rare(instances(  conveys(more(information(than(abundant(instances(  (rational...
Rarity(assumption          H=P (E|C)P (C|E)     ★ We(assume(the(effect(in(focus(and(the(candidate(=       cause(to(be(rare:...
The(pARIs(rule★ C(and(E(are(both(generally(assumed(to(be(rare((P(C)(and(P(E)(low).★ pARIs(=(proportion(of(assumedGtoGbe(ra...
The(pARIs(rule★ C(and(E(are(both(assumed(to(be(rare((P(C)(and(P(E)(low)★ pARIs(=(proportion(of(assumedGtoGbe(rare(instance...
Why(ignore(the(dGcell?★ Hempels(paradox  ★ All(ravens(are(black.(    ★ =(If(something(is(a(raven,(then(it(is(black.      ★...
toc★ New$paradigm$psychology$of$reasoning★ Reasoning$and$conditional★ Conditional$and$biconditional$event★ Biconditional$e...
DataGfit(of(pARIs(and(PowerPC                                  AS95                                        BCC03exp1generat...
MetaGanalysis★ Fit(with(experiments((the(same(as(HaSori(&(Oaksford,(2007)★ pARIs(fits(the(data(set(with(the(lowest(correlat...
correlation7.00         0.90    0.85         0.93    0.95                             0.555.13                            ...
toc★ New$paradigm$psychology$of$reasoning★ Reasoning$and$conditional★ Conditional$and$biconditional$event★ Biconditional$e...
Experiments★ Experiment,1 ★ To,test,the,validity,of,rarity,assumption,in,ordinary,   causal,induction,from,2x2,covariation...
Experiment(1:(         c(and(d(in(2x2(table★ 27(undergraduates,(9(  stimuli.                        stim. a   b   c   d★ p...
Result(of(exp.(1stim. a   b   c   d  1 1     9   1   9                                     c cell                         ...
Experiment(2:            c(and(d(in(3x2(table★ 54,undergraduates,,2,  stimuli.                                    stimulus...
Experiment(2:           c(and(d(in(3x2(table★ Each(participant(  estimates(the(intensity(  of(causal(relationship(     sti...
Exp.,2:,Result                         c cell                                   d cell       13                           ...
Exp(3.(Rarity(vs.(affirmationG           negation★ Do(people(respond(to(the(rarity((hence(  informativeness)(or(more(simply(...
Exp(3.(Rarity(vs.(affirmationG               negation★ Participants,are,randomly,divided,into,4x4=16,  groups,,four,forms,in...
Exp(3.(On(rarity★ Story:  ★ Mentally,unstable:,rare  ★ Dropout:,rare★ In,the,sample,(stimuli)  ★ Whether,the,sample,P(unst...
Exp(3.(The(combinations(of(        affirmation(and(negation                dropped not dropped                       graduat...
Exp.(3(Result((coinciding(               condition)              coinciding yes/yes                               coincidi...
Exp.(3(Result((contradicting(               condition)              contradicting yes/yes                         contradi...
Exp(3.(Discussion★ In(both(of(the(two(conditions,(coinciding(and(  contradicting, ★ Participants(responded(to(the(rarity((...
toc★ New$paradigm$psychology$of$reasoning★ Reasoning$and$conditional★ Conditional$and$biconditional$event★ Biconditional$e...
Theoretical(background(of( biconditional(event(and(pARIs★ Angelo,Gilio,and,Giuseppe,Sanfilippo,(manuscript,  under,review),...
Simulation                           Correlation of the population is 0.20.8"0.7"0.6"0.5"                                 ...
Indices(equivalent(to(the(probability(of(biconditional#event★ Psychology ★ Tversky$index$of$similarity,$Tversky((1977)   ★...
Tversky(indexPsychological Review       J                                                                             330 ...
Conjunctive     MP                                                                   Def Bicond      Other                ...
Conclusion★ Our,intuition,for,generative,causality,from,co@occurrence,  data,is,the$probability$of$biconditional$event,(or...
Future(Issues★ Information,theoretical,analysis,of,the,efficiency,to,compute,pARIs,,  defective,biconditional,or,bicondition...
Conditionals(in(development★ Development,of,understanding,of,conditionals,(Gauffroy,&,  Barouillet,,2009)★ Four,development...
C. Gauffroy, P. Barrouillet / Developmental Review 29 (2009) 249–282Indicative conditional                                ...
younger participants (third graders), explaining the age-related increase in ‘‘false” r    Causal conditional             ...
‘‘If I pour out pink liquid in the vase then stars appear on it”.                                                         ...
Probability judgment         in development             C. Gauffroy, P. Barrouillet / Developmental Review 29 (2009) 249–2...
LS(part
LS(and(pARIs★ pARIs(almost(coincides(with(LS(under(extreme(  rarity((lim(d→∞).       LSR(q|p) = lim LS(q|p) ⇡ pARIs       ...
Dilemma(and(tradeoff           The.dilemma.between. exploitation.(information(utilization)(and.   exploration.(information(...
Dilemma(and(tradeoffWe(cant(locally.optimize(while(broadening.the.      range.of.JlocalJ.at(the(same(time.        choosing(...
n@armed,bandit,problems★ The(simplest(framework(exhibiting(the(  dilemma(and(tradeoff.★ It(is(to(maximize(the(total(reward(...
n@armed,bandit,problems★ In(this(study,(we(let(the(reward(be(binary,(1(  (win)(or(0((lose). ★ This(form(is(the(most(import...
Exploitation(vs.(exploration(in((            bandits★ Exploitation(is(to(utilize(the(existing(information,(  trying(the(lo...
Exploitation(vs.(exploitation(in((            bandits★ ...(Hence(exploitation.and.exploration.is.  mutually.exclusive.and(...
_Policies_,to,handle,the,dilemma★ Basically,designed,to,_balance_,exploitation,and,  exploration,,accepting,the,incompatib...
Speed–Accuracy,Trade@off                                                        Accurate      Accuracy,                    ...
Models(for(bandits★ PolicyGbased(models                               policy                                     value of ...
The,currently,best,model,for,bandits                                                            Auer(et(al.,(             ...
Illustration(of(UCB1(                       as the arms                       are chosen                       many times ...
Current,model,for,bandits                                        Speedy & Accurate                                  0.9   ...
Problems(of(UCB1★ Worse(in(the(initial(stage((the(speed.is(low)(  compared(with(other(valid(models. ★ It(must(be(both(fast...
What(do(we(do?★ Propose,a,new,model,for,overcoming,the,speed–  accuracy,tradeoff,by,weakening,the,dilemma,between,  greedy,...
Three,cognitive,properties★ A.(Satisficing( ★ coined(as(RsatisfyR(+(RsufficeR ★ Simon,(Psy.#Rev.,(1956(★ B.(Risk(aSitude( ★ K...
Irrationality(of(the(three(       cognitive(properties★ A.(Satisficing( ★ No(optimization(but(falling(into(a(local(optimum....
Rationality(of(the(three(       cognitive(properties★ A.,Satisficing, ★ Not(optimize(but(look(for(and(choose(a(satisfactory...
Brain                                   Property$A:$Satisficing                                                            ...
Relative,evaluation(is(especially(            important★ Relative(evaluation:(  ★ is(what(even(slime(molds((粘菌)(and(real(n...
The(framework(of(models(of(the(        three(properties★ Let(there(only(be(two(arms(A1(  and(A2.★ On(the(2x2(contingency(t...
A(model((RRSR)(of(the(three(              properties★ A(value(function(VRS(equipped(with(the(  three(properties(can(be(giv...
RS(heuristics★ Property(C((relative(estimation(of(value): ★ Failing(to(get(reward(with(arm(A2,means(A1(is(   relatively,go...
RS(heuristics                                                     Reward★ Property(B((risk(aSitude)                       ...
RS(heuristics★ Property(A((satisficing)  ★ Efficiently(realized(by(property(C(&(    B,(with(reference(r,=0.5.                ...
Result(by(RS                                  1.0       RS                                            LS                  ...
The(problem(of(RS★ The,naive,relative,evaluation,of,RS,works,only,  with,2,arms.,★ With,n,arms,,RS,is,not,definable,or,any,...
LS(model★ The(performance(of(LS(in(2G                        Reward  armed(bandit(problems(is(the(                      1 ...
LS(describing(causal(intuition★ LS,fits,the,experiment,data,of,causal,induction,  (inductive,inference,of,causal,relationsh...
The(properties(of(LS#★ Figure–ground(segregation(and(invariance(  of(ground(against(change(in(focus((figure).  ★ As(the(bac...
How do cognitive agents handle the tradeoff between speed and accuracy?
How do cognitive agents handle the tradeoff between speed and accuracy?
How do cognitive agents handle the tradeoff between speed and accuracy?
How do cognitive agents handle the tradeoff between speed and accuracy?
How do cognitive agents handle the tradeoff between speed and accuracy?
