Lower Previsions

449 Pages·2014·2.99 MB·English

by  Troffaes

Lower Previsions
WILEYSERIESINPROBABILITYANDSTATISTICS
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Lower Previsions
Matthias C.M. Troffaes
DepartmentofMathematicalSciences,DurhamUniversity,UK
Gert de Cooman
SYSTeMSResearchGroup,GhentUniversity,Belgium
Thiseditionfirstpublished2014
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LibraryofCongressCataloging-in-PublicationData
Troffaes,MatthiasC.M.,1977–
Lowerprevisions/MatthiasC.M.Troffaes.GertdeCooman.
pagescm
Includesbibliographicalreferencesandindex.
ISBN978-0-470-72377-7(cloth)
1. Probabilities.2. Statisticaldecision.3. Gamesofchance(Mathematics) I.Cooman,Gertde.II.Title.
QA273.T7552014
519.2–dc23
2014002217
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
ISBN:978-0-470-72377-7
Setin10/12ptTimesLTStdbyLaserwordsPrivateLimited,Chennai,India
1 2014
MatthiasdedicatesthisbooktoSabine,Nausicaa,andLinus;
GerttoWietse,LindeandReine.

Contents
Preface xv
Acknowledgements xvii
1 Preliminarynotionsanddefinitions 1
1.1 Setsofnumbers 1
1.2 Gambles 2
1.3 Subsetsandtheirindicators 5
1.4 Collectionsofevents 5
1.5 DirectedsetsandMoore–Smithlimits 7
1.6 Uniformconvergenceofboundedgambles 9
1.7 Setfunctions,chargesandmeasures 10
1.8 Measurabilityandsimplegambles 12
1.9 Realfunctionals 17
1.10 Ausefullemma 19
PARTI LOWERPREVISIONSON
BOUNDEDGAMBLES 21
2 Introduction 23
3 Setsofacceptableboundedgambles 25
3.1 Randomvariables 26
3.2 Beliefandbehaviour 27
3.3 Boundedgambles 28
3.4 Setsofacceptableboundedgambles 29
3.4.1 Rationalitycriteria 29
3.4.2 Inference 32
4 Lowerprevisions 37
4.1 Lowerandupperprevisions 38
4.1.1 Fromsetsofacceptableboundedgamblestolower
previsions 38
4.1.2 Lowerandupperprevisionsdirectly 40
viii CONTENTS
4.2 Consistencyforlowerprevisions 41
4.2.1 Definitionandjustification 41
4.2.2 Amoredirectjustificationfortheavoidingsureloss
condition 44
4.2.3 Avoidingsurelossandavoidingpartialloss 45
4.2.4 Illustratingtheavoidingsurelosscondition 45
4.2.5 Consequencesofavoidingsureloss 46
4.3 Coherenceforlowerprevisions 46
4.3.1 Definitionandjustification 46
4.3.2 Amoredirectjustificationforthecoherencecondition 50
4.3.3 Illustratingthecoherencecondition 51
4.3.4 Linearprevisions 51
4.4 Propertiesofcoherentlowerprevisions 53
4.4.1 Interestingconsequencesofcoherence 53
4.4.2 Coherenceandconjugacy 56
4.4.3 Easierwaystoprovecoherence 56
4.4.4 Coherenceandmonotoneconvergence 63
4.4.5 Coherenceandaseminorm 64
4.5 Thenaturalextensionofalowerprevision 65
4.5.1 Naturalextensionasleast-committalextension 65
4.5.2 Naturalextensionandequivalence 66
4.5.3 Naturalextensiontoaspecificdomain 66
4.5.4 Transitivityofnaturalextension 67
4.5.5 Naturalextensionandavoidingsureloss 67
4.5.6 Simplerwaysofcalculatingthenaturalextension 69
4.6 Alternativecharacterisationsforavoidingsureloss,coherence,
andnaturalextension 70
4.7 Topologicalconsiderations 74
5 Specialcoherentlowerprevisions 76
5.1 Linearprevisionsonfinitespaces 77
5.2 Coherentlowerprevisionsonfinitespaces 78
5.3 Limitsaslinearprevisions 80
5.4 Vacuouslowerprevisions 81
5.5 {0,1}-valuedlowerprobabilities 82
5.5.1 Coherenceandnaturalextension 82
5.5.2 Thelinkwithclassicalpropositionallogic 88
5.5.3 Thelinkwithlimitsinferior 90
5.5.4 Monotoneconvergence 91
5.5.5 Loweroscillationsandneighbourhoodfilters 93
5.5.6 Extendingalowerprevisiondefinedonallcontinuous
boundedgambles 98
6 n-Monotonelowerprevisions 101
6.1 n-Monotonicity 102
6.2 n-Monotonicityandcoherence 107