{"product_id":"lower-previsions-9780470723777","title":"Lower Previsions","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eWritten 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.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003ePreface xv  \u003cp\u003eAcknowledgements xvii\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Preliminary notions and definitions 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 Sets of numbers 1\u003c\/p\u003e \u003cp\u003e1.2 Gambles 2\u003c\/p\u003e \u003cp\u003e1.3 Subsets and their indicators 5\u003c\/p\u003e \u003cp\u003e1.4 Collections of events 5\u003c\/p\u003e \u003cp\u003e1.5 Directed sets and Moore–Smith limits 7\u003c\/p\u003e \u003cp\u003e1.6 Uniform convergence of bounded gambles 9\u003c\/p\u003e \u003cp\u003e1.7 Set functions, charges and measures 10\u003c\/p\u003e \u003cp\u003e1.8 Measurability and simple gambles 12\u003c\/p\u003e \u003cp\u003e1.9 Real functionals 17\u003c\/p\u003e \u003cp\u003e1.10 A useful lemma 19\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePART I LOWER PREVISIONS ON BOUNDED GAMBLES 21\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Introduction 23\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Sets of acceptable bounded gambles 25\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Random variables 26\u003c\/p\u003e \u003cp\u003e3.2 Belief and behaviour 27\u003c\/p\u003e \u003cp\u003e3.3 Bounded gambles 28\u003c\/p\u003e \u003cp\u003e3.4 Sets of acceptable bounded gambles 29\u003c\/p\u003e \u003cp\u003e3.4.1 Rationality criteria 29\u003c\/p\u003e \u003cp\u003e3.4.2 Inference 32\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Lower previsions\u003c\/b\u003e \u003cb\u003e37\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Lower and upper previsions 38\u003c\/p\u003e \u003cp\u003e4.1.1 From sets of acceptable bounded gambles to lower previsions 38\u003c\/p\u003e \u003cp\u003e4.1.2 Lower and upper previsions directly 40\u003c\/p\u003e \u003cp\u003e4.2 Consistency for lower previsions 41\u003c\/p\u003e \u003cp\u003e4.2.1 Definition and justification 41\u003c\/p\u003e \u003cp\u003e4.2.2 A more direct justification for the avoiding sure loss condition 44\u003c\/p\u003e \u003cp\u003e4.2.3 Avoiding sure loss and avoiding partial loss 45\u003c\/p\u003e \u003cp\u003e4.2.4 Illustrating the avoiding sure loss condition 45\u003c\/p\u003e \u003cp\u003e4.2.5 Consequences of avoiding sure loss 46\u003c\/p\u003e \u003cp\u003e4.3 Coherence for lower previsions 46\u003c\/p\u003e \u003cp\u003e4.3.1 Definition and justification 46\u003c\/p\u003e \u003cp\u003e4.3.2 A more direct justification for the coherence condition 50\u003c\/p\u003e \u003cp\u003e4.3.3 Illustrating the coherence condition 51\u003c\/p\u003e \u003cp\u003e4.3.4 Linear previsions 51\u003c\/p\u003e \u003cp\u003e4.4 Properties of coherent lower previsions 53\u003c\/p\u003e \u003cp\u003e4.4.1 Interesting consequences of coherence 53\u003c\/p\u003e \u003cp\u003e4.4.2 Coherence and conjugacy 56\u003c\/p\u003e \u003cp\u003e4.4.3 Easier ways to prove coherence 56\u003c\/p\u003e \u003cp\u003e4.4.4 Coherence and monotone convergence 63\u003c\/p\u003e \u003cp\u003e4.4.5 Coherence and a seminorm 64\u003c\/p\u003e \u003cp\u003e4.5 The natural extension of a lower prevision 65\u003c\/p\u003e \u003cp\u003e4.5.1 Natural extension as least-committal extension 65\u003c\/p\u003e \u003cp\u003e4.5.2 Natural extension and equivalence 66\u003c\/p\u003e \u003cp\u003e4.5.3 