SlideShare a Scribd company logo
1 of 116
Download to read offline
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.(2012
Matsumoto(lab.,(NAIST(松本研(奈良先端科学技術大学院大学
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/
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.
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.
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)
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)
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.
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.
pARIs(part
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
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
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.
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.
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
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
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).
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$index(and(some(other(popular(indices(in(
    mathematics,(statistics(and(machine(learning.
toc
★ New$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$background
New(paradigm(psychology(of(
         reasoning
★ Very(naively(expressed...
  ★ Old(paradigm:
    ★ The(normative(theory(is(the(classical(bivalent(logic(with(
      conditionals(modeled(by(material(implication(P(if(p(then(q)(
      =(P(p(⊃(q)(=(P(¬p(or(q).
    ★ Doesnt(fit(the(data(in(many(areas:(from(this(some(said(
      humans(are(irrational(or(the(intelligence(is(quite(limited.
  ★ New(paradigm:
   ★ Probability(logic(with(P(if(p(then(q)(=(P(q|p)
    ★ de(FineSi(gives(the(appropriate(theory(of(subjective(
      probability.
    ★ Fits(the(data;(human(cognition(is(designed(to(treat(
      uncertainty(by(nature.(It(is(formed(through(evolution.
toc
★ New$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
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
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
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$background
Causal(induction
★ Example:(We(want(to(know(the(cause(of(a(health(problem,(
  right(now(just(from(pure(observation,(no(intervention.
★ I(sometimes(have(stiff(shoulders(and(a(headache.(Whats(
  the(cause?(How(about(coffee?
  ★ a:.(cause=present/effect=present)$
    ★ 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$no(headache(without(coffee?(
Causal(induction(experiment(
Stimulus$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 ¬E
C a       b
¬C c      d
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).
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?
∆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$Heuristics)((HaSori(
         ∆P = P (E|C) − P (E|¬C)
    &(Oaksford(2007) ∆P= ad − bc
   PowerPC =
                    ∆P
      PowerPC − P (E|¬C) (a + b)d
               1=
                    1 − P = ad − bc
  ∆P = P (E|C) − P (E|¬C)
                           (E|¬C)
                              (a + b)(c + d)
                ∆P        ad − bc                 E ¬E
   PowerPC =   ∆P       = ad − bc
 PowerPC = 1 − P (E|¬C) = (a + b)d
                          ad − bc
  ∆P = P (E|C) − P (E|¬C) =
                                               C a    b
             1 − P (E|¬C)         (a + b)d
                              (a + b)(c + d)
                                   a           ¬C c   d
  H=    P (E|C)P (C|E) =
                  ∆P    ∆P (a +ad −+ c)
                               b)(a bc
The(pARIs(rule
★ The(frequency(information(of(rare(instances(
  conveys(more(information(than(abundant(instances(
  (rational$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.
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 c
extreme          coefficient
  lim φ =
  rarity       P (E|C)P (C|E) = H                    d
 d→∞
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
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
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.
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
DataGfit(of(pARIs(and(PowerPC
                                  AS95                                        BCC03exp1generative                                              BCC03exp3                                                       H03
               100                                                      100                                                      100                                                       100

               80                                                        80                                                       80                                                        80
Human 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                                                        80
Human 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
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 otherwise
experiment  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.
correlation
7.00
         0.90    0.85
         0.93    0.95                             0.55
5.13                                                                          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      0.70
3.25 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                0.93
1.38 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
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
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)
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.
Result(of(exp.(1
stim. 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.
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.
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.
Exp.,2:,Result
                         c cell                                   d cell

       13                                       14

       10                                       11

        7                                       7

        3                                       4

        0                                       0

r (=(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.
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.
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
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)
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           d
orange : 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
Exp.(3(Result((coinciding(
               condition)
              coinciding yes/yes                               coinciding yes/no
100                                             100

75                                               75

50                                               50

25                                               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/no
100                                             100

 75                                              75

 50                                              50

 25                                              25

  0                                               0
           Mean                   pARIs                     Mean                  pARIs
                                           49
Exp.(3(Result((contradicting(
               condition)
              contradicting yes/yes                         contradicting yes/no
 100                                             100

  75                                                75

  50                                                50

  25                                                25

   0                                                 0
            Mean                pARIs                     Mean                pARIs


stimuli :          (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
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$bias?
toc
★ New$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$background
Theoretical(background(of(
 biconditional(event(and(pARIs
★ Angelo,Gilio,and,Giuseppe,Sanfilippo,(manuscript,
  under,review),are,studying,biconditional#event,p⟛q#
  (named,by,Andy,Fugard),in,relation,to,quasi<conjunction.
★ Bart,Kosko,(2004),studied,probable$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.
Simulation
                           Correlation of the population is 0.2
0.8"


