Lower Previsions

Matthias Troffaes, Department of Mathematical Sciences, Durham University, UKSince 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, BelgiumWith 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.

• Preface xv Acknowledgements xvii1 Preliminary notions and definitions 11.1 Sets of numbers 11.2 Gambles 21.3 Subsets and their indicators 51.4 Collections of events 51.5 Directed sets and Moore–Smith limits 71.6 Uniform convergence of bounded gambles 91.7 Set functions, charges and measures 101.8 Measurability and simple gambles 121.9 Real functionals 171.10 A useful lemma 19PART I LOWER PREVISIONS ON BOUNDED GAMBLES 212 Introduction 233 Sets of acceptable bounded gambles 253.1 Random variables 263.2 Belief and behaviour 273.3 Bounded gambles 283.4 Sets of acceptable bounded gambles 293.4.1 Rationality criteria 293.4.2 Inference 324 Lower previsions 374.1 Lower and upper previsions 384.1.1 From sets of acceptable bounded gambles to lower previsions 384.1.2 Lower and upper previsions directly 404.2 Consistency for lower previsions 414.2.1 Definition and justification 414.2.2 A more direct justification for the avoiding sure loss condition 444.2.3 Avoiding sure loss and avoiding partial loss 454.2.4 Illustrating the avoiding sure loss condition 454.2.5 Consequences of avoiding sure loss 464.3 Coherence for lower previsions 464.3.1 Definition and justification 464.3.2 A more direct justification for the coherence condition 504.3.3 Illustrating the coherence condition 514.3.4 Linear previsions 514.4 Properties of coherent lower previsions 534.4.1 Interesting consequences of coherence 534.4.2 Coherence and conjugacy 564.4.3 Easier ways to prove coherence 564.4.4 Coherence and monotone convergence 634.4.5 Coherence and a seminorm 644.5 The natural extension of a lower prevision 654.5.1 Natural extension as least-committal extension 654.5.2 Natural extension and equivalence 664.5.3 Natural extension to a specific domain 664.5.4 Transitivity of natural extension 674.5.5 Natural extension and avoiding sure loss 674.5.6 Simpler ways of calculating the natural extension 694.6 Alternative characterisations for avoiding sure loss, coherence, and natural extension 704.7 Topological considerations 745 Special coherent lower previsions 765.1 Linear previsions on finite spaces 775.2 Coherent lower previsions on finite spaces 785.3 Limits as linear previsions 805.4 Vacuous lower previsions 815.5 {0, 1}-valued lower probabilities 825.5.1 Coherence and natural extension 825.5.2 The link with classical propositional logic 885.5.3 The link with limits inferior 905.5.4 Monotone convergence 915.5.5 Lower oscillations and neighbourhood filters 935.5.6 Extending a lower prevision defined on all continuous bounded gambles 986 n-Monotone lower previsions 1016.1 n-Monotonicity 1026.2 n-Monotonicity and coherence 1076.2.1 A few observations 1076.2.2 Results for lower probabilities 1096.3 Representation results 1137 Special n-monotone coherent lower previsions 1227.1 Lower and upper mass functions 1237.2 Minimum preserving lower previsions 1277.2.1 Definition and properties 1277.2.2 Vacuous lower previsions 1287.3 Belief functions 1287.4 Lower previsions associated with proper filters 1297.5 Induced lower previsions 1317.5.1 Motivation 1317.5.2 Induced lower previsions 1337.5.3 Properties of induced lower previsions 1347.6 Special cases of induced lower previsions 1387.6.1 Belief functions 1397.6.2 Refining the set of possible values for a random variable 1397.7 Assessments on chains of sets 1427.8 Possibility and necessity measures 1437.9 Distribution functions and probability boxes 1477.9.1 Distribution functions 1477.9.2 Probability boxes 1498 Linear previsions, integration and duality 1518.1 Linear extension and integration 1538.2 Integration of probability charges 1598.3 Inner and outer set function, completion and other extensions 1638.4 Linear previsions and probability charges 1668.5 The S-integral 1688.6 The Lebesgue integral 1718.7 The Dunford integral 1728.8 Consequences of duality 1779 Examples of linear extension 1819.1 Distribution functions 1819.2 Limits inferior 1829.3 Lower and upper oscillations 1839.4 Linear extension of a probability measure 1839.5 Extending a linear prevision from continuous bounded gambles 1879.6 Induced lower previsions and random sets 18810 Lower previsions and symmetry 19110.1 Invariance for lower previsions 19210.1.1 Definition 19210.1.2 Existence of invariant lower previsions 19410.1.3 Existence of strongly invariant lower previsions 19510.2 An important special case 20010.3 Interesting examples 20510.3.1 Permutation invariance on finite spaces 20510.3.2 Shift invariance and Banach limits 20810.3.3 Stationary random processes 21011 Extreme lower previsions 21411.1 