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  • Whatdoesfrequentistmean?GlennShafer

    WorkshoponFusionLearning,BFFinferencesandStatisticalFoundationsApril11,2016

    DepartmentofStatisticsRutgersUniversity NewBrunswick

    TerminventedbyphilosopherErnestNagelin1936.FirstusedinstatisticsbyMauriceKendallin1949.EmbracedbyJerzyNeyman andothernonBayesiansinthe1960s.Philosophersandstatisticians useitdifferently.

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  • FREQUENTISM=

    frequencyinterpretationofprobability

    2

    Part1.Philosophers Probability=relativefrequency Distinctfrompropensityinterpretation Nave

    Part2.Statisticians SOMEprobabilitiesarerelativefrequencies. Probabilitymodelsmustbetestablebyobservations. TestusingCournots principle.

    Part3.Generalizingtogametheoreticprobability ThegametheoreticCournot principle Implicationsforconvergenceoffrequencies ClassicalCournot principleasspecialcase

  • Inventionofthewords

    Frequentist andfrequentism notwidelyuseduntil1960s.

    SofarasIknow,frequentist was firstusedbyphilosopherErnestNagel1936, thenbystatisticianMauriceKendall1949.

    ThephilosopherDonaldWilliamsusedfrequentism in1945.

    ErnestNagel,TheMeaningofProbability,JournaloftheAmericanStatisticalAssociation31(193):1030,1936.(SeealsoErnestNagel,PrinciplesoftheTheoryofProbability,UniversityofChicagoPress,1939.)

    MauriceG.Kendall,OntheReconciliationofTheoriesofProbability,Biometrika 36(1/2):101116,1949.DonaldWilliams,TheChallengingSituationinthePhilosophyofProbability,PhilosophyandPhenomenologicalResearch 6(1):6786.

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  • FREQUENTISM

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    PartI.Whatitmeanstophilosophers

    Probability=relativefrequency Distinctfrompropensityinterpretation Nave

    ErnestNagel19011985

    EmigratedtoUSfromSlovakiaatage10.SpentcareeratColumbia.

    Coinedfrequentist.

  • Part1.Whatfrequentismmeanstophilosophers

    5

    OxfordEnglishDictionary

    Frequentist:Onewhobelievesthattheprobabilityofaneventshouldbedefinedasthelimitofitsrelativefrequencyinalargenumberoftrials.

    ThisdefinitionwasgivenbyNagelin1936andhasbeenusedbyphilosopherssincethen.

  • Part1.Whatfrequentismmeanstophilosophers

    6

    NagelanticipatedbyJohnVenn(18341923).

    wemaydefinetheprobabilityoftheeventhappeningthatparticularwayasthenumericalfractionwhichrepresentstheproportionbetweenthetwodifferentclassesinthelongrun.

    1886editionofTheLogicofChance (page163)

  • Part1.Whatfrequentismmeanstophilosophers

    Nagellistedthreeinterpretationsofprobability:

    1. Belief(e.g.,DeMorgan)2. Logical(e.g.,Keynes)3. Frequency

    Savage(1954):

    1. Personalistic2. Necessary3. Objectivistic

    Inmathematicalstatisticsin1954,objectivisticviewsweredominant,butfrequentistwasnotyetwidelyused.

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  • Part1.Whatfrequentismmeanstophilosophers

    Nagelonthefrequencyinterpretation:

    alreadyimplicitinAristotle,buthasbecomeprominentonlywithinthelastcenturyasaconsequenceofapplyingtheprobabilitycalculustostatisticsandphysics.Itscentralideaisthatbytheprobabilityofapropositionoran"event"ismeanttherelativefrequencyofthe"event"inanindefiniteclassofevents.

    onthisinterpretationeverystatementinvolvingprobabilitiesisamaterialpropositionwhosetruthorfalsityistobediscoveredbyexaminingobjectiverelativefrequencies.