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 the
0.9"
                                                                              population correlation with a
                                                                                   very small sample
0.8"

0.7"
                                                                                                                                                                                          pARIs"


                                                                                                                                                                                                         HaSori(&(
0.6"
                                                                                                                                                                                          DFH"
0.5"
                                                                                                                                                                                          Delta"P"
0.4"


                 sd value
                                                                                                                                                                                          Phi"
0.3"

0.2"
                                                                                                                                                                                          PowerPC"
                                                                                                                                                                                                       Oaksford,(2007
0.1"

  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"
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
Tversky(index
Psychological 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                                                                                and
The 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,  Mor
represented 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
                                                                                                                          either
imply the contrast model, which expresses the similarity between objects as process of extraction and compilation.
                                                                        prior a
linear 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 the
Several 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
                                                                        voiceless
distinctive 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
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
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.
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)
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 conditional
p q         p|q           q|p          p⟛q           p⊃q
T   T        T             T             T             T
T   F        F             F             F             F
F   T        F             V             F             T
F   F        F             V             V             T
C. Gauffroy, P. Barrouillet / Developmental Review 29 (2009) 249–282




Indicative conditional                                                                                                    Conjunctive
                                                                                                                          Def Bicond
                                                                                                                                            MP
                                                                                                                                            Other




   in development
280                                  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
                                                                                                                                   Grades

NN 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+FF
Weak 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.
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 predomin
NN 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       adults
Weak 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%
                 name
Promises 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       adults
Threats 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”.
‘‘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
                                                                                                                                                       Other



conditionals 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
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 interv
re 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
LS(part
LS(and(pARIs


★ pARIs(almost(coincides(with(LS(under(extreme(
  rarity((lim(d→∞).

       LSR(q|p) = lim LS(q|p) ⇡ pARIs
                 d!1
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)
Dilemma(and(tradeoff
We(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)
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.(
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.
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.
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?
_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.
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    1000



Speed,and,accuracy,
                                                                Steps




  are,usually,not,                                        Step,
    compatible.,                              (the(number(of(choice)
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
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.
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.
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
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.(
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.
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
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
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
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 reference
Expected 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
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
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)
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[
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
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).
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.
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).
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.
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
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                                           effect
Experiment   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
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 b
LS(1|A2 ) =                                ground
                                                  A2      RS:,A1≠A2,,
                    d               c       (A2C)
              c+   d+b b   +d+     a+c a                  ,,,,,A1C=A2,
The(properties(of(LS#
              a                     c
 P (1|A1 ) =           P (1|A2 ) =
             a+b                   c+d
                            b                    ground
                    a+     b+d d            A1
                                                  (A1C)
LS(1|A1 ) =         b               a
              a+   b+d d   +b+     a+c c
                             =                            P:,A1≠A2,,
               = c + d+b b
                      d
                                     =     ground
                                                          ,,,,A1C≠A2C
LS(1|A2 ) =         d               c       (A2C)
                                                  A2
                                                          LS:,A1≠A2,,
              c+   d+b b   +d+     a+c a
                                                          ,,,,A1C=A2C
                              a + d≠
               RS(1|A1 ) =                                RS:,A1≠A2,,
                           a+d+b+c                        ,,,,,A1C=A2,
                            ≠ c+b    ≠
               RS(1|A2 ) =                                ,,,,,A2C=A1
                           c+b+d+a
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?

More Related Content

What's hot (20)

Econometrics, PhD Course, #1 Nonlinearities
Econometrics, PhD Course, #1 NonlinearitiesEconometrics, PhD Course, #1 Nonlinearities
Econometrics, PhD Course, #1 Nonlinearities
 
Slides sales-forecasting-session2-web
Slides sales-forecasting-session2-webSlides sales-forecasting-session2-web
Slides sales-forecasting-session2-web
 
Introduction to modern Variational Inference.
Introduction to modern Variational Inference.Introduction to modern Variational Inference.
Introduction to modern Variational Inference.
 