How do cognitive agents handle the tradeoff between speed and accuracy?
How do cognitive agents handle the tradeoff between speed and accuracy?
How do cognitive agents handle the tradeoff between speed and accuracy?
How do cognitive agents handle the tradeoff between speed and accuracy?
How do cognitive agents handle the tradeoff between speed and accuracy?
How do cognitive agents handle the tradeoff between speed and accuracy?
How do cognitive agents handle the tradeoff between speed and accuracy?
How do cognitive agents handle the tradeoff between speed and accuracy?
How do cognitive agents handle the tradeoff between speed and accuracy?
How do cognitive agents handle the tradeoff between speed and accuracy?
How do cognitive agents handle the tradeoff between speed and accuracy?
How do cognitive agents handle the tradeoff between speed and accuracy?
How do cognitive agents handle the tradeoff between speed and accuracy?
How do cognitive agents handle the tradeoff between speed and accuracy?
How do cognitive agents handle the tradeoff between speed and accuracy?
How do cognitive agents handle the tradeoff between speed and accuracy?
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How do cognitive agents handle the tradeoff between speed and accuracy?

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How do cognitive agents handle the tradeoff between speed and accuracy?

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How do cognitive agents handle the tradeoff between speed and accuracy?

  1. 1. How$do$cognitive$agents$handle$the$tradeoff$between$ speed$and$accuracy? Tatsuji(Takahashi(高橋(達二( ( Tokyo(Denki(University((東京電機大学 tatsujit@mail.dendai.ac.jp 28(Dec.(2012Matsumoto(lab.,(NAIST(松本研(奈良先端科学技術大学院大学
  2. 2. Tatsuji(Takahashi(高橋達二★ Studied(philosophy(and(history(of(science(with( KomachiGsan.(★ Got(a(Ph.D.(in(science(of(complex(systems(at( Kobe(university((supervisor:(Yukio–Pegio( Gunji(郡司ペギオ幸夫教授).★ Teaching(at(Tokyo(Denki(University((Hiki,( Saitama(campus),(running(a(lab(of(Rinternal( measurementR(内部観測研究室(and(gradually( changing(the(research(area(to(cognitive(science. ★ hSp://takalabo.rd.dendai.ac.jp/
  3. 3. Purpose(and(metaGtheory★ Purpose: ★ To(analytically(and(constructively(understand(the( flexibility(and(creativity(of(human(mind, ★ under(ambiguity,(uncertainty(or(even(indeterminacy,( in(this(interminable(world, ★ which(can(work(in(face(of(the$frame$problem(and( self:referential$paradox. ★ To(this(end,(we(treat(the(frame(problem(and(selfG referential(paradoxes(as(empirical(as(possible ★ in(cognitive(psychology,(machine(learning(and( robotics;(not(in(philosophy(itself.
  4. 4. The(problem(★ How$do$cognitive$agents$like$us$handle$the$ speed–accuracy$tradeoff$that$is$inevitable$in$ this$uncertain$world?★ There(should(be(many(things(we(can(learn( from(ourselves(in(understanding(and( engineering(cleverer(systems.
  5. 5. Illogical(biases(in(cognition★ In(our(classroom(experience:( ★ we(have(difficulty(in(understanding(material( implication((Rif(thenR(in(logic),(with(which(Rif(p(then( qR(is(true(if(p(is(false(or(q(is(true. ★ we(confuse(necessary(and(sufficient(conditions( ★ (Rif(p(then(qR(read(also(as(Rif(q(then(p,R(or(in(effect(Rp(iff(qR) ★ we(judge(the(probability(and(gain(from(a(situation( differently,(dependent(on(the(expression(of(the(state( description) ★ this(is(called(Rthe(Framing(effectR((popular(in(behavioral( economics,(by(Tversky(&(Kahneman)
  6. 6. Illogical(biases(in(cognition★ We(dont(follow(P(if(p(then(q)(=(P(notGp(or(q)((material( implication) ★ Generally(P(p)(is(small(hence(P(notGp)(is(big,(making(the(probability( of(P(notGp#or#q)#too(big(to(be(informative.★ We(consider(conditionals((if)(as(biGconditionals((if(and(only(if;( iff)(and(often(loosely(identify(necessary(and(sufficient(conditions ★ Merits(in(information(acquisition(using(conditionals((Oaksford(&( Chater,(1994;(HaSori,(2002) ★ Merits(in(causal(learning(for(not(strictly(distinguishing(forward( prediction(and(backward(diagnosis((with(Markov(equivalence)?★ The(Framing(effect ★ The(expression(in(state(description(represents(the(past(history(and( the(speakers(prediction(of(the(state.((McKenzie(&(Mikkelsen,(2000)
  7. 7. Illogical(biases(can(be(rational( and(even(logical★ The(illogical(biases(in(human(cognition(can(be( rationalized(when(considered(in(an(appropriate( context. ★ Sometimes(our(theory(at(hand(is(too(old(or(primitive( to(understand(the(rationality(in(human(cognition.★ Then,(it(should(be(possible(to(analyze(human( cognitive(biases(and(apply(them(to(machine( learning(or(artificial(intelligence.
  8. 8. Two(topics(of(this(talk:(★ (pARIs(part)(Study(of(how(we(reason,(with( emphasis(on(conditionals((sentences(of(the( form(Rif(p(then(qR). ★ Humans(seem(illogical(and(irrational(but(actually( the(form(of(our(reasoning(follows(some(newly( invented(theories.★ (LS$part)$Application$of$cognitive$properties$ of$human$to$machine$learning. ★ The(adapativeGness(of(some(biases(and(heuristics( in(human(cognition(can(be(actually(applied.
  9. 9. pARIs(part
  10. 10. Reasoning(and(conditional★ Three(forms(of(reasoning:(deduction,(induction,(abduction ★ Deduction(uses(conditionals ★ p(and(Rif(p(then(qR(→((q((modus(ponens) ★ Induction(forms(conditionals ★ coGoccurrence(of(p(and(q(→(Rif(p(then(qR ★ Abduction(retrogresses(conditionals(and(form(explanation ★ q(and(Rif(p(then(qR(→((p((affirmation(of(consequent) deduction induction abduction premise 1 p p q premise 2 p→q q p→q conclusion q p→q p
  11. 11. Causality(and(conditional★ Causal(relationship(is(usually(expressed(by(conditional. ★ If(global(warming(continues((W)(then(London(will(be(flooded((L). ★ (If(cause(then(effect)★ We(can(also(use(conditionals(of(the(form((If(effect.then(cause) ★ The(utility(of(confusing(the(two(forms:$ ★ We(should(test(independence(to(find(a(causal(relationship,(before( considering(the(directionality. ★ If(we(allow(for(directionality,(we(need(two(Bayes(networks,(test(and( choose(one(from(the(two.(This(is(cognitively(heavy(for(intuition. directed mode undirected mode Model 1 C E Model C ? E Model 2 C E
  12. 12. Material(implication★ Modeling(conditional(by(material.implication ★ Rif(p,(then(qR(⇔Rnot(p,(or(qR A(⊃(C C=T C=F A=T T F ★ Paradoxes(of(material(implication(1 A=F T T ★ If(there(is(no(gravity,(then(I(am(the(king(of(Japan. ★ If(p((antecedent)(is(false,(Rif(p(then(qR(is(true(no(maSer(what(q(is. ★ Paradoxes(of(material(implication(2 ★ If(I(am(the(king(of(Japan,(then(Tokyo(is(the(capital(of(Japan. ★ If(q((consequent)(is(true,(Rif(p(then(qR(is(true(no(maSer(what(p(is.★ Experiments(show(that(humans(do(not(follow( material(implication.
  13. 13. Material(implication A(⊃(C A=T C=T C=F T F A=F T T ★ Why(humans(dont(follow(material(implication? ★ Old(paradigm(psychology(of(reasoning:(Its(because(human(are( irrational(or(effortless((e.(g.,(mental(models(theory) ★ New(paradigm(psychology(of(reasoning:(Humans(reason( factoring(the(uncertainty(and(the(context((environment(structure)( into(their(reasoning. ★ Considering(uncertainty((the(truth(value(of(a(proposition(as( probability(in([0,1](with(1((true)(and(0((false)),( ★ With(the(probability(of(an(event((proposition)(usually(being(very( small,(material(implication(doesnt(work. ★ Humans(reason(allowing(for(uncertainty. ★ The(meaning(of(Rif(p(then(qR(by(humans(is(modeled(not(by(p#⊃# q(but(by(q|p. ★ With(q|p,#¬p(cases(are(ignored.