Preliminary results concerning real functionals 21511.2 Inequality preserving functionals 21711.2.1 Definition 21711.2.2 Linear functionals 21711.2.3 Monotone functionals 21811.2.4 n-Monotone functionals 21811.2.5 Coherent lower previsions 21911.2.6 Combinations 22011.3 Properties of inequality preserving functionals 22011.4 Infinite non-negative linear combinations of inequality preserving functionals 22111.4.1 Definition 22111.4.2 Examples 22211.4.3 Main result 22311.5 Representation results 22411.6 Lower previsions associated with proper filters 22511.6.1 Belief functions 22511.6.2 Possibility measures 22611.6.3 Extending a linear prevision defined on all continuous bounded gambles 22611.6.4 The connection with induced lower previsions 22711.7 Strongly invariant coherent lower previsions 228PART II EXTENDING THE THEORY TO UNBOUNDED GAMBLES 23112 Introduction 23313 Conditional lower previsions 23513.1 Gambles 23613.2 Sets of acceptable gambles 23613.2.1 Rationality criteria 23613.2.2 Inference 23813.3 Conditional lower previsions 24013.3.1 Going from sets of acceptable gambles to conditional lower previsions 24013.3.2 Conditional lower previsions directly 25213.4 Consistency for conditional lower previsions 25413.4.1 Definition and justification 25413.4.2 Avoiding sure loss and avoiding partial loss 25713.4.3 Compatibility with the definition for lower previsions on bounded gambles 25813.4.4 Comparison with avoiding sure loss for lower previsions on bounded gambles 25813.5 Coherence for conditional lower previsions 25913.5.1 Definition and justification 25913.5.2 Compatibility with the definition for lower previsions on bounded gambles 26413.5.3 Comparison with coherence for lower previsions on bounded gambles 26413.5.4 Linear previsions 26413.6 Properties of coherent conditional lower previsions 26613.6.1 Interesting consequences of coherence 26613.6.2 Trivial extension 26913.6.3 Easier ways to prove coherence 27013.6.4 Separate coherence 27813.7 The natural extension of a conditional lower prevision 27913.7.1 Natural extension as least-committal extension 28013.7.2 Natural extension and equivalence 28113.7.3 Natural extension to a specific domain and the transitivity of natural extension 28213.7.4 Natural extension and avoiding sure loss 28313.7.5 Simpler ways of calculating the natural extension 28513.7.6 Compatibility with the definition for lower previsions on bounded gambles 28613.8 Alternative characterisations for avoiding sure loss, coherence and natural extension 28713.9 Marginal extension 28813.10 Extending a lower prevision from bounded gambles to conditional gambles 29513.10.1 General case 29513.10.2 Linear previsions and probability charges 29713.10.3 Vacuous lower previsions 29813.10.4 Lower previsions associated with proper filters 30013.10.5 Limits inferior 30013.11 The need for infinity? 30114 Lower previsions for essentially bounded gambles 30414.1 Null sets and null gambles 30514.2 Null bounded gambles 31014.3 Essentially bounded gambles 31114.4 Extension of lower and upper previsions to essentially bounded gambles 31614.5 Examples 32214.5.1 Linear previsions and probability charges 32214.5.2 Vacuous lower previsions 32314.5.3 Lower previsions associated with proper filters 32314.5.4 Limits inferior 32414.5.5 Belief functions 32514.5.6 Possibility measures 32515 Lower previsions for previsible gambles 32715.1 Convergence in probability 32815.2 Previsibility 33115.3 Measurability 34015.4 Lebesgue’s dominated convergence theorem 34315.5 Previsibility by cuts 34815.6 A sufficient condition for previsibility 35015.7 Previsibility for 2-monotone lower previsions 35215.8 Convex combinations 35515.9 Lower envelope theorem 35515.10 Examples 35815.10.1 Linear previsions and probability charges 35815.10.2 Probability density functions: The normal density 35915.10.3 Vacuous lower previsions 36015.10.4 Lower previsions associated with proper filters 36115.10.5 Limits inferior 36115.10.6 Belief functions 36215.10.7 Possibility measures 36215.10.8 Estimation 365Appendix A Linear spaces, linear lattices and convexity 368Appendix B Notions and results from topology 371B.1 Basic definitions 371B.2 Metric spaces 372B.3 Continuity 373B.4 Topological linear spaces 374B.5 Extreme points 374Appendix C The Choquet integral 376C.1 Preliminaries 376C.1.1 The improper Riemann integral of a non-increasing function 376C.1.2 Comonotonicity 378C.2 Definition of the Choquet integral 378C.3 Basic properties of the Choquet integral 379C.4 A simple but useful equality 387C.5 A simplified version of Greco’s representation theorem 389Appendix D The extended real calculus 391D.1 Definitions 391D.2 Properties 392Appendix E Symbols and notation 396References 398Index 407

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