Lundi 16h15-copules-charpentier
Lundi 16h15-copules-charpentierLundi 16h15-copules-charpentier
Lundi 16h15-copules-charpentier
 
Slides lln-risques
Slides lln-risquesSlides lln-risques
Slides lln-risques
 
Slides lausanne-2013-v2
Slides lausanne-2013-v2Slides lausanne-2013-v2
Slides lausanne-2013-v2
 
File 4 a k-tung 2 radial na printed 12-11
File 4 a k-tung 2 radial na printed 12-11File 4 a k-tung 2 radial na printed 12-11
File 4 a k-tung 2 radial na printed 12-11
 
Slides amsterdam-2013
Slides amsterdam-2013Slides amsterdam-2013
Slides amsterdam-2013
 
Slides edf-1
Slides edf-1Slides edf-1
Slides edf-1
 
Slides ihp
Slides ihpSlides ihp
Slides ihp
 
Side 2019 #3
Side 2019 #3Side 2019 #3
Side 2019 #3
 
Slides univ-van-amsterdam
Slides univ-van-amsterdamSlides univ-van-amsterdam
Slides univ-van-amsterdam
 
Inverse-Trigonometric-Functions.pdf
Inverse-Trigonometric-Functions.pdfInverse-Trigonometric-Functions.pdf
Inverse-Trigonometric-Functions.pdf
 
Reinforcement Learning in Economics and Finance
Reinforcement Learning in Economics and FinanceReinforcement Learning in Economics and Finance
Reinforcement Learning in Economics and Finance
 
transformations and nonparametric inference
transformations and nonparametric inferencetransformations and nonparametric inference
transformations and nonparametric inference
 
pres_coconat
pres_coconatpres_coconat
pres_coconat
 
Slides simplexe
Slides simplexeSlides simplexe
Slides simplexe
 
Convergence of ABC methods
Convergence of ABC methodsConvergence of ABC methods
Convergence of ABC methods
 
Sildes buenos aires
Sildes buenos airesSildes buenos aires
Sildes buenos aires
 
Lecture1
Lecture1Lecture1
Lecture1
 

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

09 - 27 Jan - Recursion Part 1
09 - 27 Jan - Recursion Part 109 - 27 Jan - Recursion Part 1
09 - 27 Jan - Recursion Part 1Neeldhara Misra
 
Logic and proof
Logic and proofLogic and proof
Logic and proofSuresh Ram
 
Assignement of discrete mathematics
Assignement of discrete mathematicsAssignement of discrete mathematics
Assignement of discrete mathematicsSyed Umair
 
Assignement of discrete mathematics
Assignement of discrete mathematicsAssignement of discrete mathematics
Assignement of discrete mathematicsSyed Umair
 

Similar to How do cognitive agents handle the tradeoff between speed and accuracy? (6)

09 - 27 Jan - Recursion Part 1
09 - 27 Jan - Recursion Part 109 - 27 Jan - Recursion Part 1
09 - 27 Jan - Recursion Part 1
 
Boolean algebra
Boolean algebraBoolean algebra
Boolean algebra
 
4. symbolic logic
4. symbolic logic4. symbolic logic
4. symbolic logic
 
Logic and proof
Logic and proofLogic and proof
Logic and proof
 
Assignement of discrete mathematics
Assignement of discrete mathematicsAssignement of discrete mathematics
Assignement of discrete mathematics
 
Assignement of discrete mathematics
Assignement of discrete mathematicsAssignement of discrete mathematics
Assignement of discrete mathematics
 

More from Tatsuji Takahashi

Tenenbaum review-20140704-1600
Tenenbaum review-20140704-1600Tenenbaum review-20140704-1600
Tenenbaum review-20140704-1600Tatsuji Takahashi
 
2014 02 feb-18-tue-hri-jp-高橋達二-認知バイアスと双条件付確率
2014 02 feb-18-tue-hri-jp-高橋達二-認知バイアスと双条件付確率2014 02 feb-18-tue-hri-jp-高橋達二-認知バイアスと双条件付確率
2014 02 feb-18-tue-hri-jp-高橋達二-認知バイアスと双条件付確率Tatsuji Takahashi
 
A toy model of human cognition: Utilizing fluctuation in uncertain and non-s...
A toy model of  human cognition: Utilizing fluctuation in uncertain and non-s...A toy model of  human cognition: Utilizing fluctuation in uncertain and non-s...
A toy model of human cognition: Utilizing fluctuation in uncertain and non-s...Tatsuji Takahashi
 
内部観測研究室説明スライド (2013年07月27日)
内部観測研究室説明スライド (2013年07月27日)内部観測研究室説明スライド (2013年07月27日)
内部観測研究室説明スライド (2013年07月27日)Tatsuji Takahashi
 
不確実性の下での論理と「双条件付確率」の導入
不確実性の下での論理と「双条件付確率」の導入不確実性の下での論理と「双条件付確率」の導入
不確実性の下での論理と「双条件付確率」の導入Tatsuji Takahashi
 
Jpa2012 david-over-2012-09 sep-13-tatsujit-jp
Jpa2012 david-over-2012-09 sep-13-tatsujit-jpJpa2012 david-over-2012-09 sep-13-tatsujit-jp
Jpa2012 david-over-2012-09 sep-13-tatsujit-jpTatsuji Takahashi
 