Natural extension to a specific domain 66\u003c\/p\u003e \u003cp\u003e4.5.4 Transitivity of natural extension 67\u003c\/p\u003e \u003cp\u003e4.5.5 Natural extension and avoiding sure loss 67\u003c\/p\u003e \u003cp\u003e4.5.6 Simpler ways of calculating the natural extension 69\u003c\/p\u003e \u003cp\u003e4.6 Alternative characterisations for avoiding sure loss, coherence, and natural extension 70\u003c\/p\u003e \u003cp\u003e4.7 Topological considerations 74\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Special coherent lower previsions 76\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Linear previsions on finite spaces 77\u003c\/p\u003e \u003cp\u003e5.2 Coherent lower previsions on finite spaces 78\u003c\/p\u003e \u003cp\u003e5.3 Limits as linear previsions 80\u003c\/p\u003e \u003cp\u003e5.4 Vacuous lower previsions 81\u003c\/p\u003e \u003cp\u003e5.5 {0, 1}-valued lower probabilities 82\u003c\/p\u003e \u003cp\u003e5.5.1 Coherence and natural extension 82\u003c\/p\u003e \u003cp\u003e5.5.2 The link with classical propositional logic 88\u003c\/p\u003e \u003cp\u003e5.5.3 The link with limits inferior 90\u003c\/p\u003e \u003cp\u003e5.5.4 Monotone convergence 91\u003c\/p\u003e \u003cp\u003e5.5.5 Lower oscillations and neighbourhood filters 93\u003c\/p\u003e \u003cp\u003e5.5.6 Extending a lower prevision defined on all continuous bounded gambles 98\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 n-Monotone lower previsions 101\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 n-Monotonicity 102\u003c\/p\u003e \u003cp\u003e6.2 n-Monotonicity and coherence 107\u003c\/p\u003e \u003cp\u003e6.2.1 A few observations 107\u003c\/p\u003e \u003cp\u003e6.2.2 Results for lower probabilities 109\u003c\/p\u003e \u003cp\u003e6.3 Representation results 113\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Special n-monotone coherent lower previsions 122\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Lower and upper mass functions 123\u003c\/p\u003e \u003cp\u003e7.2 Minimum preserving lower previsions 127\u003c\/p\u003e \u003cp\u003e7.2.1 Definition and properties 127\u003c\/p\u003e \u003cp\u003e7.2.2 Vacuous lower previsions 128\u003c\/p\u003e \u003cp\u003e7.3 Belief functions 128\u003c\/p\u003e \u003cp\u003e7.4 Lower previsions associated with proper filters 129\u003c\/p\u003e \u003cp\u003e7.5 Induced lower previsions 131\u003c\/p\u003e \u003cp\u003e7.5.1 Motivation 131\u003c\/p\u003e \u003cp\u003e7.5.2 Induced lower previsions 133\u003c\/p\u003e \u003cp\u003e7.5.3 Properties of induced lower previsions 134\u003c\/p\u003e \u003cp\u003e7.6 Special cases of induced lower previsions 138\u003c\/p\u003e \u003cp\u003e7.6.1 Belief functions 139\u003c\/p\u003e \u003cp\u003e7.6.2 Refining the set of possible values for a random variable 139\u003c\/p\u003e \u003cp\u003e7.7 Assessments on chains of sets 142\u003c\/p\u003e \u003cp\u003e7.8 Possibility and necessity measures 143\u003c\/p\u003e \u003cp\u003e7.9 Distribution functions and probability boxes 147\u003c\/p\u003e \u003cp\u003e7.9.1 Distribution functions 147\u003c\/p\u003e \u003cp\u003e7.9.2 Probability boxes 149\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Linear previsions, integration and duality 151\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Linear extension and integration 153\u003c\/p\u003e \u003cp\u003e8.2 Integration of probability charges 159\u003c\/p\u003e \u003cp\u003e8.3 Inner and outer set function, completion and other extensions 