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  • Part1.Whatfrequentismmeanstophilosophers

    IntheStanfordEncyclopediaofPhilosophy,AlanHjek listssixinterpretationsofprobability:1. Classical(Bernoulli,Laplace)2. Logical(Johnson,Keynes,Jeffreys,Carnap)3. Subjective(Ramsey,deFinetti,4. Frequency(Venn,Reichenbach,vonMises)5. Propensity(Peirce,Popper)6. Bestsystem(DavidLewis)

    In2005,MariaCarlaGalavotti listedthefirst5.

    In2009,Hjek gave15argumentsagainstfrequentism.

    PhilosophicalIntroductiontoProbability,byMariaCarlaGalavotti,2005."MisesRedux" Redux:FifteenArgumentsagainstFiniteFrequentism,byAlanHjek,Erkenntnis 45(2/3):209227,1996

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  • Part1.Whatfrequentismmeanstophilosophers

    Philosophersconsiderfrequentismhopelesslynave.

    Manystatisticianscallthemselvesfrequentists.

    Hypothesis:Thetwogroupsdontreallyunderstandthewordinthesameway.

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  • FREQUENTISM

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    Part2.Whatitmeanstostatisticians SOMEprobabilitiesarerelativefrequencies. Probabilitymodelsmustbetestablebyobservations. TestusingCournots principle.

    MauriceG.Kendall19071983Britishstatistician

    Apparentlyfirststatisticiantousefrequentist.

    In1949,hesoughtreconciliationwithsubjectivists.

    OntheReconciliationofTheoriesofProbability,"Biometrika, 36:104,1949.

  • Part2.Whatfrequentismmeanstostatisticians

    StatisticiansproblemwithfrequentismasdefinedbyNagel:

    BernoullisTheorem: Inalargenumberofindependenttrialsofaneventwithprobabilityp,Probability(relativefrequencyp) 1.

    Therearetwoprobabilitiesinthetheorem.Aretheybothfrequencies?

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  • Part2.Whatfrequentismmeanstostatisticians

    BernoullisTheorem(1713): Inalargenumberofindependenttrialsofaneventwithprobabilityp,

    Probability(relativefrequencyp)1.

    Borels StrongLawofLargeNumbers(1909): Inaninfinitesequenceofindependenttrialsofaneventwithprobabilityp,

    Probability(relativefrequencyp)=1.

    Eachstatementinvolvestwoprobabilities.

    1. Ifprobability=relativefrequency bydefinition,whydoyouneedthetheorem?

    2. Howdoyoumakesenseofthesecondprobability?(Infiniteregress?)

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  • Part2.Whatfrequentismmeanstostatisticians

    JohnVennclearedthedeckbyrejectingBernoullistheorem.

    ifthereisanyvalueintheforegoingcriticism,thebasisonwhichthemathematicsrestisfaulty,owingtotherebeingreallynothingwhichwecanwithproprietycallanobjectiveprobability.

    TheLogicofChance,1886,page91

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    JohnVenn18341923

  • Part2.Whatfrequentismmeanstostatisticians

    Vennsattitudewasviableinhistime,whenmathematicalprobabilityhadfallenintolowrepute.

    EventheFrench(Bertrand,Poincar)hadlostinterestintheclassicallimittheorems(Bernoulli,DeMoivre,Poisson).

    Statisticianswereexploitingfloodsofdatawithoutprobabilitytheory.(Kollektivmasslehre)

    TheBritishmostlytreatedprobabilityasasourceoftoyproblems(geometricprobability)orsyllogisms.

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  • Part2.Whatfrequentismmeanstostatisticians

    Butobjectiveprobabilitywasmorethanfrequency

    in classicalprobabilityasrevivedonthecontinent(Chebyshev,Markoc,Cantelli,Lvy,Kolmogorov),and

    inmathematicalstatisticsafteritabsorbedcontinentalprobability(Cramer,Neyman)

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  • Part2.Whatfrequentismmeanstostatisticians

    BernoullisTheorem(1713): Inalargenumberofindependenttrialsofaneventwithprobabilityp,

    Probability(relativefrequencyp)1.