  14. 14. Defective(conditional★ For(half(a(century((since(1966),(it( has(been(known(that(humans( follow(the(Rdefective(truth(tableR( Table. defective truth table when(understanding(and(using( conditionals,(as(in(the(Table. If A then C C=T C=F★ Conditional(is(not(truthGfunctional? A=T true false★ For(a(conditional(p#=(RIf(A,(then(C,R ★ If(the(truth(value(combination(of( A=F irrelevant irrelevant antecedent(A(and(consequent(C(is( TT,(p(is(true.(If(TF,(p#is(false.(When( defective (no truth value assigned) A(is(false,(participants(of( experiments(answer(that(FT(and(FF( Psychologically: Wason, 1966; Johnson-Laird and Tagart, 1969; Wason and Johnson-Laird, do(not(make(p(true(nor(false(but( 1972; Evans et al., 1993. irrelevant(to(the(truth(value(of(p. Theoretically: Strawson 1950; Quine 1952 14
  15. 15. Defective(biconditional★ There(is(our(tendency(of( If and only if C=T C=F A then C interpreting(Rif(A(then(CR( as(Rif(A(then(C,(and(if(C( A=T true false then(AR(or(RA(if(and(only(if( CR((biconditional(reading). A=F false irrelevant★ Here(the(interpreted( biconditional(is(called( defective(biconditional. conjunction★ True(for(TT,(false(for(TF( If A If C and(FT,(irrelevant(only(for( then C C=T C=F then A C=T C=F FF. A=T T F★ In(deductive(tasks,(this( A=T T I paSern(has(been(known( (Evans(&(Over,(2004). A=F I I A=F F I 15
  16. 16. From(defective(conditional(to( conditional(event★ P(if(p(then(q)(=(P(q|p) ★ Not(P(if(p(then(q)(=(P(p(⊃(q)(=(P(¬p(or(q)★ q|p(as.an.event((conditional(event) ★ Boolean(algebra((ring)(R(can(not(nonGtrivially( include(q|p((Lewis(triviality(result). ★ We(need(to(extend(R(to(R|R#(conditional#event# algebra#:#Goodman,(Nguyen,(Walker,(1991).
  17. 17. Overview★ New(paradigm(psychology(of(reasoning★ De(FineSis(conditional$and(biconditional$event★ biconditional(event(in(causal$induction: ★ the(pARIs((proportion(of(assumedGtoGbe(rare(instances)(rule★ Meta:analysis(and(three$experiments$to(confirm(the(validity( of(pARIs★ Theoretical$background(and(connections(to(other(areas,(such( as: ★ Developmental$study$of$conditionals(by(Gauffroy(and( Barouillet((2009),( ★ Amos(Tverskys(study(of(similarity((1977),(and( ★ Jaccard$similarity$index(and(some(other(popular(indices(in( mathematics,(statistics(and(machine(learning.
  18. 18. toc★ New$paradigm$psychology$of$reasoning★ Reasoning$and$conditional★ Conditional$and$biconditional$event★ Biconditional$event$in$causal$induction:$ pARIs$(proportion$of$assumed:to:be$rare$instances)★ Meta:analysis★ Three$experiments★ Theoretical$background
  19. 19. New(paradigm(psychology(of( reasoning★ Very(naively(expressed... ★ Old(paradigm: ★ The(normative(theory(is(the(classical(bivalent(logic(with( conditionals(modeled(by(material(implication(P(if(p(then(q)( =(P(p(⊃(q)(=(P(¬p(or(q). ★ Doesnt(fit(the(data(in(many(areas:(from(this(some(said( humans(are(irrational(or(the(intelligence(is(quite(limited. ★ New(paradigm: ★ Probability(logic(with(P(if(p(then(q)(=(P(q|p) ★ de(FineSi(gives(the(appropriate(theory(of(subjective( probability. ★ Fits(the(data;(human(cognition(is(designed(to(treat( uncertainty(by(nature.(It(is(formed(through(evolution.
  20. 20. toc★ New$paradigm$psychology$of$reasoning★ Reasoning$and$conditional★ Conditional$and$biconditional$event★ Biconditional$event$in$causal$induction:$ pARIs$(proportion$of$assumed:to:be$rare$instances)★ Meta:analysis★ Three$experiments★ Theoretical$background
  21. 21. defective conditional and defective biconditional★ Defective truth table in the older paradigms ★ (Wason, 1966; Johnson-Laird and Tagart, 1969; Wason and Johnson-Laird, 1972; Evans et al., 1993) ★ is normative and coherent in the new paradigm old(paradigm new(paradigm defective( conditional( → conditional event(q|p defective( biconditional( → biconditional 21 event(p⟛q
  22. 22. de(FineSis(conditional$event★ Conditional(event,(formerly(called(defective(conditional,(is(a( core(notion(in(the(new(paradigm(psychology(of(reasoning.★ The(Equation:(the(probability(of(a(conditional(is(the( conditional(probability(of(the(consequent(given(the( antecedent. ★ P(if$p$then$q)$=$P(q|p)$(the$Equation) ★ ¬p(cases(are(neglected,(and(Rq|pR(is(itself(a((conditional)(event. de Finetti material conditional conditional biconditional conditional event event event p q p⊃q q|p p|q p⟛q T T T T T conjunction T T F F F V F F T T V F F V: void case F F T V V V
  23. 23. toc★ New$paradigm$psychology$of$reasoning★ Reasoning$and$conditional★ Conditional$and$biconditional$event★ Biconditional$event$in$causal$induction:$ pARIs$(proportion$of$assumed:to:be$rare$instances)★ Meta:analysis★ Three$experiments★ Theoretical$background
  24. 24. Causal(induction★ Example:(We(want(to(know(the(cause(of(a(health(problem,( right(now(just(from(pure(observation,(no(intervention.★ I(sometimes(have(stiff(shoulders(and(a(headache.(Whats( the(cause?(How(about(coffee? ★ a:.(cause=present/effect=present)$ ★ How(frequently(I(got(a(headache(after(having(a(cup(of(coffee?( ★ b:.(present/absent)$ ★ How(frequently(I(get$no(headache(after(coffee? ★ c:.(absent/present)$ ★ How(frequently(I(got(a(headache(without(coffee?( ★ d:.(absent/absent)$ ★ How(frequently(I(get$no(headache(without(coffee?(
  25. 25. Causal(induction(experiment(Stimulus$presentation:(a( showing b-cell type joint event pair(of(two(kinds(of( pictures(illustrating(the(presence(and(absence(of( cause(and(effect,(at(left( and(right,(respectively Response:(participants( evaluate(the(causal(intensity(they(felt(from(0( to(100,(using(a(slider( E ¬EC a b¬C c d
  26. 26. Causal((intensity)(induction★ Two(phases(of(causal(induction((HaSori(&(Oaksford( 2007) ★ Phase$1:$observational((statistical) ★ Phase$2:$interventional((experimental)★ We(focus(on(causal(induction(of(the(phase$1( for(generative$cause(because(preventive( causes(are(confusing(and(hard(to(treat( especially(in(the(observation(phase((HaSori(&( Oaksford,(2007).
  27. 27. Causal(Induction★ Here(we(study(the(causal(intensity.★ Recent(studies(emphasize(the(structure((the( topology(of(Bayes(network)(rather(than(the( intensity((node(weight).(However,(structure( and(intensity(have(a(mutual(relationship.(In(an( unknown(situation,(intensity(is(what(maSers( since(structure(is(not(known.★ Many(problems(about(intensity(remain( untouched. ★ Why.normative.models.such.as.∆P.and.Power.PC. donBt.fit.the.data?
  28. 28. ∆P = P (E|C) − P (E|¬C) = (a + b)(c + d) (a + b)(c + d) Framework(and(models(of(causal( PowerPC = induction + d) ad − bc∆P = P (E|C) − P (E|¬C)∆P = (a + b)(c 1 − P (E|¬C) ★ The(data((input)(is(coGoccurrence(of(the(target( effect((E)(and(a(candidate(cause((C). ∆P = P (E|C) − P (E|¬C) ∆P ∆P PowerPC = PowerPC = ★ Normative:(Delta:P(and(Power$PC((Cheng,(1997) 1 − P (E|¬C) 1 − P (E|¬C) ★ Descriptive:(H((Dual$Factor$Heuristics)((HaSori( ∆P = P (E|C) − P (E|¬C) &(Oaksford(2007) ∆P= ad − bc PowerPC = ∆P PowerPC − P (E|¬C) (a + b)d 1= 1 − P = ad − bc ∆P = P (E|C) − P (E|¬C) (E|¬C) (a + b)(c + d) ∆P ad − bc E ¬E PowerPC = ∆P = ad − bc PowerPC = 1 − P (E|¬C) = (a + b)d ad − bc ∆P = P (E|C) − P (E|¬C) = C a b 1 − P (E|¬C) (a + b)d (a + b)(c + d) a ¬C c d H= P (E|C)P (C|E) = ∆P ∆P (a +ad −+ c) b)(a bc
  29. 29. The(pARIs(rule★ The(frequency(information(of(rare(instances( conveys(more(information(than(abundant(instances( (rational$analysis(and(rarity$assumption,(see(esp.( McKenzie(2007).★ Because(of(the(frame(problemGlike(aspect,(the(dGcell( information(can(be(unreliable((depends(strongly(on( how(we(frame(and(count).★ Hence(we(calculate(the(causal(intensity(only(by(the( proportion(of(assumedGtoGbe(rare(instances((pARIs) ★ named(after(pCI:.proportion.of.confirmatory. instances,(White(2003.