More from Tatsuji Takahashi (6)

Tenenbaum review-20140704-1600
Tenenbaum review-20140704-1600Tenenbaum review-20140704-1600
Tenenbaum review-20140704-1600
 
2014 02 feb-18-tue-hri-jp-高橋達二-認知バイアスと双条件付確率
2014 02 feb-18-tue-hri-jp-高橋達二-認知バイアスと双条件付確率2014 02 feb-18-tue-hri-jp-高橋達二-認知バイアスと双条件付確率
2014 02 feb-18-tue-hri-jp-高橋達二-認知バイアスと双条件付確率
 
A toy model of human cognition: Utilizing fluctuation in uncertain and non-s...
A toy model of  human cognition: Utilizing fluctuation in uncertain and non-s...A toy model of  human cognition: Utilizing fluctuation in uncertain and non-s...
A toy model of human cognition: Utilizing fluctuation in uncertain and non-s...
 
内部観測研究室説明スライド (2013年07月27日)
内部観測研究室説明スライド (2013年07月27日)内部観測研究室説明スライド (2013年07月27日)
内部観測研究室説明スライド (2013年07月27日)
 
不確実性の下での論理と「双条件付確率」の導入
不確実性の下での論理と「双条件付確率」の導入不確実性の下での論理と「双条件付確率」の導入
不確実性の下での論理と「双条件付確率」の導入
 
Jpa2012 david-over-2012-09 sep-13-tatsujit-jp
Jpa2012 david-over-2012-09 sep-13-tatsujit-jpJpa2012 david-over-2012-09 sep-13-tatsujit-jp
Jpa2012 david-over-2012-09 sep-13-tatsujit-jp
 

Recently uploaded

Diploma in Nursing Admission Test Question Solution 2023.pdf
Diploma in Nursing Admission Test Question Solution 2023.pdfDiploma in Nursing Admission Test Question Solution 2023.pdf
Diploma in Nursing Admission Test Question Solution 2023.pdfMohonDas
 
P4C x ELT = P4ELT: Its Theoretical Background (Kanazawa, 2024 March).pdf
P4C x ELT = P4ELT: Its Theoretical Background (Kanazawa, 2024 March).pdfP4C x ELT = P4ELT: Its Theoretical Background (Kanazawa, 2024 March).pdf
P4C x ELT = P4ELT: Its Theoretical Background (Kanazawa, 2024 March).pdfYu Kanazawa / Osaka University
 
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRA
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRADUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRA
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRATanmoy Mishra
 
Presentation on the Basics of Writing. Writing a Paragraph
Presentation on the Basics of Writing. Writing a ParagraphPresentation on the Basics of Writing. Writing a Paragraph
Presentation on the Basics of Writing. Writing a ParagraphNetziValdelomar1
 
How to Add a New Field in Existing Kanban View in Odoo 17
How to Add a New Field in Existing Kanban View in Odoo 17How to Add a New Field in Existing Kanban View in Odoo 17
How to Add a New Field in Existing Kanban View in Odoo 17Celine George
 
The Singapore Teaching Practice document
The Singapore Teaching Practice documentThe Singapore Teaching Practice document
The Singapore Teaching Practice documentXsasf Sfdfasd
 
Clinical Pharmacy Introduction to Clinical Pharmacy, Concept of clinical pptx
Clinical Pharmacy  Introduction to Clinical Pharmacy, Concept of clinical pptxClinical Pharmacy  Introduction to Clinical Pharmacy, Concept of clinical pptx
Clinical Pharmacy Introduction to Clinical Pharmacy, Concept of clinical pptxraviapr7
 
Patterns of Written Texts Across Disciplines.pptx
Patterns of Written Texts Across Disciplines.pptxPatterns of Written Texts Across Disciplines.pptx
Patterns of Written Texts Across Disciplines.pptxMYDA ANGELICA SUAN
 
How to Manage Cross-Selling in Odoo 17 Sales
How to Manage Cross-Selling in Odoo 17 SalesHow to Manage Cross-Selling in Odoo 17 Sales
How to Manage Cross-Selling in Odoo 17 SalesCeline George
 
How to Show Error_Warning Messages in Odoo 17
How to Show Error_Warning Messages in Odoo 17How to Show Error_Warning Messages in Odoo 17
How to Show Error_Warning Messages in Odoo 17Celine George
 
How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17Celine George
 
HED Office Sohayok Exam Question Solution 2023.pdf
HED Office Sohayok Exam Question Solution 2023.pdfHED Office Sohayok Exam Question Solution 2023.pdf
HED Office Sohayok Exam Question Solution 2023.pdfMohonDas
 
