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UncertainInference
Copingwithuncertaintyisanecessarypartofordinarylifeandiscrucialtoanunderstanding
ofhowthemindworks.Forexample,itisavitalelementindevelopingartificialintelligence
thatwillnotbeunderminedbyitsownrigidities.Therehavebeenmanyapproachestothe
problemofuncertaininference,rangingfromprobabilitytoinductivelogictononmonotonic
logic. This book seeks to provide a clear exposition of these approaches within a unified
framework.
The principal market for the book will be students and professionals in philosophy,
computer science, and artificial intelligence. Among the special features of the book are
achapteronevidentialprobability,aninterpretationofprobabilityspecificallydeveloped
withaneyetoinductiveanduncertaininference,whichhasnotreceivedabasicexposition
before; chapters on nonmonotonic reasoning and theory replacement that concern mat-
tersrarelyaddressedinstandardphilosophicaltexts;andchaptersonMill’smethodsand
statistical inference that cover material sorely lacking in the usual treatments of artificial
intelligenceandcomputerscience.
HenryE.Kyburg,Jr.istheBurbankProfessorofPhilosophyandComputerScience,atthe
University of Rochester and a research scientist at the Institute for Human and Machine
CognitionattheUniversityofWestFlorida.Heistheauthororeditorofanumberofbooks
oninduction,probability,andstatisticalinference,aswellasaneditor(withRonLouiand
GregCarlson)ofKnowledgeRepresentationandDefeasibleReasoning.HeisaFellowofthe
AmericanAcademyofArtsandSciencesandaFellowoftheAmericanAssociationforthe
AdvancementofScience.
ChohManTengisaresearchscientistattheInstituteforHumanandMachineCognition
attheUniversityofWestFlorida.Sheisinterestedinuncertainreasoningandinmachine
learning.
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Uncertain Inference
HENRY E. KYBURG, Jr.
UniversityofRochesterand
InstituteforHumanandMachineCognition,
UniversityofWestFlorida
and
CHOH MAN TENG
InstituteforHumanandMachineCognition,
UniversityofWestFlorida
PUBLISHED BY CAMBRIDGE UNIVERSITY PRESS (VIRTUAL PUBLISHING)
FOR AND ON BEHALF OF THE PRESS SYNDICATE OF THE UNIVERSITY OF
CAMBRIDGE
The Pitt Building, Trumpington Street, Cambridge CB2 IRP
40 West 20th Street, New York, NY 10011-4211, USA
477 Williamstown Road, Port Melbourne, VIC 3207, Australia
http://www.cambridge.org
© Henry E. Kyburg, Jr., and Choh Man Teng 2001
This edition © Henry E. Kyburg, Jr., and Choh Man Teng 2003
First published in printed format 2001
A catalogue record for the original printed book is available
from the British Library and from the Library of Congress
Original ISBN 0 521 80064 1 hardback
Original ISBN 0 521 00101 3 paperback
ISBN 0 511 02068 6 virtual (netLibrary Edition)
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Contents
Preface page xi
1 HistoricalBackground 1
1.1 Introduction 1
1.2 Inference 1
1.3 RootsinthePast 6
1.4 FrancisBacon 7
1.5 TheDevelopmentofProbability 10
1.6 JohnStuartMill 13
1.7 G.H.vonWright 17
1.8 BibliographicalNotes 19
1.9 Exercises 19
Bibliography 20
2 FirstOrderLogic 21
2.1 Introduction 21
2.2 Syntax 24
2.3 Semantics 28
2.4 W.V.O.Quine’sMathematicalLogic 30
2.5 ArgumentsfromPremises 33
2.6 Limitations 34
2.7 Summary 39
2.8 BibliographicalNotes 39
2.9 Exercises 39
Bibliography 41
3 TheProbabilityCalculus 42
3.1 Introduction 42
3.2 ElementaryProbability 44
3.2.1 CombinationsandPermutations 44
3.2.2 TheProbabilityCalculus 47
3.2.3 ElementaryTheorems 49
v
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vi CONTENTS
3.3 ConditionalProbability 50
3.3.1 TheAxiomofConditionalProbability 51
3.3.2 Bayes’Theorem 53
3.4 ProbabilityDistributions 54
3.4.1 FrequencyFunctionsandDistributionFunctions 55
3.4.2 PropertiesofDistributions 59
3.5 SamplingDistributions 61
3.6 UsefulDistributions 62
3.7 Summary 64
3.8 BibliographicalNotes 65
3.9 Exercises 65
Bibliography 67
4 InterpretationsofProbability 68
4.1 Introduction 68
4.2 TheClassicalView 68
4.3 EmpiricalInterpretationsofProbability 71
4.3.1 TheLimitingFrequencyInterpretation 72
