Philosophical Lectures on Probability: Collected, edited, and annotated by Alberto Mura

239 Pages·2008·2.89 MB·English

by  Bruno de Finetti

PHILOSOPHICALLECTURESONPROBABILITY
SYNTHESE LIBRARY
STUDIESINEPISTEMOLOGY,
LOGIC,METHODOLOGY,ANDPHILOSOPHYOFSCIENCE
Editor-in-Chief:
VINCENTF.HENDRICKS,RoskildeUniversity,Roskilde,Denmark
JOHNSYMONS,UniversityofTexasatElPaso,U.S.A.
HonoraryEditor:
JAAKKOHINTIKKA,BostonUniversity,U.S.A.
Editors:
DIRKVANDALEN,UniversityofUtrecht,TheNetherlands
THEOA.F.KUIPERS,UniversityofGroningen,TheNetherlands
TEDDYSEIDENFELD,CarnegieMellonUniversity,U.S.A.
PATRICKSUPPES,StanfordUniversity,California,U.S.A.
JANWOLEN´SKI,JagiellonianUniversity,Krako´w,Poland
VOLUME340
Bruno de Finetti
PHILOSOPHICAL
LECTURES ON
PROBABILITY
COLLECTED, EDITED,
AND ANNOTATED
BY
AlbertoMura
UniversityofSassari,Italy
With an Introductory Essay by
MariaCarlaGalavotti
Translated by
HykelHosni
123
Author
Editor
†
BrunodeFinetti
AlbertoMura
Universita`degliStudidiSassari
Dipto.diTeorieericerchedeisistemi
culturali
PiazzaContediMoriana,8
07100Sassari
Italy
[email protected]
ThistranslationwassupportedbytheAlexandervonHumboldtFoundation,theGermanFederalMin-
istryofEducationandResearchandtheProgramfortheInvestmentintheFuture(ZIP)oftheGerman
GovernmentthroughaSofjaKovalevskajaAward,aswellastheItalianuniversitiesofPisaandSassari.
ISBN:978-1-4020-8201-6 e-ISBN:978-1-4020-8202-3
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Contents
Preface............................................................ ix
Editor’sNotice..................................................... xiii
DeFinetti’sPhilosophyofProbability................................. xv
1 IntroductoryLecture ............................................ 1
AgainsttheAxiomaticApproach ................................... 1
Subjectivism .................................................... 3
DefiningProbability .............................................. 4
ProperScoringRules ............................................. 4
2 DecisionsandProperScoringRules ............................... 15
WhyProperScoringRulesAreProper............................... 15
ProbabilityDependsontheSubject’sStateofInformation .............. 18
SequentialDecisions.............................................. 19
SubjectivismversusObjectivism.................................... 21
ForanOmniscientBeingProbabilityWouldNotExist ................. 23
3 GeometricRepresentationofBrier’sRule.......................... 27
EnvelopeFormedbyStraightLines ................................. 27
OperationalDefinitionofProbability ................................ 28
4 Bayes’Theorem................................................. 31
Bayes’TheoremandLinearity...................................... 31
StatisticsandInitialProbabilities ................................... 34
BayesianUpdatingisNotaCorrectiveRevision....................... 35
“Adhockeries” ................................................... 36
Bayes’TheoremforRandomQuantities ............................. 37
InexpressibleEvidence............................................ 39
5 PhysicalProbabilityandComplexity .............................. 47
“Perfect”Dice ................................................... 47
TheLotteryParadox .............................................. 49
v
vi Contents
ProbabilityasFrequency .......................................... 50
ProbabilityandPhysicalLaws...................................... 51
ProbabilisticTheoriesasInstruments ................................ 53
RandomSequences............................................... 54
6 StochasticIndependenceandRandomSequences ................... 59
LogicalandStochasticIndependence................................ 59
Propensities ..................................................... 60
IndependenceandFrequentism ..................................... 61
VonMisesCollectives............................................. 62
7 SuperstitionandFrequentism..................................... 69
TheFrequentistFallacy ........................................... 69
IdealizedFrameworks............................................. 71
TheFallacyofHypothesisTesting .................................. 72
8 Exchangeability................................................. 75
UrnDrawingswithReplacementbutWithoutIndependence............. 75
Inductionand“Unknown”Probabilities.............................. 78
ExchangeableRandomQuantities................................... 80
AllegedObjectivityandConvergenceofSubjectiveProbabilities......... 81
9 Distributions.................................................... 87
IntroductoryConcepts ............................................ 87
CumulativeDistributions .......................................... 89
ContinuousDistributionsWithoutDensity............................ 90
TheGeneralCase ................................................ 93