The basics of sentences session 10pptx.pptx
The basics of sentences session 10pptx.pptxThe basics of sentences session 10pptx.pptx
The basics of sentences session 10pptx.pptxheathfieldcps1
 
How to Solve Singleton Error in the Odoo 17
How to Solve Singleton Error in the  Odoo 17How to Solve Singleton Error in the  Odoo 17
How to Solve Singleton Error in the Odoo 17Celine George
 
Practical Research 1 Lesson 9 Scope and delimitation.pptx
Practical Research 1 Lesson 9 Scope and delimitation.pptxPractical Research 1 Lesson 9 Scope and delimitation.pptx
Practical Research 1 Lesson 9 Scope and delimitation.pptxKatherine Villaluna
 
Patient Counselling. Definition of patient counseling; steps involved in pati...
Patient Counselling. Definition of patient counseling; steps involved in pati...Patient Counselling. Definition of patient counseling; steps involved in pati...
Patient Counselling. Definition of patient counseling; steps involved in pati...raviapr7
 
M-2- General Reactions of amino acids.pptx
M-2- General Reactions of amino acids.pptxM-2- General Reactions of amino acids.pptx
M-2- General Reactions of amino acids.pptxDr. Santhosh Kumar. N
 
Education and training program in the hospital APR.pptx
Education and training program in the hospital APR.pptxEducation and training program in the hospital APR.pptx
Education and training program in the hospital APR.pptxraviapr7
 
UKCGE Parental Leave Discussion March 2024
UKCGE Parental Leave Discussion March 2024UKCGE Parental Leave Discussion March 2024
UKCGE Parental Leave Discussion March 2024UKCGE
 

Recently uploaded (20)

Diploma in Nursing Admission Test Question Solution 2023.pdf
Diploma in Nursing Admission Test Question Solution 2023.pdfDiploma in Nursing Admission Test Question Solution 2023.pdf
Diploma in Nursing Admission Test Question Solution 2023.pdf
 
P4C x ELT = P4ELT: Its Theoretical Background (Kanazawa, 2024 March).pdf
P4C x ELT = P4ELT: Its Theoretical Background (Kanazawa, 2024 March).pdfP4C x ELT = P4ELT: Its Theoretical Background (Kanazawa, 2024 March).pdf
P4C x ELT = P4ELT: Its Theoretical Background (Kanazawa, 2024 March).pdf
 
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRA
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRADUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRA
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRA
 
Presentation on the Basics of Writing. Writing a Paragraph
Presentation on the Basics of Writing. Writing a ParagraphPresentation on the Basics of Writing. Writing a Paragraph
Presentation on the Basics of Writing. Writing a Paragraph
 
How to Add a New Field in Existing Kanban View in Odoo 17
How to Add a New Field in Existing Kanban View in Odoo 17How to Add a New Field in Existing Kanban View in Odoo 17
How to Add a New Field in Existing Kanban View in Odoo 17
 
The Singapore Teaching Practice document
The Singapore Teaching Practice documentThe Singapore Teaching Practice document
The Singapore Teaching Practice document
 
Clinical Pharmacy Introduction to Clinical Pharmacy, Concept of clinical pptx
Clinical Pharmacy  Introduction to Clinical Pharmacy, Concept of clinical pptxClinical Pharmacy  Introduction to Clinical Pharmacy, Concept of clinical pptx
Clinical Pharmacy Introduction to Clinical Pharmacy, Concept of clinical pptx
 
Patterns of Written Texts Across Disciplines.pptx
Patterns of Written Texts Across Disciplines.pptxPatterns of Written Texts Across Disciplines.pptx
Patterns of Written Texts Across Disciplines.pptx
 
How to Manage Cross-Selling in Odoo 17 Sales
How to Manage Cross-Selling in Odoo 17 SalesHow to Manage Cross-Selling in Odoo 17 Sales
How to Manage Cross-Selling in Odoo 17 Sales
 
Personal Resilience in Project Management 2 - TV Edit 1a.pdf
Personal Resilience in Project Management 2 - TV Edit 1a.pdfPersonal Resilience in Project Management 2 - TV Edit 1a.pdf
Personal Resilience in Project Management 2 - TV Edit 1a.pdf
 
How to Show Error_Warning Messages in Odoo 17
How to Show Error_Warning Messages in Odoo 17How to Show Error_Warning Messages in Odoo 17
How to Show Error_Warning Messages in Odoo 17
 
How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17
 
HED Office Sohayok Exam Question Solution 2023.pdf
HED Office Sohayok Exam Question Solution 2023.pdfHED Office Sohayok Exam Question Solution 2023.pdf
HED Office Sohayok Exam Question Solution 2023.pdf
 
