Lower Previsions

Content: 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 functions 128     7.4 Lower previsions associated with proper filters 129     7.5 Induced lower previsions 131     7.5.1 Motivation 131     7.5.2 Induced lower previsions 133     7.5.3 Properties of induced lower previsions 134     7.6 Special cases of induced lower previsions 138     7.6.1 Belief functions 139     7.6.2 Refining the set of possible values for a random variable 139     7.7 Assessments on chains of sets 142     7.8 Possibility and necessity measures 143     7.9 Distribution functions and probability boxes 147     7.9.1 Distribution functions 147     7.9.2 Probability boxes 149     8 Linear previsions, integration and duality 151     8.1 Linear extension and integration 153     8.2 Integration of probability charges 159     8.3 Inner and outer set function, completion and other extensions 163     8.4 Linear previsions and probability charges 166     8.5 The S-integral 168     8.6 The Lebesgue integral 171     8.7 The Dunford integral 172     8.8 Consequences of duality 177     9 Examples of linear extension 181     9.1 Distribution functions 181     9.2 Limits inferior 182     9.3 Lower and upper oscillations 183     9.4 Linear extension of a probability measure 183     9.5 Extending a linear prevision from continuous bounded gambles 187     9.6 Induced lower previsions and random sets 188     10 Lower previsions and symmetry 191     10.1 Invariance for lower previsions 192     10.1.1 Definition 192     10.1.2 Existence of invariant lower previsions 194     10.1.3 Existence of strongly invariant lower previsions 195     10.2 An important special case 200     10.3 Interesting examples 205     10.3.1 Permutation invariance on finite spaces 205     10.3.2 Shift invariance and Banach limits 208     10.3.3 Stationary random processes 210     11 Extreme lower previsions 214     11.1 Preliminary results concerning real functionals 215     11.2 Inequality preserving functionals 217     11.2.1 Definition 217     11.2.2 Linear functionals 217     11.2.3 Monotone functionals 218     11.2.4 n-Monotone functionals 218     11.2.5 Coherent lower previsions 219     11.2.6 Combinations 220     11.3 Properties of inequality preserving functionals 220     11.4 Infinite non-negative linear combinations of inequality preserving functionals 221     11.4.1 Definition 221     11.4.2 Examples 222     11.4.3 Main result 223     11.5 Representation results 224     11.6 Lower previsions associated with proper filters 225     11.6.1 Belief functions 225     11.6.2 Possibility measures 226     11.6.3 Extending a linear prevision defined on all continuous bounded gambles 226     11.6.4 The connection with induced lower previsions 227     11.7 Strongly invariant coherent lower previsions 228     PART II EXTENDING THE THEORY TO UNBOUNDED GAMBLES 231     12 Introduction 233     13 Conditional lower previsions 235     13.1 Gambles 236     13.2 Sets of acceptable gambles 236     13.2.1 Rationality criteria 236     13.2.2 Inference 238     13.3 Conditional lower previsions 240     13.3.1 Going from sets of acceptable gambles to conditional lower previsions 240     13.3.2 Conditional lower previsions directly 252     13.4 Consistency for conditional lower previsions 254     13.4.1 Definition and justification 254     13.4.2 Avoiding sure loss and avoiding partial loss 257     13.4.3 Compatibility with the definition for lower previsions on bounded gambles 258     13.4.4 Comparison with avoiding sure loss for lower previsions on bounded gambles 258     13.5 Coherence for conditional lower previsions 259     13.5.1 Definition and justification 259     13.5.2 Compatibility with the definition for lower previsions on bounded gambles 264     13.5.3 Comparison with coherence for lower previsions on bounded gambles 264     13.5.4 Linear previsions 264     13.6 Properties of coherent conditional lower previsions 266     13.6.1 Interesting consequences of coherence 266     13.6.2 Trivial extension 269     13.6.3 Easier ways to prove coherence 270     13.6.4 Separate coherence 278     13.7 The natural extension of a conditional lower prevision 279     13.7.1 Natural extension as least-committal extension 280     13.7.2 Natural extension and equivalence 281     13.7.3 Natural extension to a specific domain and the transitivity of natural extension 282     13.7.4 Natural extension and avoiding sure loss 283     13.7.5 Simpler ways of calculating the natural extension 285     13.7.6 Compatibility with the definition for lower previsions on bounded gambles 286     13.8 Alternative characterisations for avoiding sure loss, coherence and natural extension 287     13.9 Marginal extension 288     13.10 Extending a lower prevision from bounded gambles to conditional gambles 295     13.10.1 General case 295     13.10.2 Linear previsions and probability charges 297     13.10.3 Vacuous lower previsions 298     13.10.4 Lower previsions associated with proper filters 300     13.10.5 Limits inferior 300     13.11 The need for infinity? 301     14 Lower previsions for essentially bounded gambles 304     14.1 Null sets and null gambles 305     14.2 Null bounded gambles 310     14.3 Essentially bounded gambles 311     14.4 Extension of lower and upper previsions to essentially bounded gambles 316     14.5 Examples 322     14.5.1 Linear previsions and probability charges 322     14.5.2 Vacuous lower previsions 323     14.5.3 Lower previsions associated with proper filters 323     14.5.4 Limits inferior 324     14.5.5 Belief functions 325     14.5.6 Possibility measures 325     15 Lower previsions for previsible gambles 327     15.1 Convergence in probability 328     15.2 Previsibility 331     15.3 Measurability 340     15.4 Lebesgue's dominated convergence theorem 343     15.5 Previsibility by cuts 348     15.6 A sufficient condition for previsibility 350     15.7 Previsibility for 2-monotone lower previsions 352     15.8 Convex combinations 355     15.9 Lower envelope theorem 355     15.10 Examples 358     15.10.1 Linear previsions and probability charges 358     15.10.2 Probability density functions: The normal density 359     15.10.3 Vacuous lower previsions 360     15.10.4 Lower previsions associated with proper filters 361     15.10.5 Limits inferior 361     15.10.6 Belief functions 362     15.10.7 Possibility measures 362     15.10.8 Estimation 365     Appendix A Linear spaces, linear lattices and convexity 368     Appendix B Notions and results from topology 371     B.1 Basic definitions 371     B.2 Metric spaces 372     B.3 Continuity 373     B.4 Topological linear spaces 374     B.5 Extreme points 374     Appendix C The Choquet integral 376     C.1 Preliminaries 376     C.1.1 The improper Riemann integral of a non-increasing function 376     C.1.2 Comonotonicity 378     C.2 Definition of the Choquet integral 378     C.3 Basic properties of the Choquet integral 379     C.4 A simple but useful equality 387     C.5 A simplified version of Greco's representation theorem 389     Appendix D The extended real calculus 391     D.1 Definitions 391     D.2 Properties 392     Appendix E Symbols and notation 396     References 398     Index 407