    Tomakesenseofthesecondprobability:Interpretaprobabilityclosetoone,singledoutinadvance,notasafrequencybutaspracticalcertainty.

    Bernoullibroughtthisnotionofpracticalcertaintyintomathematicalprobability.

    AntoineAugustinCournot (18011877)addedthatitistheonlytoconnectprobabilitywithphenomena.

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  • Part2.Whatfrequentismmeanstostatisticians

    ...Thephysicallyimpossibleeventisthereforetheonethathasinfinitelysmallprobability,andonlythisremarkgivessubstanceobjectiveandphenomenalvaluetothetheoryofmathematicalprobability.[1843,p.78]

    (Kanthaddistinguishedbetweenthenoumenon,orthinginitself,andthephenomenon,orobjectofexperience.)

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    AntoineAugustinCournot18011877

    FromCournots principletomarketefficiency,byGlennShafer.WorkingPaper#15,2006,atwww.probabilityandfinance.com.

  • Part2.Whatfrequentismmeanstostatisticians

    AnotherwayofexpressingCournots principle:contentofaprobabilitymodel

    =itstestablepredictions

    Commonplaceamongstatisticiansandprobabilists.

    Evenwhentheyacceptthelabelfrequentist.

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  • Part2.Whatfrequentismmeanstostatisticians

    FoundersofmodernprobabilityandstatisticsadoptedCournots principleasthebasisoftheirfrequentism.

    OskarAnderson(founderofLMUstatistics)

    JacquesHadamard (thelastuniversalmathematician)

    PaulLvy,EmileBorel,MauriceFrchet (whorevivedFrenchprobabilityinthe20th century)

    AndreiKolmogorov(founderofmodernprobability)

    JerzyNeyman (founderofclassicalfrequentiststatistics)

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  • Part2.Whatfrequentismmeanstostatisticians

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    Alexandr Chuprov18741926

    OskarAnderson18871960

    Chuprovs 1911RussianbookonthephilosophyofstatisticsemphasizedCournots principleasthebridgebetweenBernoullismathematicsandtheconclusionthatfrequencieswillmatchprobabilities.

    Anderson,Chuprovs studentatSaintPetersburg,hadalongcareerpromotinghisideas.HisGermantextbookonmathematicalstatisticsappearedin1935.

  • Part2.Whatfrequentismmeanstostatisticians

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    HansRichter19121978

    1955wurdeerordentlicherProfessorander LudwigMaximiliansUniversittMnchen mitdemneuenLehrstuhlfrMathematischeStatistikundWirtschaftsmathematik.

    Gemeinsammit LeopoldSchmetterer grndeteRichterdie ZeitschriftfrWahrscheinlichkeitstheorieundverwandteGebiete (seit1986 ProbabilityTheoryandRelatedFields).

    RichteremphasizedtheCournotschePrinzip inhis1956textbookandhiscontributionstoDialectica.

  • Hadamards twofundamentalnotionsofprobability1. Equallyprobableevents.The(subjective)basisforthe

    mathematicsofprobability.

    2. Eventofverysmallprobability.Onlywaytoprovideanobjectivevaluetoinitiallysubjectiveprobabilities.

    231. Evnements galement probables2. Evnement trspeu probable

    Part2.Whatfrequentismmeanstostatisticians

    JacquesHadamard18651963

    PaulLvy18861971

  • Wecanonlydiscusstheobjectivevalueofthenotionofprobabilitywhenweknowthetheorysverifiableconsequences.Theyallflowfromthisprinciple:asufficientlysmallprobabilitycanbeneglected.Inotherwords:aneventsufficientlyunlikelycanbeconsideredpracticallyimpossible.

    Thorie deladdition desvariablesalatoire (p.3):