The basics of sentences session 10pptx.pptx
The basics of sentences session 10pptx.pptxThe basics of sentences session 10pptx.pptx
The basics of sentences session 10pptx.pptx
 
How to Solve Singleton Error in the Odoo 17
How to Solve Singleton Error in the  Odoo 17How to Solve Singleton Error in the  Odoo 17
How to Solve Singleton Error in the Odoo 17
 
Practical Research 1 Lesson 9 Scope and delimitation.pptx
Practical Research 1 Lesson 9 Scope and delimitation.pptxPractical Research 1 Lesson 9 Scope and delimitation.pptx
Practical Research 1 Lesson 9 Scope and delimitation.pptx
 
Patient Counselling. Definition of patient counseling; steps involved in pati...
Patient Counselling. Definition of patient counseling; steps involved in pati...Patient Counselling. Definition of patient counseling; steps involved in pati...
Patient Counselling. Definition of patient counseling; steps involved in pati...
 
M-2- General Reactions of amino acids.pptx
M-2- General Reactions of amino acids.pptxM-2- General Reactions of amino acids.pptx
M-2- General Reactions of amino acids.pptx
 
Education and training program in the hospital APR.pptx
Education and training program in the hospital APR.pptxEducation and training program in the hospital APR.pptx
Education and training program in the hospital APR.pptx
 
UKCGE Parental Leave Discussion March 2024
UKCGE Parental Leave Discussion March 2024UKCGE Parental Leave Discussion March 2024
UKCGE Parental Leave Discussion March 2024
 