163\u003c\/p\u003e \u003cp\u003e8.4 Linear previsions and probability charges 166\u003c\/p\u003e \u003cp\u003e8.5 The S-integral 168\u003c\/p\u003e \u003cp\u003e8.6 The Lebesgue integral 171\u003c\/p\u003e \u003cp\u003e8.7 The Dunford integral 172\u003c\/p\u003e \u003cp\u003e8.8 Consequences of duality 177\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Examples of linear extension 181\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 Distribution functions 181\u003c\/p\u003e \u003cp\u003e9.2 Limits inferior 182\u003c\/p\u003e \u003cp\u003e9.3 Lower and upper oscillations 183\u003c\/p\u003e \u003cp\u003e9.4 Linear extension of a probability measure 183\u003c\/p\u003e \u003cp\u003e9.5 Extending a linear prevision from continuous bounded gambles 187\u003c\/p\u003e \u003cp\u003e9.6 Induced lower previsions and random sets 188\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Lower previsions and symmetry 191\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Invariance for lower previsions 192\u003c\/p\u003e \u003cp\u003e10.1.1 Definition 192\u003c\/p\u003e \u003cp\u003e10.1.2 Existence of invariant lower previsions 194\u003c\/p\u003e \u003cp\u003e10.1.3 Existence of strongly invariant lower previsions 195\u003c\/p\u003e \u003cp\u003e10.2 An important special case 200\u003c\/p\u003e \u003cp\u003e10.3 Interesting examples 205\u003c\/p\u003e \u003cp\u003e10.3.1 Permutation invariance on finite spaces 205\u003c\/p\u003e \u003cp\u003e10.3.2 Shift invariance and Banach limits 208\u003c\/p\u003e \u003cp\u003e10.3.3 Stationary random processes 210\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Extreme lower previsions 214\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1 Preliminary results concerning real functionals 215\u003c\/p\u003e \u003cp\u003e11.2 Inequality preserving functionals 217\u003c\/p\u003e \u003cp\u003e11.2.1 Definition 217\u003c\/p\u003e \u003cp\u003e11.2.2 Linear functionals 217\u003c\/p\u003e \u003cp\u003e11.2.3 Monotone functionals 218\u003c\/p\u003e \u003cp\u003e11.2.4 n-Monotone functionals 218\u003c\/p\u003e \u003cp\u003e11.2.5 Coherent lower previsions 219\u003c\/p\u003e \u003cp\u003e11.2.6 Combinations 220\u003c\/p\u003e \u003cp\u003e11.3 Properties of inequality preserving functionals 220\u003c\/p\u003e \u003cp\u003e11.4 Infinite non-negative linear combinations of inequality preserving functionals 221\u003c\/p\u003e \u003cp\u003e11.4.1 Definition 221\u003c\/p\u003e \u003cp\u003e11.4.2 Examples 222\u003c\/p\u003e \u003cp\u003e11.4.3 Main result 223\u003c\/p\u003e \u003cp\u003e11.5 Representation results 224\u003c\/p\u003e \u003cp\u003e11.6 Lower previsions associated with proper filters 225\u003c\/p\u003e \u003cp\u003e11.6.1 Belief functions 225\u003c\/p\u003e \u003cp\u003e11.6.2 Possibility measures 226\u003c\/p\u003e \u003cp\u003e11.6.3 Extending a linear prevision defined on all continuous bounded gambles 226\u003c\/p\u003e \u003cp\u003e11.6.4 The connection with induced lower previsions 227\u003c\/p\u003e \u003cp\u003e11.7 Strongly invariant coherent lower previsions 228\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePART II EXTENDING THE THEORY TO UNBOUNDED GAMBLES 231\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12 Introduction 233\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e13 Conditional lower previsions 235\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e13.1 Gambles 236\u003c\/p\u003e \u003cp\u003e13.2 Sets of acceptable gambles 