CharacteristicFunctions........................................... 93
AboutMeans .................................................... 95
10 TheConceptofMean ............................................ 97
Chisini’sSerendipity.............................................. 97
Γ-MeansandtheNagumo-KolmogorovTheorem .....................100
StatisticalTheoryofExtremesandAssociativeMeans..................102
InequalitiesAmongAssociativeMeans ..............................104
ConcludingRemarks..............................................105
11 InductionandSampleRandomization .............................109
ExchangeabilityandConvergencetotheObservedFrequency ...........109
BayesianStatisticsandSampleRandomization........................110
12 CompleteAdditivityandZeroProbabilities ........................113
TheBettingFrameworkandItsLimits...............................113
FiniteandCountableAdditivity.....................................114
‘Strict’Coherence................................................118
Contents vii
ConditioningonEventsofZeroProbability...........................120
Allais’Paradox ..................................................123
13 TheDefinitionsofProbability.....................................127
Axiomatic,Classical,andFrequentisticApproaches ...................127
IndistinguishableEventsandEqualProbability........................130
FrequentismandExchangeability ...................................131
VonMises’“Regellosigkeitsaxiom” .................................134
14 TheGambler’sFallacy...........................................137
AgainsttheMeasure-TheoreticApproach ............................137
Gambler’sFallacyandFrequentistFallacy............................140
EventsandPropositions ...........................................144
15 “Facts”and“Events” ............................................149
APragmaticViewofEvents .......................................149
OnElementaryFacts..............................................152
Eventsand“Phenomena”..........................................155
16 “Facts”and“Events”:AnExample ...............................159
ASequenceofCoinTosses ........................................159
AGraphicalRepresentation........................................161
17 Prevision,RandomQuantities,andTrievents .......................165
ProbabilityasaSpecialCaseofPrevision ............................165
TheConglomerativeProperty ......................................167
Trievents........................................................169
18 De´sire´ Andre´’sArgument ........................................177
HeadsandTails:TheGambler’sRuin................................177
TheWiener-Le´vyProcess .........................................180
AgainonGambler’sRuin..........................................180
TheBallotProblem...............................................182
ThePowerofDe´sire´ Andre´’sArgumentativeStrategy ..................183
19 CharacteristicFunctions .........................................185
PrevisionandLinearity............................................185
RotationsintheComplexPlane.....................................186
SomeImportantCharacteristicFunctions.............................188
ConcludingRemarks..............................................189
CitedLiterature....................................................191
NameIndex........................................................209
Preface
This book is a substantially revised and updated version of the volume Filosofia
dellaprobabilita` (PhilosophyofProbability)publishedinItalianin1995,tenyears
afterthedeathofBrunodeFinetti.ItisnotaworkleftinwrittenformbydeFinetti
butcomesfromthetranscriptionsofthetapesrecordingacourseforgraduatestu-
dentsheldbydeFinettiin1979atthe“Istitutonazionaledialtamatematica”(Na-
tionalInstituteforAdvancedMathematics)inRome.Toproperlyreflecttheorigin
ofthepublishedmaterial,anewtitlehasbeenchosenforthepresentedition.
Thelectures—whichwereheldorally,withoutthesupportofanywrittentext—
wererecordedwiththeaimofhavingthempublished.ForthatreasondeFinettiand
I had agreed on the recordingof the lectures and his later editing of the resulting
text.With thisprojectin mind,alongwiththe recordingontape,I tooknumerous
notesand,aboveall,InotedcarefullyeverythingdeFinettiwroteontheblackboard
duringthelectures.
Unfortunately, the original project could not be realized: other commitments
forcedmetoputoffthetranscriptionofthetapes.Later,thepoorhealthofdeFinetti
madethisimpossible.Onlymanyyearslater,ontheoccasionofthetenthanniver-
saryofthedeathofdeFinettididIhavetheideaoftranscribingthoseoldtapesand
publishingthem with commentsand notes. Giventhe rather informalcharacterof
deFinetti’slectures,thetranscriptionrequiredsomeeditinginordertobepublished
asawrittentext.Thisworkwasundertakenscrupulouslyinordertoavoidaltering
deFinetti’sideas.