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

  • 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.(2012 Matsumoto(lab.,(NAIST(松本研(奈良先端科学技術大学院大学
  • 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. 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$paradox. ★ To(this(end,(we(treat(the(frame(problem(and(selfG referential(paradoxes(as(empirical(as(possible ★ in(cognitive(psychology,(machine(learning(and( robotics;(not(in(philosophy(itself.
  • 4. The(problem( ★ How$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. 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. 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. 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. 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.
  • 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. 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. 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. 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. 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. 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. 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. 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$index(and(some(other(popular(indices(in( mathematics,(statistics(and(machine(learning.
  • 18. toc ★ New$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$background
  • 19. New(paradigm(psychology(of( reasoning ★ Very(naively(expressed... ★ Old(paradigm: ★ The(normative(theory(is(the(classical(bivalent(logic(with( conditionals(modeled(by(material(implication(P(if(p(then(q)( =(P(p(⊃(q)(=(P(¬p(or(q). ★ Doesnt(fit(the(data(in(many(areas:(from(this(some(said( humans(are(irrational(or(the(intelligence(is(quite(limited. ★ New(paradigm: ★ Probability(logic(with(P(if(p(then(q)(=(P(q|p) ★ de(FineSi(gives(the(appropriate(theory(of(subjective( probability. ★ Fits(the(data;(human(cognition(is(designed(to(treat( uncertainty(by(nature.(It(is(formed(through(evolution.
  • 20. toc ★ New$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$background
  • 21. defective conditional and defective biconditional ★ Defective truth table in the older paradigms ★ (Wason, 1966; Johnson-Laird and Tagart, 1969; Wason and Johnson-Laird, 1972; Evans et al., 1993) ★ is normative and coherent in the new paradigm old(paradigm new(paradigm defective( conditional( → conditional event(q|p defective( biconditional( → biconditional 21 event(p⟛q
  • 22. de(FineSis(conditional$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$Equation) ★ ¬p(cases(are(neglected,(and(Rq|pR(is(itself(a((conditional)(event. de Finetti material conditional conditional biconditional conditional event event event p q p⊃q q|p p|q p⟛q T T T T T conjunction T T F F F V F F T T V F F V: void case F F T V V V
  • 23. toc ★ New$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$background
  • 24. Causal(induction ★ Example:(We(want(to(know(the(cause(of(a(health(problem,( right(now(just(from(pure(observation,(no(intervention. ★ I(sometimes(have(stiff(shoulders(and(a(headache.(Whats( the(cause?(How(about(coffee? ★ a:.(cause=present/effect=present)$ ★ 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$no(headache(without(coffee?(
  • 25. Causal(induction(experiment( Stimulus$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 ¬E C a b ¬C c d
  • 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. 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. ∆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$Heuristics)((HaSori( ∆P = P (E|C) − P (E|¬C) &(Oaksford(2007) ∆P= ad − bc PowerPC = ∆P PowerPC − P (E|¬C) (a + b)d 1= 1 − P = ad − bc ∆P = P (E|C) − P (E|¬C) (E|¬C) (a + b)(c + d) ∆P ad − bc E ¬E PowerPC = ∆P = ad − bc PowerPC = 1 − P (E|¬C) = (a + b)d ad − bc ∆P = P (E|C) − P (E|¬C) = C a b 1 − P (E|¬C) (a + b)d (a + b)(c + d) a ¬C c d H= P (E|C)P (C|E) = ∆P ∆P (a +ad −+ c) b)(a bc
  • 29. The(pARIs(rule ★ The(frequency(information(of(rare(instances( conveys(more(information(than(abundant(instances( (rational$analysis(and(rarity$assumption,(see(esp.( McKenzie(2007). ★ Because(of(the(frame(problemGlike(aspect,(the(dGcell( information(can(be(unreliable((depends(strongly(on( how(we(frame(and(count). ★ Hence(we(calculate(the(causal(intensity(only(by(the( proportion(of(assumedGtoGbe(rare(instances((pARIs) ★ named(after(pCI:.proportion.of.confirmatory. instances,(White(2003.
  • 30. Rarity(assumption H=P (E|C)P (C|E) ★ We(assume(the(effect(in(focus(and(the(candidate( = cause(to(be(rare:(P(C)(and(P(E)(to(be(small. P (E|C)P (C|E) = a ★ Originally(in(Oaksford(&(Chater,(1994,( (a + b)(a + c) ★ then(in(HaSori(&(Oaksford,(2007,(McKenzie(2007,( a in(the(study(of(causal(induction = P (E|C)P (C|E) = (a + b)(a + c) ★ C(and(E(to(take(small(proportion(in(U. U lim φ = d→∞ P (E|C)P (C|E) = H C E ϕ: correlation ba c extreme coefficient lim φ = rarity P (E|C)P (C|E) = H d d→∞
  • 31. The(pARIs(rule ★ C(and(E(are(both(generally(assumed(to(be(rare((P(C)(and(P(E)(low). ★ pARIs(=(proportion(of(assumedGtoGbe(rare(instances((a,#b,#and(c).( pARIs = P(p⟛q) = a / (a+b+c) U E -E C E C a b ba c d -C c d conditional event biconditional event infering causal intensity C E E|C C⟛E pARIs T T T T positive T F F F negative F T V F negative F F V V irrelevant
  • 32. The(pARIs(rule ★ C(and(E(are(both(assumed(to(be(rare((P(C)(and(P(E)(low) ★ pARIs(=(proportion(of(assumedGtoGbe(rare(instances((a,#b,#and(c).( ★ The(probability(of(the(conjunction(of(cause(and(effect(given(the( disjunction(of(cause(and(effect((conditioned(on(the(disjunction).( pARIs = P(C iff E) = P(C and E | C or E) P(C and E) a = = P(C or E) a+b+c E -E U C E C a b ba c d -C c d
  • 33. Why(ignore(the(dGcell? ★ Hempels(paradox ★ All(ravens(are(black.( ★ =(If(something(is(a(raven,(then(it(is(black. ★ Is$a$non:black$non:raven$confirmatory? ★ If(a(nonGraven(that(is(not(black(is(rare,(it(is( informative(hence(not(ignored.((McKenzie(&( Mikkelsen,(2000) ★ If(Raven:nonGraven(=(5:5(and(black/nonGblack(=(5:5: ★ RAll(men(are(stupid(than(the(average(of(human( beings.R((RIf(one(is(a(man,(then.he(is(relatively(stupid.R) ★ A(thoughtful(woman(can(be(confirmatory.
  • 34. toc ★ New$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. DataGfit(of(pARIs(and(PowerPC AS95 BCC03exp1generative BCC03exp3 H03 100 100 100 100 80 80 80 80 Human 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 80 Human 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. 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 otherwise experiment 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. correlation 7.00 0.90 0.85 0.93 0.95 0.55 5.13 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 0.70 3.25 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 0.93 1.38 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. 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. 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. 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. Result(of(exp.(1 stim. 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. 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. 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. Exp.,2:,Result c cell d cell 13 14 10 11 7 7 3 4 0 0 r (=(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. 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. 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. 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. 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 d orange : 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. Exp.(3(Result((coinciding( condition) coinciding yes/yes coinciding yes/no 100 100 75 75 50 50 25 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/no 100 100 75 75 50 50 25 25 0 0 Mean pARIs Mean pARIs 49