4.3.2 ThePropensityInterpretation 78
4.4 LogicalInterpretationsofProbability 80
4.5 SubjectiveInterpretationsofProbability 87
4.5.1 DutchBook 87
4.5.2 Conditionalization 89
4.6 Summary 93
4.7 BibliographicalNotes 95
4.8 Exercises 95
Bibliography 96
5 NonstandardMeasuresofSupport 98
5.1 Support 98
5.2 KarlPopper 99
5.2.1 Corroboration 100
5.2.2 Levi’sCriticism 102
5.3 OtherMeasures 103
5.4 Dempster–ShaferBeliefFunctions 104
5.4.1 BeliefFunctionsandMassFunctions 105
5.4.2 ReductiontoSetsofProbabilities 106
5.4.3 CombiningEvidence 108
5.4.4 SpecialCases 110
5.4.5 AssessmentofBeliefFunctions 111
5.5 SetsofProbabilityFunctions 113
5.6 Summary 114
5.7 BibliographicalNotes 114
5.8 Exercises 115
Bibliography 116
6 NonmonotonicReasoning 117
6.1 Introduction 117
6.2 Logicand(Non)monotonicity 117
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CONTENTS vii
6.3 DefaultLogic 121
6.3.1 Preliminaries 121
6.3.2 TransformationofOpenDefaultTheories 123
6.3.3 Extensions 124
6.3.4 NeedforaFixedPoint 126
6.3.5 NumberofExtensions 127
6.3.6 Representation 128
6.3.7 VariantsofDefaultLogic 131
6.4 AutoepistemicLogic 134
6.4.1 ModalLogic 134
6.4.2 AutoepistemicReasoningvsDefaultReasoning 136
6.4.3 StableExpansions 138
6.4.4 AlternativeFixed-PointFormulation 140
6.4.5 Groundedness 142
6.5 Circumscription 143
6.6 UnresolvedIssues 146
6.6.1 “Intuition”:BasisofDefaults 146
6.6.2 ComputationalComplexity 147
6.6.3 MultipleExtensions 147
6.7 Summary 148
6.8 BibliographicalNotes 148
6.9 Exercises 149
Bibliography 150
7 TheoryReplacement 152
7.1 Introduction 152
7.2 TheoryChange 153
7.2.1 Expansion 153
7.2.2 Contraction 154
7.2.3 Revision 155
7.3 RationalityConsiderations 156
7.4 TheAGMPostulates 158
7.4.1 Expansion 158
7.4.2 Contraction 159
7.4.3 Revision 161
7.5 Connections 163
7.6 SelectingaContractionFunction 164
7.7 EpistemicEntrenchment 166
7.8 MustItBe? 168
7.8.1 BeliefBases 168
7.8.2 Updates 169
7.8.3 RationalityRevisited 170
7.8.4 IteratedChange 171
7.9 Summary 171
7.10 BibliographicalNotes 172
7.11 Exercises 172
Bibliography 173
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viii CONTENTS
8 StatisticalInference 175
8.1 Introduction 175
8.2 ClassicalStatistics 178
8.2.1 SignificanceTests 179
8.2.2 HypothesisTesting 182
8.2.3 ConfidenceIntervals 186
8.3 BayesianStatistics 192
8.4 Summary 195
8.5 BibliographicalNotes 197
8.6 Exercises 197
Bibliography 198
9 EvidentialProbability 200
9.1 Introduction 200
9.2 BackgroundIssuesandAssumptions 201
9.3 TheSyntaxofStatisticalKnowledge 203
9.4 ReferenceClassesandTargetClasses 205
9.4.1 ReferenceFormulas 205
9.4.2 TargetFormulas 208
9.5 PrimaFacieSupport 209
9.5.1 IndefiniteProbabilities 210
9.5.2 DefiniteProbabilities 210
9.6 Sharpening 212
9.6.1 Precision 213
9.6.2 Specificity 213
9.6.3 Richness 216
9.6.4 Sharpens 217
9.7 PartialProof 219
9.8 ExtendedExample 220
9.9 AUsefulAlgorithm 225
9.10 RelationstoOtherInterpretations 225
9.11 Summary 226
9.12 BibliographicalNotes 227
9.13 Exercises 227
Bibliography 228
10 Semantics 230
10.1 Introduction 230
10.2 ModelsandTruth 232
10.3 ModelRatios 235
10.4 RelevantModels 236
10.5 PartialValidity 240
10.6 Remarks 242
10.7 Summary 244
10.8 BibliographicalNotes 245
10.9 Exercises 245
Bibliography 245
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CONTENTS ix
11 Applications 247
11.1 Introduction 247
11.2 ElementaryResults 248
11.3 InferencefromSamples 254
11.4 Example 259
11.5 StatisticalInduction 261
11.6 BayesianInduction 262
11.7 SequencesofDraws 264
11.8 Summary 266
11.9 BibliographicalNotes 267
11.10 Exercises 268
Bibliography 268
12 ScientificInference 270
12.1 Introduction 270
12.1.1 Objectivity 270
12.1.2 EvidentialandPracticalCertainty 272
12.1.3 StatisticalInference 273
12.2 DemonstrativeInduction 274
12.3 DirectMeasurement 275
12.4 IndirectMeasurement 280
12.5 Theory,Language,andError 285
12.6 Summary 286
12.7 BibliographicalNotes 287
12.7.1 Measurement 287
12.7.2 Theories 287
12.7.3 Datamining 287
12.8 Exercises 288
Bibliography 288
NamesIndex 291
Index 293
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