Althoughoriginallyitwasaimedatanaudiencecomposedmainlyofmathemati-
cians,thecoursedoesnothave,excepttoasmalldegree,thecharacterofamathe-
maticscourse:insteaditisratheraseriesoflecturesinphilosophyofprobability,in
whichdeFinettidealsindetailwithalmostallthethemesofhissubject.
Thisworkthereforepresentsanexposition,alsoaccessibletoreadersnottrained
in mathematics,ofthe ideasofthe fatherof oneof themostimportantphilosoph-
ical and scientific schools of thought, which emerged in the past century and are
becomingmoreandmorepopularinthepresentone:theneo-Bayesianschool.
The intended spirit of the course is well illustrated in the “notice” which de
FinettihimselfwrotetobedisplayedontheInstitute’snotice-board:
ix
x Preface
Thecourse,withadeliberatelygenerictitle[“OnProbability”]willdealwiththeconceptual
andcontroversialquestionsonthesubjectofprobability:questionswhichitisnecessaryto
resolve,onewayoranother,sothatthedevelopmentofreasoningisnotreducedtoamere
formalistic game of mathematical expressions or tovacuous and simplisticpseudophilo-
sophicalstatementsorallegedlypracticalclaims.
Thenoticemade itveryclearwhatthe threetargetsof de Finetti’scriticism were:
(1) the formalism of those who see in the theory of probability a branch of an
abstract systems of axioms; (2) the “vacuous and simplistic pseudophilosophi-
cal statements”: an allusion to the various “definitions” of probability (such as
frequency-limit,or “ratio between the number of favorable cases and the number
of equally possible cases,” and so forth) which he judged as superficialand with-
outcontent;(3) “allegedly practical” behaviorof those who consider as irrelevant
thephilosophicalquestions(or“conceptual”asde Finettipreferredto say)forthe
applicationofthecalculusofprobabilityinconcretesituations.
In the first few daysof the course there were aboutten people attending.Most
of them were graduate students of the ordinary courses of the Institute, but there
werealsootherstudents.Thenumberreducedwithtime.Thepresenceofalimited
audiencemeantthatthecoursetookonthecharacterofaninformalandinteractive
seminar.Usually,asthereadercanrealize,theseminarswereaseriesofdiscussions
aboutvariousaspectsofthephilosophyofprobability.
Thelecturescontainnumerousobservationsandcomments,whichareinteresting
indefiningdeFinetti’sphilosophyofprobability.Forexample,notetheinteresting
observationsoninductivedynamics(Chapter4)orphysicalprobability(Chapter5,
thedistinction—newandimportantinseveralrespects—betweenfactsandevents
(Chapters15 and 16), the thesis accordingto which the meaning of a proposition
candependonempiricalconditions(Chapter17).
AlthoughIaddedormodifiednumerousnotestothetextwithrespecttothefirst
Italian edition, the text from de Finetti’s lectures has been translated into English
withoutfurtherediting.
When de Finetti ran the course I had known him for about a year. Around the
end of 1977, Professor Francesco Barone (1923–2001),whose student I was, had
the good idea of putting me in touch with de Finetti. When I went to visit him at
hishomeinRome,deFinettiwelcomedmewarmlyandIhadaconversationwith
him which lasted for more than five hours. That was the first of a long series of
meetings,thankstowhichIhadtheprivilegeoftalkingtohimabouteveryaspectof
myresearch.Atoneofthesemeetings,whichtookplaceatthebeginningof1979,
hetoldmehehadacceptedtheinvitationtoholdthecoursepresentedinthisvolume.
Onthatoccasionhewarmlyaskedmetotakepart,“alsobecause—hesaid—the
presenceofapersonalreadyintroducedtothephilosophicalproblemsofprobability
couldbeinteresting.”
IwouldliketotakethisoccasiontoexpressthedebtIfeelIhavetowardsBruno
de Finetti. The impact that the meeting with him had on my education was deci-
sive,hisideaofmathematicsasaninstrumenttobeusedinconceptualclarification
and—aboveall—theconcreteexamplewithwhichhegavesubstancetothisidea