  • 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 pARIs stimuli : (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. 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$bias?
  • 52. toc ★ New$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$background
  • 53. Theoretical(background(of( biconditional(event(and(pARIs ★ Angelo,Gilio,and,Giuseppe,Sanfilippo,(manuscript, under,review),are,studying,biconditional#event,p⟛q# (named,by,Andy,Fugard),in,relation,to,quasi<conjunction. ★ Bart,Kosko,(2004),studied,probable$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. Simulation Correlation of the population is 0.2 0.8" 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 the 0.9" population correlation with a very small sample 0.8" 0.7" pARIs" HaSori(&( 0.6" DFH" 0.5" Delta"P" 0.4" sd value Phi" 0.3" 0.2" PowerPC" Oaksford,(2007 0.1" 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. 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. Tversky(index Psychological 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 and The 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, Mor represented 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 either imply the contrast model, which expresses the similarity between objects as process of extraction and compilation. prior a linear 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 the Several 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 voiceless distinctive 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. 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. 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. 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. 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 conditional p q p|q q|p p⟛q p⊃q T T T T T T T F F F F F F T F V F T F F F V V T
  • 61. C. Gauffroy, P. Barrouillet / Developmental Review 29 (2009) 249–282 Indicative conditional Conjunctive Def Bicond MP Other in development 280 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 Grades NN 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+FF Weak 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. 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 predomin NN 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 adults Weak 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% name Promises 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 adults Threats 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. ‘‘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 Other conditionals 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. 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 interv re 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
  • 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. 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. Dilemma(and(tradeoff We(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. 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. 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. 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. 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. _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. 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 1000 Speed,and,accuracy, Steps are,usually,not, Step, compatible., (the(number(of(choice)
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 reference Expected 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$evaluation Try arms other than A1 by relative value value evaluation (see-saw) Daw et Tversky & if absolute of A1 of A2 if relative al., Kahneman, Nature, Science, Choose A1 and lose 2006 1974 value value value value of A1 of A2 of A1 of A2
  • 85. Relative,evaluation(is(especially( important ★ Relative(evaluation:( ★ is(what(even(slime(molds((粘菌)(and(real(neural(networks( (conservation(of(synaptic(weights)(do.(Behavioral(economics(found( that(humans(comparatively(evaluate(actions(and(states. ★ weakens,the,dilemma,between,exploitation,and,exploration,with, the,see2saw,game,like,competition,among,arms:( ★ Through,failure,(low,reward),,choice,of,greedy,action,may,quickly, trigger,to,the,next,choice,of,the,previously,second,best,,non@greedy,arm. ★ Through,success,(high,reward),,choice,of,greedy,action,may,quickly, trigger,to,focussing,on,the,currently,greedy,action,,lessening,the, possibility,of,choosing,non@greedy,arms,by,decreasing,the,value,of,other, arms. Try arms other than A1 by relative value value evaluation (see-saw) if absolute of A1 of A2 if relative Choose A1 and lose value value value value of A1 of A2 of A1 of A2
  • 86. The(framework(of(models(of(the( three(properties ★ Let(there(only(be(two(arms(A1( and(A2. ★ On(the(2x2(contingency(table( Reward of(two(actions(and(two( 1 0 reward(levels(in(the(right,( ★ The(expected(reward(value( A1 a b for(each(is A2 c d ★ V(A1)=E(A1)=P(1|A1)=(a/(a+b) ★ V(A2)=E(A2)=P(1|A2)=(c/(c+d)
  • 87. A(model((RRSR)(of(the(three( properties ★ A(value(function(VRS(equipped(with(the( three(properties(can(be(given(as:( ★ VRS(A1)(=((a+d)/(a+d+b+c),( ★ VRS(A2)(=((b+c)/(b+c+a+d). Reward ★ with(the(denominator(identical, 1 0 ((((((((((((((((((((((((((((((((is(simply( argmax V (Ai ) Ai A1 a b the(sign(of((a+d)G(b+c) A2 c d ★ This(is(the(RS(heuristics:( ★ [if$(a+d$>$b+c)$then$choose$A1,$else$choose$A2[
  • 88. 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. 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. 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. 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. 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. 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. 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 effect Experiment 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. 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 b LS(1|A2 ) = ground A2 RS:,A1≠A2,, d c (A2C) c+ d+b b +d+ a+c a ,,,,,A1C=A2,
  • 96. The(properties(of(LS# a c P (1|A1 ) = P (1|A2 ) = a+b c+d b ground a+ b+d d A1 (A1C) LS(1|A1 ) = b a a+ b+d d +b+ a+c c = P:,A1≠A2,, = c + d+b b d = ground ,,,,A1C≠A2C LS(1|A2 ) = d c (A2C) A2 LS:,A1≠A2,, c+ d+b b +d+ a+c a ,,,,A1C=A2C a + d≠ RS(1|A1 ) = RS:,A1≠A2,, a+d+b+c ,,,,,A1C=A2, ≠ c+b ≠ RS(1|A2 ) = ,,,,,A2C=A1 c+b+d+a