  30. 30. Rarity(assumption H=P (E|C)P (C|E) ★ We(assume(the(effect(in(focus(and(the(candidate(= cause(to(be(rare:(P(C)(and(P(E)(to(be(small. P (E|C)P (C|E) = a ★ Originally(in(Oaksford(&(Chater,(1994,( (a + b)(a + c) ★ then(in(HaSori(&(Oaksford,(2007,(McKenzie(2007,( a in(the(study(of(causal(induction= P (E|C)P (C|E) = (a + b)(a + c) ★ C(and(E(to(take(small(proportion(in(U. U lim φ = d→∞ P (E|C)P (C|E) = H C E ϕ: correlation ba cextreme coefficient lim φ = rarity P (E|C)P (C|E) = H d d→∞
  31. 31. The(pARIs(rule★ C(and(E(are(both(generally(assumed(to(be(rare((P(C)(and(P(E)(low).★ pARIs(=(proportion(of(assumedGtoGbe(rare(instances((a,#b,#and(c).( pARIs = P(p⟛q) = a / (a+b+c) U E -E C E C a b ba c d -C c d conditional event biconditional event infering causal intensity C E E|C C⟛E pARIs T T T T positive T F F F negative F T V F negative F F V V irrelevant
  32. 32. The(pARIs(rule★ C(and(E(are(both(assumed(to(be(rare((P(C)(and(P(E)(low)★ pARIs(=(proportion(of(assumedGtoGbe(rare(instances((a,#b,#and(c).(★ The(probability(of(the(conjunction(of(cause(and(effect(given(the( disjunction(of(cause(and(effect((conditioned(on(the(disjunction).( pARIs = P(C iff E) = P(C and E | C or E) P(C and E) a = = P(C or E) a+b+c E -E U C E C a b ba c d -C c d
  33. 33. Why(ignore(the(dGcell?★ Hempels(paradox ★ All(ravens(are(black.( ★ =(If(something(is(a(raven,(then(it(is(black. ★ Is$a$non:black$non:raven$confirmatory?★ If(a(nonGraven(that(is(not(black(is(rare,(it(is( informative(hence(not(ignored.((McKenzie(&( Mikkelsen,(2000)★ If(Raven:nonGraven(=(5:5(and(black/nonGblack(=(5:5: ★ RAll(men(are(stupid(than(the(average(of(human( beings.R((RIf(one(is(a(man,(then.he(is(relatively(stupid.R) ★ A(thoughtful(woman(can(be(confirmatory.
  34. 34. toc★ New$paradigm$psychology$of$reasoning★ Reasoning$and$conditional★ Conditional$and$biconditional$event★ Biconditional$event$in$causal$induction:$ pARIs$(proportion$of$assumed:to:be$rare$instances)★ Meta:analysis★ Three$experiments★ Theoretical$background
  35. 35. DataGfit(of(pARIs(and(PowerPC AS95 BCC03exp1generative BCC03exp3 H03 100 100 100 100 80 80 80 80Human rating Human rating Human rating Human rating 60 60 60 60 40 40 40 40 20 20 20 20 0 0 0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Model prediction Model prediction Model prediction Model prediction H06 LS00exp123 W03JEPexp2 W03JEPexp6 100 100 100 100 80 80 80 80Human rating Human rating Human rating Human rating 60 60 60 60 40 40 40 40 20 20 20 20 0 0 0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Model prediction Model prediction Model prediction Model prediction
  36. 36. MetaGanalysis★ Fit(with(experiments((the(same(as(HaSori(&(Oaksford,(2007)★ pARIs(fits(the(data(set(with(the(lowest(correlation(r(<(0.9,(the( highest(average(correlation(in(almost(all(the(data,(and(the( smallest(average(error. best next best bad otherwiseexperiment model pARIs DFH PowerPC ∆P Phi P(E|C) P(C|E) pCI AS95 0.94 0.95 0.95 0.88 0.89 0.91 0.76 0.87 BCC03: exp1 0.98 0.97 0.89 0.92 0.91 0.82 0.51 0.92 BCC03: exp3 0.99 0.99 0.98 0.93 0.93 0.95 0.88 0.93 H03 0.99 0.98 -0.09 0.01 0.70 -0.01 0.98 0.40 H06 0.97 0.96 0.74 0.71 0.71 0.89 0.58 0.70 LS00 0.93 0.95 0.86 0.83 0.84 0.58 0.34 0.83 W03.2 0.90 0.85 0.44 0.29 0.55 0.47 0.18 0.77 W03.6 0.93 0.90 0.46 0.46 0.46 0.77 0.56 0.54 average r 0.95 0.94 0.65 0.63 0.75 0.67 0.60 0.75 average error 11.97 18.48 33.39 24.30 27.18 27.78 24.75 29.93 Values other than in error row are correlation coefficient r.
  37. 37. correlation7.00 0.90 0.85 0.93 0.95 0.555.13 0.77 0.44 0.29 0.84 0.97 0.96 0.47 0.83 0.86 0.18 0.83 0.71 0.58 0.34 0.58 0.703.25 0.99 0.98 0.74 0.71 0.70 0.89 0.01 0.40 0.99 0.99 0.98 0.98 0.93 0.93 0.95 0.931.38 0.98 0.88 0.97 0.89 0.92 0.91 0.82 0.92 0.51 0.94 0.95 0.95 0.88 0.89 0.91 0.76 0.87 -0.09 -0.01-0.50 pARIs DFH PowerPC ΔP Phi P(E|C) P(C|E) pCI AS95 BCC03exp1 BCC03exp3 H03 H06 LS00 W03.2 300 225 150 75 average,error 0
  38. 38. toc★ New$paradigm$psychology$of$reasoning★ Reasoning$and$conditional★ Conditional$and$biconditional$event★ Biconditional$event$in$causal$induction:$ pARIs$(proportion$of$assumed:to:be$rare$instances)★ Meta:analysis★ Three$experiments★ Theoretical$background
  39. 39. Experiments★ Experiment,1 ★ To,test,the,validity,of,rarity,assumption,in,ordinary, causal,induction,from,2x2,covariation,information★ Experiment,2 ★ To,test,the,validity,of,rarity,assumption,in,causal, induction,from,3x2,covariation,information ★ Difference,in,the,cognition,between,rare,events,(a,#b,,and, c@type),and,non@rare,d@type,event,,people,just,vaguely, recognize,and,memorize,the,occurrence,of,d@type,events.★ Experiment,3 ★ Rarity,vs.,presence@absence,(yes@no)
  40. 40. Experiment(1:( c(and(d(in(2x2(table★ 27(undergraduates,(9( stimuli. stim. a b c d★ p:(to(give(artificial(diet(to( 1 1 9 1 9 2 1 9 5 5 your(horse,(q:(your(horse( 3 1 9 9 1 gets(ill.( 4 5 5 1 9★ After(the(presentation(of( 5 5 5 5 5 (a,b,c,d),(participants(are( 6 5 5 9 1 7 9 1 1 9 asked(the(causal(intensity( 8 9 1 5 5 and(then(the(frequency(of(cG( 9 9 1 9 1 and(dGtype(event.
  41. 41. Result(of(exp.(1stim. a b c d 1 1 9 1 9 c cell d cell 2 1 9 5 5 10 10 3 1 9 9 1 8 8 4 5 5 1 9 6 5 5 5 5 5 5 4 6 5 5 9 1 2 3 7 9 1 1 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 8 9 1 5 5 real c estimated c real d estimated d 9 9 1 9 1★ Participants(estimation(of(c(and(d(occurrence(was( basically(faithful,(but(d(is(estimated(larger(than(the( real(stimuli.
  42. 42. Experiment(2: c(and(d(in(3x2(table★ 54,undergraduates,,2, stimuli. stimulus A q not-q★ As,a,medical,scientist,,p:,to, p1 6 4 give,a,medicine,(three,types,, p1,,p2,and,p3),to,a,patient,q:, p2 9 1 the,patient,develops, p3 2 8 antibodies,against,a,virus.★ After,the,presentation,of,six, stimulus B q not-q kinds,of,events,,participants, p1 5 5 are,asked,the,causal, p2 8 2 intensity,of,p1,to,q,and,p2,to, p3 1 9 q,,and,then,the,frequency,of, c<,and,d@type,event.
  43. 43. Experiment(2: c(and(d(in(3x2(table★ Each(participant( estimates(the(intensity( of(causal(relationship( stimulus A q not-q from(p1(to(q. p1 6 a 4 b★ Then(asked(the(value(of( focus p2 9 c 1 d c,(as(RHow(often(q( + + happened(in(the( p3 2 8 absence(of(p1?.R(The( given(value(of(c(is( 9+2=11.