236\u003c\/p\u003e \u003cp\u003e13.2.1 Rationality criteria 236\u003c\/p\u003e \u003cp\u003e13.2.2 Inference 238\u003c\/p\u003e \u003cp\u003e13.3 Conditional lower previsions 240\u003c\/p\u003e \u003cp\u003e13.3.1 Going from sets of acceptable gambles to conditional lower previsions 240\u003c\/p\u003e \u003cp\u003e13.3.2 Conditional lower previsions directly 252\u003c\/p\u003e \u003cp\u003e13.4 Consistency for conditional lower previsions 254\u003c\/p\u003e \u003cp\u003e13.4.1 Definition and justification 254\u003c\/p\u003e \u003cp\u003e13.4.2 Avoiding sure loss and avoiding partial loss 257\u003c\/p\u003e \u003cp\u003e13.4.3 Compatibility with the definition for lower previsions on bounded gambles 258\u003c\/p\u003e \u003cp\u003e13.4.4 Comparison with avoiding sure loss for lower previsions on bounded gambles 258\u003c\/p\u003e \u003cp\u003e13.5 Coherence for conditional lower previsions 259\u003c\/p\u003e \u003cp\u003e13.5.1 Definition and justification 259\u003c\/p\u003e \u003cp\u003e13.5.2 Compatibility with the definition for lower previsions on bounded gambles 264\u003c\/p\u003e \u003cp\u003e13.5.3 Comparison with coherence for lower previsions on bounded gambles 264\u003c\/p\u003e \u003cp\u003e13.5.4 Linear previsions 264\u003c\/p\u003e \u003cp\u003e13.6 Properties of coherent conditional lower previsions 266\u003c\/p\u003e \u003cp\u003e13.6.1 Interesting consequences of coherence 266\u003c\/p\u003e \u003cp\u003e13.6.2 Trivial extension 269\u003c\/p\u003e \u003cp\u003e13.6.3 Easier ways to prove coherence 270\u003c\/p\u003e \u003cp\u003e13.6.4 Separate coherence 278\u003c\/p\u003e \u003cp\u003e13.7 The natural extension of a conditional lower prevision 279\u003c\/p\u003e \u003cp\u003e13.7.1 Natural extension as least-committal extension 280\u003c\/p\u003e \u003cp\u003e13.7.2 Natural extension and equivalence 281\u003c\/p\u003e \u003cp\u003e13.7.3 Natural extension to a specific domain and the transitivity of natural extension 282\u003c\/p\u003e \u003cp\u003e13.7.4 Natural extension and avoiding sure loss 283\u003c\/p\u003e \u003cp\u003e13.7.5 Simpler ways of calculating the natural extension 285\u003c\/p\u003e \u003cp\u003e13.7.6 Compatibility with the definition for lower previsions on bounded gambles 286\u003c\/p\u003e \u003cp\u003e13.8 Alternative characterisations for avoiding sure loss, coherence and natural extension 287\u003c\/p\u003e \u003cp\u003e13.9 Marginal extension 288\u003c\/p\u003e \u003cp\u003e13.10 Extending a lower prevision from bounded gambles to conditional gambles 295\u003c\/p\u003e \u003cp\u003e13.10.1 General case 295\u003c\/p\u003e \u003cp\u003e13.10.2 Linear previsions and probability charges 297\u003c\/p\u003e \u003cp\u003e13.10.3 Vacuous lower previsions 298\u003c\/p\u003e \u003cp\u003e13.10.4 Lower previsions associated with proper filters 300\u003c\/p\u003e \u003cp\u003e13.10.5 Limits inferior 300\u003c\/p\u003e \u003cp\u003e13.11 The need for infinity? 301\u003c\/p\u003e \u003cp\u003e\u003cb\u003e14 Lower previsions for essentially bounded gambles 304\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e14.1 Null sets and null gambles 305\u003c\/p\u003e \u003cp\u003e14.2 Null bounded gambles 310\u003c\/p\u003e \u003cp\u003e14.3 Essentially bounded gambles 311\u003c\/p\u003e \u003cp\u003e14.4 Extension of lower and upper previsions to essentially bounded gambles 316\u003c\/p\u003e \u003cp\u003e14.5 Examples 322\u003c\/p\u003e \u003cp\u003e14.5.1 Linear previsions and probability charges 