Lower Previsions

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.

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