  44. 44. Exp.,2:,Result c cell d cell 13 14 10 11 7 7 3 4 0 0r (=(0.99 2 1 2 3 4 1 2 3 4 r2(=(0.49 real c estimated c real d estimated d ★ ParticipantsN,estimation,of,c,and,d,occurrence,were,very, different.,The,correlation,between,the,estimated,d,and,the, real,,given,value,of,d,was,significantly,smaller,than,for,c.
  45. 45. Exp(3.(Rarity(vs.(affirmationG negation★ Do(people(respond(to(the(rarity((hence( informativeness)(or(more(simply((as(in( matching(heuristics/bias)(to(yes/no((presence/ absence(of(cause(and(effect)?★ 132(undergraduates,(4(stimuli(x(2(conditions.★ Participants(evaluates(the(causal(relationship( from(mental$unstableness(to(dropout(in( college(students.
  46. 46. Exp(3.(Rarity(vs.(affirmationG negation★ Participants,are,randomly,divided,into,4x4=16, groups,,four,forms,in,two,conditions,(coinciding,and, contraditing) ★ Group(1(:(Yes/Yes(means(Runstable(and(dropped(outR ★ Group(2(:(Yes/No(means(Runstable(and(not(graduatedR ★ Group(3(:(No/Yes(means(Rnot(healthy(and(dropped(outR ★ Group(4(:(No/No(means(Rnot(healthy(and(not(graduatedR
  47. 47. Exp(3.(On(rarity★ Story: ★ Mentally,unstable:,rare ★ Dropout:,rare★ In,the,sample,(stimuli) ★ Whether,the,sample,P(unstable),is,small,or,not ★ Whether,the,sample,P(dropout),is,small,or,not★ Two,conditions: ★ Coinciding$condition$:,the,sample,P(unstable),and, P(dropout),are,both,small,(coincides,with,the,story/prior, knowledge) ★ Contradicting$condition$:,the,sample,P(unstable),and, P(dropout),are,both,large,(contradicts,with,the,story/prior, knowledge)
  48. 48. Exp(3.(The(combinations(of( affirmation(and(negation dropped not dropped graduated not out out graduated unstable a b unstable a b not unstable c d not unstable c dorange : confirmatory instances, yellow : disconfirmatory instances, white : irrelevant dropped not dropped graduated not out out graduated mentally healthy a b mentally healthy a b not mentally healthy c d not mentally healthy c d Participants evaluate the intensity of the causal relationship from the cause unstableness to the effect dropout is evaluated. 48
  49. 49. Exp.(3(Result((coinciding( condition) coinciding yes/yes coinciding yes/no100 10075 7550 5025 25 0 0 Mean pARIs Mean pARIs (2,2,2,8) (1,1,3,10) (1,1,1,15) (1,1,3,14) coinciding no/yes coinciding no/no100 100 75 75 50 50 25 25 0 0 Mean pARIs Mean pARIs 49
  50. 50. Exp.(3(Result((contradicting( condition) contradicting yes/yes contradicting yes/no 100 100 75 75 50 50 25 25 0 0 Mean pARIs Mean pARIsstimuli : (6,1,1,1) (8,1,2,3) (7,3,1,3) (6,2,2,3) contradicting no/yes contradicting no/no 100 100 75 75 50 50 25 25 0 0 Mean pARIs Mean pARIs 50
  51. 51. Exp(3.(Discussion★ In(both(of(the(two(conditions,(coinciding(and( contradicting, ★ Participants(responded(to(the(rarity((hence( informativeness). ★ Not(to(mere(yes/no((presence/absence(of(cause( and(effect). ★ If(they(had(responded(to(yes/no,(rather(than(the( rarity,(then(we(would(observe(something(like( matching$bias?
  52. 52. toc★ New$paradigm$psychology$of$reasoning★ Reasoning$and$conditional★ Conditional$and$biconditional$event★ Biconditional$event$in$causal$induction:$ pARIs$(proportion$of$assumed:to:be$rare$instances)★ Meta:analysis★ Three$experiments★ Theoretical$background
  53. 53. Theoretical(background(of( biconditional(event(and(pARIs★ Angelo,Gilio,and,Giuseppe,Sanfilippo,(manuscript, under,review),are,studying,biconditional#event,p⟛q# (named,by,Andy,Fugard),in,relation,to,quasi<conjunction.★ Bart,Kosko,(2004),studied,probable$equivalence,, equivalent,idea,in,his,fuzzy,probability,theory.★ There,are,some,equivalent,indices,defined,for, computing,similarity.★ Computer,simulations,shows,that,pARIs,is,very, efficient,,reconciling,speed,and,accuracy,or,variance,and, bias,(their,tradeoff),in,inferring,the,correlation,of,the, population,from,a,small,sample,set,,with,the,highest, reliability,and,precision.
  54. 54. Simulation Correlation of the population is 0.20.8"0.7"0.6"0.5" pARIs" DFH"0.4" Delta"P" Phi"0.3" PowerPC"0.2"0.1" 0" 1" 2" mean value through MC sim. 3" 4" 5" 6" 7" 8" 9" 10" 11" 12" 13" 14" 15" 16" 17" 18" 19" 20" 21" 22" 23" 24" 25" 26" 27" 28" 29" DFH:(accurate(but(slow pARIs both speedily and ΔP:(fast(but(inaccurate 1" accurately grasps the0.9" population correlation with a very small sample0.8"0.7" pARIs" HaSori(&(0.6" DFH"0.5" Delta"P"0.4" sd value Phi"0.3"0.2" PowerPC" Oaksford,(20070.1" 0" 1" 2" 3" 4" 5" 6" 7" 8" 54 9" 10" 11" 12" 13" 14" 15" 16" 17" 18" 19" 20" 21" 22" 23" 24" 25" 26" 27" 28" 29" 30" 31"
  55. 55. Indices(equivalent(to(the(probability(of(biconditional#event★ Psychology ★ Tversky$index$of$similarity,$Tversky((1977) ★ Asymmetric(similarity(measure(comparing(a(variant(to( a(prototype.(Also(in:(Gregson((1975)(and(Sjöberg((1972)★ Mathematics,(machine(learning(and(statistics:( ★ Probable$equivalence,$or(the(probabilistic( indentity(of(two(sets(A(and(B,$P(A=B)(by( Kosko((2004) ★ Tanimoto$similarity$coefficient ★ Jaccard$similarity$measure
  56. 56. Tversky(indexPsychological Review J 330 Copyright © 1977 C_? by the American Psychological Association, Inc. AMOS TVERSKY V O L U M E 84 NUMBER 4 JULY 1977 not to compo A-B 2. M Features of Similarity APIB whene Amos Tversky FEATURES OF SIMILAR Hebrew University B-A Jerusalem, Israel andThe metric and dimensional assumptions that underlie the geometric Figure 1. A graphical illustration of the relation between represen- matching function of interest is the ratio model,tation of similarity are questioned on both theoretical and empirical two feature sets. grounds.A new set-theoretical approach to similarity is developed in which objects are Hence, Morrepresented as collections of features, and similarity is described as a feature- _matching process. Specifically, a set of qualitative assumptions is shown to of features is viewed as a product of a lection f(A - Tha B eitherimply the contrast model, which expresses the similarity between objects as process of extraction and compilation. prior alinear combination of the measures of their common and distinctive features. . , - ( nB)+af(A-B)+^f(B-A) f(B),ofpro Second, the term, feature usually denotes theSeveral predictions of the contrast model are tested in studies of similarity with com tive fe f A model is used to uncover,value of a binary variable (e.g., voiced vs.both semantic and perceptual stimuli. The analyze, symmetr object «,/3>0,and explain a variety of empirical phenomena such as the role of common and consonants) or the value of a nominal voicelessdistinctive features, the relations between judgments of similarity and differ-ence, the presence of asymmetric similarities, and the effects of context on (e.g., eye color). Feature representa- variable axiom in measu letters
  57. 57. Conjunctive MP Def Bicond Other Def Cond Weak 90%Biconditional,event 80% 70% 60% 50%★ Developmental 40% 30% 20% ★ Merely(transient(in(the( 10% process(of(narrowing( 0% 3 6 9 adults Grades the(scope,(between( Conjunctive MP conjunctive(and( Def Bicond Other Def Cond Strong conditional?((Gauffroy( 90% 80% and(Barouillet,(2009) 70% 60% ★ Probably(there(are( 50% 40% theoretical(reasons(for( 30% the(dominance(of( 20% 10% defective(biconditional( 0% 3 6 9 adults (biconditional(event). Grades Gauffroy and Barouillet, 2009 Fig. 3. Percent of response patterns categorized as conjunctive, defective biconditional (Def Bicond), defective cond Cond), matching (MP), and others as a function of grades for strong and weak causal conditionals in Experiment 2
  58. 58. Conclusion★ Our,intuition,for,generative,causality,from,co@occurrence, data,is,the$probability$of$biconditional$event,(or, defective$biconditional). ★ Conditional,event,is,the,conditional,in,the,new,paradigm. ★ Biconditional$event$is,the,biconditional,in,the,new, paradigm.★ In,causal,induction,,biconditional,event,focuses,on,rare$ events,and,neglects,abundant,events,,in,the,uncertain, world. ★ pARIs:,proportion,of,assumed@to@be,rare,instances★ Defective,biconditional,is,turning,out,to,have,some, normative,nature,and,theoretical,grounds,as, biconditional,event.
