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Informationen zum Autor Matthias Troffaes, Department of Mathematical Sciences, Durham University, UK Since gaining his PhD, Dr Troffaes has conducted research in Belgium and the US in imprecise probabilities, before becoming a lecturer in statistics at Durham. He has published papers in a variety of journals, and written two book chapters. Gert de Cooman, SYSTeMS Research Group, Ghent University, Belgium With many years' research and teaching experience, Professor de Cooman serves/has served on the Editorial Boards of many statistical journals. He has published over 40 journal articles, and is an editor of the Imprecise Probabilities Project. He has also written chapters for six books, and has co-edited four. Klappentext This book has two main purposes. On the one hand, it provides aconcise and systematic development of the theory of lower previsions,based on the concept of acceptability, in spirit of the work ofWilliams and Walley. On the other hand, it also extends this theory todeal with unbounded quantities, which abound in practicalapplications.Following Williams, we start out with sets of acceptable gambles. Fromthose, we derive rationality criteria---avoiding sure loss andcoherence---and inference methods---natural extension---for(unconditional) lower previsions. We then proceed to study variousaspects of the resulting theory, including the concept of expectation(linear previsions), limits, vacuous models, classical propositionallogic, lower oscillations, and monotone convergence. We discussn-monotonicity for lower previsions, and relate lower previsions withChoquet integration, belief functions, random sets, possibilitymeasures, various integrals, symmetry, and representation theoremsbased on the Bishop-De Leeuw theorem.Next, we extend the framework of sets of acceptable gambles to consideralso unbounded quantities. As before, we again derive rationalitycriteria and inference methods for lower previsions, this time alsoallowing for conditioning. We apply this theory to constructextensions of lower previsions from bounded random quantities to alarger set of random quantities, based on ideas borrowed from thetheory of Dunford integration.A first step is to extend a lower prevision to random quantities thatare bounded on the complement of a null set (essentially boundedrandom quantities). This extension is achieved by a natural extensionprocedure that can be motivated by a rationality axiom stating thatadding null random quantities does not affect acceptability.In a further step, we approximate unbounded random quantities by asequences of bounded ones, and, in essence, we identify those forwhich the induced lower prevision limit does not depend on the detailsof the approximation. We call those random quantities 'previsible'. Westudy previsibility by cut sequences, and arrive at a simplesufficient condition. For the 2-monotone case, we establish a Choquetintegral representation for the extension. For the general case, weprove that the extension can always be written as an envelope ofDunford integrals. We end with some examples of the theory. Zusammenfassung Written by authorities in the field, Lower Previsions illustrates how the theory of Lower Previsions can be extended to cover a larger set of random quantities. The text highlights a crucial problem in the theory of imprecise probability and provides a detailed theory on how to resolve it. Inhaltsverzeichnis Preface xv Acknowledgements xvii 1 Preliminary notions and definitions 1 1.1 Sets of numbers 1 1.2 Gambles 2 1.3 Subsets and their indicators 5 1.4 Collections of events 5 1.5 Directed sets and Moore-Smith limits 7 1.6 Uniform convergence of bounded gambles 9 1.7 Set functions, charges and measures 10 1.8 Measurability and simple gambles 12 1.9 Real functionals 17 1.10 A useful lemma ...
List of contents
Preface xv
Acknowledgements xvii
1 Preliminary notions and definitions 1
1.1 Sets of numbers 1
1.2 Gambles 2
1.3 Subsets and their indicators 5
1.4 Collections of events 5
1.5 Directed sets and Moore-Smith limits 7
1.6 Uniform convergence of bounded gambles 9
1.7 Set functions, charges and measures 10
1.8 Measurability and simple gambles 12
1.9 Real functionals 17
1.10 A useful lemma 19
PART I LOWER PREVISIONS ON BOUNDED GAMBLES 21
2 Introduction 23
3 Sets of acceptable bounded gambles 25
3.1 Random variables 26
3.2 Belief and behaviour 27
3.3 Bounded gambles 28
3.4 Sets of acceptable bounded gambles 29
3.4.1 Rationality criteria 29
3.4.2 Inference 32
4 Lower previsions 37
4.1 Lower and upper previsions 38
4.1.1 From sets of acceptable bounded gambles to lower previsions 38
4.1.2 Lower and upper previsions directly 40
4.2 Consistency for lower previsions 41
4.2.1 Definition and justification 41
4.2.2 A more direct justification for the avoiding sure loss condition 44
4.2.3 Avoiding sure loss and avoiding partial loss 45
4.2.4 Illustrating the avoiding sure loss condition 45
4.2.5 Consequences of avoiding sure loss 46
4.3 Coherence for lower previsions 46
4.3.1 Definition and justification 46
4.3.2 A more direct justification for the coherence condition 50
4.3.3 Illustrating the coherence condition 51
4.3.4 Linear previsions 51
4.4 Properties of coherent lower previsions 53
4.4.1 Interesting consequences of coherence 53
4.4.2 Coherence and conjugacy 56
4.4.3 Easier ways to prove coherence 56
4.4.4 Coherence and monotone convergence 63
4.4.5 Coherence and a seminorm 64
4.5 The natural extension of a lower prevision 65
4.5.1 Natural extension as least-committal extension 65
4.5.2 Natural extension and equivalence 66
4.5.3 Natural extension to a specific domain 66
4.5.4 Transitivity of natural extension 67
4.5.5 Natural extension and avoiding sure loss 67
4.5.6 Simpler ways of calculating the natural extension 69
4.6 Alternative characterisations for avoiding sure loss, coherence, and natural extension 70
4.7 Topological considerations 74
5 Special coherent lower previsions 76
5.1 Linear previsions on finite spaces 77
5.2 Coherent lower previsions on finite spaces 78
5.3 Limits as linear previsions 80
5.4 Vacuous lower previsions 81
5.5 {0, 1}-valued lower probabilities 82
5.5.1 Coherence and natural extension 82
5.5.2 The link with classical propositional logic 88
5.5.3 The link with limits inferior 90
5.5.4 Monotone convergence 91
5.5.5 Lower oscillations and neighbourhood filters 93
5.5.6 Extending a lower prevision defined on all continuous bounded gambles 98
6 n-Monotone lower previsions 101
6.1 n-Monotonicity 102
6.2 n-Monotonicity and coherence 107
6.2.1 A few observations 107
6.2.2 Results for lower probabilities 109
6.3 Representation results 113
7 Special n-monotone coherent lower previsions 122
7.1 Lower and upper mass functions 123
7.2 Minimum preserving lower previsions 127
7.2.1 Definition and properties 127
7.2.2 Vacuous lower previsions 128
7.3 Belief fun