322\u003c\/p\u003e \u003cp\u003e14.5.2 Vacuous lower previsions 323\u003c\/p\u003e \u003cp\u003e14.5.3 Lower previsions associated with proper filters 323\u003c\/p\u003e \u003cp\u003e14.5.4 Limits inferior 324\u003c\/p\u003e \u003cp\u003e14.5.5 Belief functions 325\u003c\/p\u003e \u003cp\u003e14.5.6 Possibility measures 325\u003c\/p\u003e \u003cp\u003e\u003cb\u003e15 Lower previsions for previsible gambles 327\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e15.1 Convergence in probability 328\u003c\/p\u003e \u003cp\u003e15.2 Previsibility 331\u003c\/p\u003e \u003cp\u003e15.3 Measurability 340\u003c\/p\u003e \u003cp\u003e15.4 Lebesgue’s dominated convergence theorem 343\u003c\/p\u003e \u003cp\u003e15.5 Previsibility by cuts 348\u003c\/p\u003e \u003cp\u003e15.6 A sufficient condition for previsibility 350\u003c\/p\u003e \u003cp\u003e15.7 Previsibility for 2-monotone lower previsions 352\u003c\/p\u003e \u003cp\u003e15.8 Convex combinations 355\u003c\/p\u003e \u003cp\u003e15.9 Lower envelope theorem 355\u003c\/p\u003e \u003cp\u003e15.10 Examples 358\u003c\/p\u003e \u003cp\u003e15.10.1 Linear previsions and probability charges 358\u003c\/p\u003e \u003cp\u003e15.10.2 Probability density functions: The normal density 359\u003c\/p\u003e \u003cp\u003e15.10.3 Vacuous lower previsions 360\u003c\/p\u003e \u003cp\u003e15.10.4 Lower previsions associated with proper filters 361\u003c\/p\u003e \u003cp\u003e15.10.5 Limits inferior 361\u003c\/p\u003e \u003cp\u003e15.10.6 Belief functions 362\u003c\/p\u003e \u003cp\u003e15.10.7 Possibility measures 362\u003c\/p\u003e \u003cp\u003e15.10.8 Estimation 365\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix A Linear spaces, linear lattices and convexity 368\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix B Notions and results from topology 371\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eB.1 Basic definitions 371\u003c\/p\u003e \u003cp\u003eB.2 Metric spaces 372\u003c\/p\u003e \u003cp\u003eB.3 Continuity 373\u003c\/p\u003e \u003cp\u003eB.4 Topological linear spaces 374\u003c\/p\u003e \u003cp\u003eB.5 Extreme points 374\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix C The Choquet integral 376\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eC.1 Preliminaries 376\u003c\/p\u003e \u003cp\u003eC.1.1 The improper Riemann integral of a non-increasing function 376\u003c\/p\u003e \u003cp\u003eC.1.2 Comonotonicity 378\u003c\/p\u003e \u003cp\u003eC.2 Definition of the Choquet integral 378\u003c\/p\u003e \u003cp\u003eC.3 Basic properties of the Choquet integral 379\u003c\/p\u003e \u003cp\u003eC.4 A simple but useful equality 387\u003c\/p\u003e \u003cp\u003eC.5 A simplified version of Greco’s representation theorem 389\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix D The extended real calculus 391\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eD.1 Definitions 391\u003c\/p\u003e \u003cp\u003eD.2 Properties 392\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix E Symbols and notation 396\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eReferences 398\u003c\/p\u003e \u003cp\u003eIndex 407\u003c\/p\u003e","brand":"John Wiley \u0026 Sons Inc","offers":[{"title":"Default Title","offer_id":49402419085655,"sku":"9780470723777","price":71.06,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780470723777.jpg?v=1730480341","url":"https:\/\/bookcurl.com\/products\/lower-previsions-9780470723777","provider":"Book Curl","version":"1.0","type":"link"}