  59. 59. Future(Issues★ Information,theoretical,analysis,of,the,efficiency,to,compute,pARIs,, defective,biconditional,or,biconditional,event ★ Gilio,and,Sanfilippo,proved,biconditional,event,is,a,kind,of,norm,,and,Kosko, defined,it,as,a,measure,for,the,identity,(binary,relation),of,two,random, variables★ The,relationship,of,causal,induction,and,(causal),conditionals ★ Semantic,and,pragmatic,analysis,,and,the,conditionals,of,the,diagnostic/ abductive,form,_if,effect,,then,cause._,(Over)★ To,determine,the,scope,of,the,pARIs,rule ★ In,other,words,,when,delta@p,or,Power,PC,can,be,descriptive?,(w/,Habori,, Habori,,Over)★ To,establish,a,full,connection,with,the,new,paradigm,psychology,of, reasoning,(Over,,Evans,,...),and,the,de,Finebi,table,(Baratgin,,Policer,,...), (w/,Baratgin,,Habori,,Habori) ★ Toward,an,integration,of,conditional,reasoning,and,statistical,inference, ★ The,four,cards,in,Wason,selection,tasks,fall,into,four,cells,on,de,Finebi,table.,(Over)
  60. 60. Conditionals(in(development★ Development,of,understanding,of,conditionals,(Gauffroy,&, Barouillet,,2009)★ Four,developmental,stages:,3rd,grader,,6th,grader,,9th, grader,,adults,(respectively,,8,,11,,15,,24,years,old,in,average)★ Defective,biconditional,=,biconditional,event,shows,up. conjunctive defective defective material probability conditional biconditional conditionalp q p|q q|p p⟛q p⊃qT T T T T TT F F F F FF T F V F TF F F V V T
  61. 61. C. Gauffroy, P. Barrouillet / Developmental Review 29 (2009) 249–282Indicative conditional Conjunctive Def Bicond MP Other in development280 C. Gauffroy, P. Barrouillet / Developmental Review 29 (2009) Def Cond 249–282 NN 90% 80%Appendix 70% 60%BB conditionals used in Experiment 1 50% 40% ‘‘If the pupil is a boy then he wears glasses”. 30% ‘‘If the door is open then the light is switched on”. 20% ‘‘If the student is a woman then she wears a shirt with long sleeves”. 10% ‘‘If the piece is big then it is pierced”. 0% 3 6 9 adults GradesNN conditionals used in Experiment 1 Conjunctive MP Def Bicond Other Def Cond ‘‘If the card is yellow then a triangle is printed on it”. BB ‘‘If there is a star on the screen then there is a circle”. 90% ‘‘If he wears a red t-shirt then he wears a green trousers”. 80% ‘‘If there is a rabbit in the cage then there is a cat”. 70% 60% name form 50%Strong causal relations used in Experiment 2 Conjunctive = TT/All 40% Def Bicond = TT/(TT+TF+FT) are switched on”. 30% ‘‘If the button 3 is turned then the blackboard’s lights 20% Def Cond = TT/(TT/TF) ‘‘If the lever 2 is down, then the rabbit’s cage is open”. 10% ‘‘If the second button of the machine is green then the machine makes sweets”. MP = (TT+FT+FF)/All ‘‘If I pour out pink liquid in the vase then stars appear on it”. 0% 3 6 9 adults Grades Other = other forms All := TT+TF+FT+FFWeak causal relations used in Experiment 2 61 Gauffroy & Barouillet, 2009 Fig. 1. Percent of response patterns categorized as conjunctive, defective biconditional (Def Bicond), defecti Cond), matching (MP) and others as a function of grades for NN and BB conditionals in Experiment 1.
  62. 62. younger participants (third graders), explaining the age-related increase in ‘‘false” r Causal conditional p :q case. First of all, as we predicted, conjunctive response patterns predominNN conditionals used in Experiment 1 Conjunctive in development MP ‘‘If the card is yellow then a triangle is printed on it”. Def Bicond Other ‘‘If there is a star on the screen then there is a circle”. Def Cond Weak ‘‘If he wears a red t-shirt then he wears a green trousers”. 90% ‘‘If there is a rabbit in the cage then there is a cat”. 80% 70% 60%Strong causal relations used in Experiment 2 50% 40% ‘‘If the button 3 is turned then the blackboard’s lights are switched on”. ‘‘If the lever 2 is down, then the rabbit’s cage is open”. 30% ‘‘If the second button of the machine is green then the machine makes sweets”. 20% ‘‘If I pour out pink liquid in the vase then stars appear on it”. 10% 0% 3 6 9 adultsWeak causal relations used in Experiment 2 Grades Conjunctive MP ‘‘If the touch F5 is pressed then the computer screen becomes black”. Def Bicond Other ‘‘If the boy eats alkali pills then his skin tans”. Def Cond Strong ‘‘If the fisherman puts flour in the water then he catches a lot of fishes”. 90% ‘‘If the gardener pours out buntil in his garden then he gathers a lot of tomatoes”. 80% 70% namePromises used in Experiment 3 form 60% 50% Conjunctive = TT/All ‘‘If you gather the leafs in the garden then I give you 5 francs”. 40% ‘‘If you score Def Bicond a goal then I name= TT/(TT+TF+FT) you captain”. 30% ‘‘If you exercise the dog then I cook you a cake for dinner”. 20% ‘‘If you clean your room then you watchTT/(TT/TF) Def Cond = the TV”. 10% MP = (TT+FT+FF)/All 0% 3 6 9 adultsThreats used in Experiment 3 Other = other forms Grades All := TT+TF+FT+FF ‘‘If you break the vase then I take your ball”. Gauffroy & Barouillet, 2009 Fig. 3. Percent of response patterns categorized as conjunctive, defective biconditional (Def Bicond), defec 62 Cond), matching (MP), and others as a function of grades for strong and weak causal conditionals in Exp ‘‘If you do not buy the bread then you do not play video games”.
  63. 63. ‘‘If I pour out pink liquid in the vase then stars appear on it”. C. Gauffroy, P. Barrouillet / Developmental Review 29 (2009) 249–282 Weak causal relations used in Experiment 2 Promise and threat Conjunctive Def Bicond Equivalence Otherconditionals in development ‘‘If the touch F5 is pressed then the computer screen becomes black”. Def Cond Promises ‘‘If the boy eats alkali pills then his skin tans”. 90% ‘‘If the fisherman puts flour in the water then he catches a lot of fishes”. 80% ‘‘If the gardener pours out buntil in his garden then he gathers a lot of tomatoes”. 70% 60% Promises used in Experiment 3 50% 40% 30% ‘‘If you gather the leafs in the garden then I give you 5 francs”. ‘‘If you score a goal then I name you captain”. 20% ‘‘If you exercise the dog then I cook you a cake for dinner”. 10% ‘‘If you clean your room then you watch the TV”. 0% 3 6 9 Adults Grades Threats used in Experiment 3 Conjunctive Equivalence Def Bicond Other ‘‘If you break the vase then I take your ball”. Def Cond Threats ‘‘If you do not buy the bread then you do not play video games”. 90% ‘‘If you do not do your homework then you do not go to the attraction park”. 80% ‘‘If you have a bad mark then you do not go to the movie”. 70% name form 60% References 50% Conjunctive = TT/All 40% Artman, L., Cahan, S., & Avni-Babad, D. (2006). Age, schooling and conditional reasoning. 30% Cognitive Development, 21(2), 131–145. Def Bicond = TT/(TT+TF+FT) Barra, B. G., Bucciarelli, M., & Johnson-Laird, P. N. (1995). Development of syllogistic reasoning. American Journal of Psychology, 20% 108(2), 157–193. Cond Def = TT/(TT/TF) 10% Barrouillet, P., Gauffroy, C., & Lecas, J. F. (2008). Mental models and the suppositional account of conditionals. Psychological 0% MP Review, 115(3), 760–771. = (TT+FT+FF)/All 3 6 9 Adults Barrouillet, P., Gavens, N., Vergauwe, E., Gaillard, V., & Camos, V. (2009). Memory span development: A time-based resource- Grades Equivalence = (TT+FF)/All sharing model account. Developmental Psychology, 45(2), 477–490. Fig. 4. Percent of response patterns categorized as conjunctive, defective biconditional (Def Bicond), defective Barrouillet, P., Grosset, N., & Lecas, J. F. (2000). Conditional reasoning by mentaland others as a Chronometric promises and threats in Experiment 3. Cond), equivalence, models: function of grades for and developmental All := TT+TF+FT+FF evidence. Cognition, 75, 237–266. 63 Gauffroy & Barouillet, 2009
  64. 64. Probability judgment in development C. Gauffroy, P. Barrouillet / Developmental Review 29 (2009) 249–282 269 272 C. Gauffroy, P. Barrouillet / Developmental Review 29 (2009) 249–282 Conjunctive Def Cond Def Bicond Other 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% 6 9 Adults Grades Fig. 5. Example of material given to participants in the probability task. of response patterns categorized as conjunctive, defective biconditional (Def Bicond), and defec Fig. 6. Percent (Def Cond) responses to the probability task in Experiment 4. could be expected from previous studies (Evans et al., 2003; Oberauer & Wilhelm, 2003) responses were very frequent, even in adults. Our interpretation is that the difficulty of t many participants to base their evaluation on the sole initial model provided by heuristic s our theory account for the way people evaluate the probability of conditional statements a consequence, it can be observed that the developmental trend resulting from the intervre its developmental predictions? Our hypothesis is that people evaluate theGauffroy & 64 probability of Barouillet, 2009 analytic system is delayed in the probability task, with sixth graders producing almost 80% tive responses, a rate never observed with the truth table task in the present study or t
  65. 65. LS(part
  66. 66. LS(and(pARIs★ pARIs(almost(coincides(with(LS(under(extreme( rarity((lim(d→∞). LSR(q|p) = lim LS(q|p) ⇡ pARIs d!1
  67. 67. Dilemma(and(tradeoff The.dilemma.between. exploitation.(information(utilization)(and. exploration.(information(acquisition) leads(to the.tradeoff.between. speed((shortGterm(reward)(and. accuracy((longGterm(reward)
  68. 68. Dilemma(and(tradeoffWe(cant(locally.optimize(while(broadening.the. range.of.JlocalJ.at(the(same(time. choosing(a(known(option(vs.( looking(for(a(new(unknown(option. leads(to While(it(is(desirable(to(be(fast(and(accurate,( quality(often(comes(at(the(cost(of(speed.( (Jiang(et(al.,(NIPS(2012)
  69. 69. n@armed,bandit,problems★ The(simplest(framework(exhibiting(the( dilemma(and(tradeoff.★ It(is(to(maximize(the(total(reward(acquired( from(n(sources(with(unknown(reward( distribution.★ OneGarmed(bandit(is(a(slot(machine(that(gives( a(reward((win)(or(not((lose).★ nGarmed(bandit(is(a(slot(machine(with(n(arms( that(have(different(probability(of(winning.(
  70. 70. n@armed,bandit,problems★ In(this(study,(we(let(the(reward(be(binary,(1( (win)(or(0((lose). ★ This(form(is(the(most(important(one(used(in( MonteGCarlo(Tree(Search(extremely(successful( and(popular(for(AIs(for(the(Game(of(Go(囲碁AI.(★ Each(arm(of(the(slot(machine(has(a(probability( of(giving(1((win). ★ n(probabilities(defines(a(nGarmed(bandit( problems.
  71. 71. Exploitation(vs.(exploration(in(( bandits★ Exploitation(is(to(utilize(the(existing(information,( trying(the(local(optimization. ★ In(bandits,(it(is(to(choose(the((greedy)(arm(with(the( highest(probability(of(winning.★ Exploration(is(to(broaden(the(range(of(information(at( hand,(trying(the(search(for(the(best(yet(unknown(arm. ★ to(choose(an((nonLgreedy)(arm(with(the(unknown(or( lower(probability(of(winning(than(the(greedy(arm.★ Hence(exploitation.and.exploration.is.mutually. exclusive.and(incompatible.
  72. 72. Exploitation(vs.(exploitation(in(( bandits★ ...(Hence(exploitation.and.exploration.is. mutually.exclusive.and(incompatible. ★ QUESTION:(Is(this(true?(On(what(ground?(Isnt( there(the,cost,of,well2definedness?
  73. 73. _Policies_,to,handle,the,dilemma★ Basically,designed,to,_balance_,exploitation,and, exploration,,accepting,the,incompatibility,between, them,,probabilistically,recombining,the,two. ★ ε2greedy,policy: ★ Given,a,parameter,ε,,choose,the,greedy,action,with, probability,1–ε,and,one,of,the,non@greedy,actions,with, probability,ε. ★ Softmax,action$selection$policy: ★ Roulebe,selection,of,action,with,the,probability,of, choosing,each,action,given,by,Gibbs,distribution,and,a, noise,(temperature),parameter,τ.
  74. 74. Speed–Accuracy,Trade@off Accurate Accuracy, 0.8 (the(rate(of(the( Speedy Accuracy rate 0.7 optimal(action( chosen) 0.6 —$softmax$1 softmax1 softmax2 0.5 —$softmax$2 0 200 400 600 800 1000Speed,and,accuracy, Steps are,usually,not, Step, compatible., (the(number(of(choice)
  75. 75. Models(for(bandits★ PolicyGbased(models policy value of action ★ ε2greedy,policy,and,Softmax, actions action,selection,rule value function action function action value value★ Value(function(models action action policy value ★ UCB1,(this,enabled,the, value of actions action current,performance,of, state Game,of,Go,AI,with,MCTS) Agent ★ LS,(our,cognitively–inspired, reward action model,implementing, cognitive,properties,that, Environment appear,to,be,illogical,and, useless) Components of reinforcement learning model
  76. 76. The,currently,best,model,for,bandits Auer(et(al.,( UCB1,: Machine#learning,( 2002 Value function considering the reliability the(term(to(suspend(judgment(and(induce(RsearchR (sample size) UCB1@tuned,:★ A,is,an,action,(arm)★ E,is,the,presence,of,reward,(E=1).★ n,is,the,current,step,(=,the,number,of,times,arms,are,chosen).★ ni,is,the,number,of,times,the,agent,chose,the,arm,Ai.
  77. 77. Illustration(of(UCB1( as the arms are chosen many times 0.6 0.6 0.4 the extra 0.4 term decays A1 < A2 A1 > A2★ The,reason,for,the,performance,of,UCB1@tuned,is,that,it,delays,the,judgement,of,value,as,long,as,possible.
  78. 78. Current,model,for,bandits Speedy & Accurate 0.9 Accuracy, Accurate 0.8 Accuracy rate 0.7 Speedy — softmax 1 0.6 — softmax 2 softmax1 softmax2 — UCB1 UCB1 UCB1.tuned — UCB1-tuned 0.5 0 200 400 600 800 1000 Step,Steps
  79. 79. Problems(of(UCB1★ Worse(in(the(initial(stage((the(speed.is(low)( compared(with(other(valid(models. ★ It(must(be(both(fast(and(accurate,(but(UCB1( pursues(accuracy(at(the(cost(of(speed. ★ UCB1(requires(so(many(steps. ★ It(doesnt(work(well(when(the(reward(is(sparse. ★ In(the(real(world,(we(cant(limitlessly(choose(actions.( We(dont(have(such(massive(resource.(Also,(the( reward(for(an(action(can(come(much(later.(
  80. 80. What(do(we(do?★ Propose,a,new,model,for,overcoming,the,speed– accuracy,tradeoff,by,weakening,the,dilemma,between, greedy,and,non2greedy,actions. ★ We,implement,our,ideas,as,a,value,function,,not,as,a, policy,,because: ★ Value,function,,such,as,expected,value,or,conditional, probability,,is,much,more,portable. ★ Policy,often,needs,many,parameters,and,therefore,requires, parameter@tuning,,and,then,becomes,specific,to,a,certain, problem.,(←,Knowledge,of,the,problem,somewhat,required,a# priori) ★ The,ideas,to,implement,are,based,on,cognitive,properties, from,cognitive,science,with,empirical,supports,from,brain, science.
  81. 81. Three,cognitive,properties★ A.(Satisficing( ★ coined(as(RsatisfyR(+(RsufficeR ★ Simon,(Psy.#Rev.,(1956(★ B.(Risk(aSitude( ★ Kahneman(&(Tversky,(Am.#Psy.,(1984★ C.(Relative(estimation( ★ Tversky(&(Kahneman,(Science,(1974
  82. 82. Irrationality(of(the(three( cognitive(properties★ A.(Satisficing( ★ No(optimization(but(falling(into(a(local(optimum.★ B.(Risk(aSitude( ★ Groundless(introduction(of(asymmetry(between( gain(and(loss.★ C.(Relative(estimation ★ Superstitious(assumption(of(the(value(of(arms( mutually(dependent
  83. 83. Rationality(of(the(three( cognitive(properties★ A.,Satisficing, ★ Not(optimize(but(look(for(and(choose(a(satisfactory( answer(over(a(reference(level,(when(global(optimization( is(intractable. ★ If,only,the,reference,is,properly,set,(just,between,the,best, and,second,best,arm),,satisficing,means,optimization.★ B.,Risk,abitude( ★ Consider(the(reliability(of(information(★ C.,Relative,estimation ★ Evaluate(the(value(of(an(action(in(comparison(with(other( actions
  84. 84. Brain Property$A:$Satisficing Psychology science value reference all arms are over reference value of A1 of A2 No pursuit of arms over the reference level given Kolling et al., Simon, Psy. reference Science, 2012 Rev., 1956 all arms are under reference value value of A1 of A2 Search hard for an arm over the reference level Property$B:$Risk$aZitude$(Reliability$consideration) Risk-avoiding over the reference Risk-seeking under the referenceExpected value 0.75 = 75% reflection effect 25% = 25% Boorman Kahneman et al., & Tversky,win (o) and lose ○×○○○ ×○××× ×○○○○ ○×○○ ○×××× ×○×× Neuron, Am. Psy., (x) in the past ○○○×○ ×××○× 2009 1984 ○○×○× ××○×○ with the boundary of 0.5 comparison considering > < reliability Rely on 15/20 than 3/4. Gamble on 1/4 rather than 5/20. Property$C:$Relative$evaluation Try arms other than A1 by relative value value evaluation (see-saw) Daw et Tversky & if absolute of A1 of A2 if relative al., Kahneman, Nature, Science, Choose A1 and lose 2006 1974 value value value value of A1 of A2 of A1 of A2
  85. 85. Relative,evaluation(is(especially( important★ Relative(evaluation:( ★ is(what(even(slime(molds((粘菌)(and(real(neural(networks( (conservation(of(synaptic(weights)(do.(Behavioral(economics(found( that(humans(comparatively(evaluate(actions(and(states. ★ weakens,the,dilemma,between,exploitation,and,exploration,with, the,see2saw,game,like,competition,among,arms:( ★ Through,failure,(low,reward),,choice,of,greedy,action,may,quickly, trigger,to,the,next,choice,of,the,previously,second,best,,non@greedy,arm. ★ Through,success,(high,reward),,choice,of,greedy,action,may,quickly, trigger,to,focussing,on,the,currently,greedy,action,,lessening,the, possibility,of,choosing,non@greedy,arms,by,decreasing,the,value,of,other, arms. Try arms other than A1 by relative value value evaluation (see-saw) if absolute of A1 of A2 if relative Choose A1 and lose value value value value of A1 of A2 of A1 of A2
  86. 86. The(framework(of(models(of(the( three(properties★ Let(there(only(be(two(arms(A1( and(A2.★ On(the(2x2(contingency(table( Reward of(two(actions(and(two( 1 0 reward(levels(in(the(right,(★ The(expected(reward(value( A1 a b for(each(is A2 c d ★ V(A1)=E(A1)=P(1|A1)=(a/(a+b) ★ V(A2)=E(A2)=P(1|A2)=(c/(c+d)
  87. 87. A(model((RRSR)(of(the(three( properties★ A(value(function(VRS(equipped(with(the( three(properties(can(be(given(as:( ★ VRS(A1)(=((a+d)/(a+d+b+c),( ★ VRS(A2)(=((b+c)/(b+c+a+d). Reward ★ with(the(denominator(identical, 1 0 ((((((((((((((((((((((((((((((((is(simply( argmax V (Ai ) Ai A1 a b the(sign(of((a+d)G(b+c) A2 c d★ This(is(the(RS(heuristics:( ★ [if$(a+d$>$b+c)$then$choose$A1,$else$choose$A2[
  88. 88. RS(heuristics★ Property(C((relative(estimation(of(value): ★ Failing(to(get(reward(with(arm(A2,means(A1(is( relatively,good,(and(vice(versa. ★ The(value(of(A1(and(A2(are(respectively(a+d(and(c+b. Reward 1 0 A1 a b A2 c d VRS(A1) a+d VRS(A2) c+b
  89. 89. RS(heuristics Reward★ Property(B((risk(aSitude) 1 0 ★ Let((a,b,c,d)(=((70,(30,(7,(3). A1 a b ★ V(A1):V(A2)(=(73:37( A2 c d ★ More(reliable((A1)(is(preferred. VRS(A1) a+d ★ Let((a,b,c,d)(=((30,(70,(3,(7). VRS(A2) c+b ★ V(A1):V(A2)(=(37:73( ★ Less(reliable((A2)(is(preferred((since(A2(has(more(chance( of(having(beSer(value(than(30%(of(giving(reward).
  90. 90. RS(heuristics★ Property(A((satisficing) ★ Efficiently(realized(by(property(C(&( B,(with(reference(r,=0.5. Reward ★ If(P(1|A1)(=(P(1|A2)(>(0.5(and(N(A1)( 1 0 >(N(A2)(then(VRS(A1)(>(VRS(A2)(and( keep(choosing(A1,(indifferently. A1 a b ★ When((a,b,c,d)(=((70,(30,(7,(3),(((( A2 c d VRS(A1):VRS(A2)(=(73:37.( VRS(A1) a+d ★ If(P(1|A1)(=(P(1|A2)(<(0.5(and(N(A1)( V (A2) >(N(A2)(then(VRS(A1)(<(VRS(A2)(and( RS c+b try(A2,(wondering(if(P(1|A2)(>(r((0.5). ★ When((a,b,c,d)(=((30,(70,(3,(7),(((( VRS(A1):VRS(A2)(=(37:73.
  91. 91. Result(by(RS 1.0 RS LS CP ToWH0.5L 0.9 SMH0.3L SMH0.7L Accuracy rate 0.8 0.7 0.6 0.5 1 5 10 50 100 500 1000 step★ The(result(shown(is(of(a(2Garmed(bandit( problems((0.6,(0.4)((the(reward(probability(of( A1(and(A2).
  92. 92. The(problem(of(RS★ The,naive,relative,evaluation,of,RS,works,only, with,2,arms.,★ With,n,arms,,RS,is,not,definable,or,any, generalization,doesnNt,work,well.★ So,,we,need,another,model,that,keeps,the,same, high,performance.★ We,introduce,our,LS,model,,first,proposed,by, Shinohara,(2007),—,kind,of,haphazardly. ★ 篠原修二,,田口亮,,桂田浩一,,&,新田恒雄.,(2007).,因果性に基づく 信念形成モデルとN本腕バンディット問題への適用.,人工知能学 会論文誌,,22(1),,58–68.
  93. 93. LS(model★ The(performance(of(LS(in(2G Reward armed(bandit(problems(is(the( 1 0 same(as(RS,(and(LS(can(be( applied(to(nGarmed(bandit( A1 a b problems. A2 c d ★ While(RS(compares(an(arm( with(the(other(arm, a P (1|A1 ) = ★ LS(compares(an(arm(with(the( a+b RgroundR(formed(from(the( whole(set(of(arms. b a+ b+d d★ LS(fits(the(intuition(of(human( LS(1|A1 ) = b a a + b+d d + b + a+c c about(causal(relationship(with( very(high,(actually(the(highest( correlation((r(>(0.85(for(all( RS(1|A1 ) = a+d experiments). a+d+b+c
  94. 94. LS(describing(causal(intuition★ LS,fits,the,experiment,data,of,causal,induction, (inductive,inference,of,causal,relationship),the,best, among,other,42,models,including,the,most,popular, ΔP=P(E|C)–P(E|¬C)., ★ Experiment,of,causal,induction: ★ Given,an,effect,E,in,focus,(e.g.,,stomachache),and,a,candidate, cause,C,(e.g.,,drinking,milk),,answer,the,causal,relationship, from,C,to,E.,The,co@occurrence,information,of,C,and,E,is,given. Meta-analysis effectExperiment AS95 BCC03.1 BCC03.3 H03 H06 LS00 W03.2 W03.6 E ¬E r2 for LS 0.9 0.96 0.96 0.97 0.94 0.73 0.91 0.72 cause C a b r2 for ΔP 0.78 0.84 0.7 0.0 0.5 0.77 0.08 0.21 ¬C c d
  95. 95. The(properties(of(LS#★ Figure–ground(segregation(and(invariance( of(ground(against(change(in(focus((figure). ★ As(the(background(stays(invariant(when(you( see(each(of(the(two(possible(objects,(a(rabbit( or(a(duck,(but(not(both(at(the(same(time. P:,A1≠A2,, b ground a+ b+d d A1 ,,,,A1C≠A2C (A1C)LS(1|A1 ) = a+ b a LS:,A1≠A2,, b+d d +b+ a+c c d ,,,,A1C=A2C c+ d+b bLS(1|A2 ) = ground A2 RS:,A1≠A2,, d c (A2C) c+ d+b b +d+ a+c a ,,,,,A1C=A2,

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