Lower Previsions (Wiley Series in Probability and Statistics)

個数:
電子版価格 ¥10,198
  • 電書あり

Lower Previsions (Wiley Series in Probability and Statistics)

  • 在庫がございません。海外の書籍取次会社を通じて出版社等からお取り寄せいたします。
    通常6~9週間ほどで発送の見込みですが、商品によってはさらに時間がかかることもございます。
    重要ご説明事項
    1. 納期遅延や、ご入手不能となる場合がございます。
    2. 複数冊ご注文の場合、分割発送となる場合がございます。
    3. 美品のご指定は承りかねます。
  • ≪洋書のご注文につきまして≫ 「海外取次在庫あり」および「国内仕入れ先からお取り寄せいたします」表示の商品でも、納期の目安期間内にお届けできないことがございます。あらかじめご了承ください。

  • 製本 Hardcover:ハードカバー版/ページ数 415 p.
  • 言語 ENG
  • 商品コード 9780470723777
  • DDC分類 519.2

Full Description


This book has two main purposes. On the one hand, it provides a concise and systematic development of the theory of lower previsions, based on the concept of acceptability, in spirit of the work of Williams and Walley. On the other hand, it also extends this theory to deal with unbounded quantities, which abound in practical applications. Following Williams, we start out with sets of acceptable gambles. From those, we derive rationality criteria---avoiding sure loss and coherence---and inference methods---natural extension---for (unconditional) lower previsions. We then proceed to study various aspects of the resulting theory, including the concept of expectation (linear previsions), limits, vacuous models, classical propositional logic, lower oscillations, and monotone convergence. We discuss n-monotonicity for lower previsions, and relate lower previsions with Choquet integration, belief functions, random sets, possibility measures, various integrals, symmetry, and representation theorems based on the Bishop-De Leeuw theorem. Next, we extend the framework of sets of acceptable gambles to consider also unbounded quantities.As before, we again derive rationality criteria and inference methods for lower previsions, this time also allowing for conditioning. We apply this theory to construct extensions of lower previsions from bounded random quantities to a larger set of random quantities, based on ideas borrowed from the theory of Dunford integration. A first step is to extend a lower prevision to random quantities that are bounded on the complement of a null set (essentially bounded random quantities). This extension is achieved by a natural extension procedure that can be motivated by a rationality axiom stating that adding null random quantities does not affect acceptability. In a further step, we approximate unbounded random quantities by a sequences of bounded ones, and, in essence, we identify those for which the induced lower prevision limit does not depend on the details of the approximation. We call those random quantities 'previsible'. We study previsibility by cut sequences, and arrive at a simple sufficient condition. For the 2-monotone case, we establish a Choquet integral representation for the extension.For the general case, we prove that the extension can always be written as an envelope of Dunford integrals. We end with some examples of the theory.

Table of Contents

Preface                                            xv
Acknowledgements xvii
1 Preliminary notions and definitions 1 (20)
1.1 Sets of numbers 1 (1)
1.2 Gambles 2 (3)
1.3 Subsets and their indicators 5 (1)
1.4 Collections of events 5 (2)
1.5 Directed sets and Moore-Smith limits 7 (2)
1.6 Uniform convergence of bounded gambles 9 (1)
1.7 Set functions, charges and measures 10 (2)
1.8 Measurability and simple gambles 12 (5)
1.9 Real functionals 17 (2)
1.10 A useful lemma 19 (2)
Part I Lower Previsions On Bounded Gambles 21 (210)
2 Introduction 23 (2)
3 Sets of acceptable bounded gambles 25 (12)
3.1 Random variables 26 (1)
3.2 Belief and behaviour 27 (1)
3.3 Bounded gambles 28 (1)
3.4 Sets of acceptable bounded gambles 29 (8)
3.4.1 Rationality criteria 29 (3)
3.4.2 Inference 32 (5)
4 Lower previsions 37 (39)
4.1 Lower and upper previsions 38 (3)
4.1.1 From sets of acceptable bounded 38 (2)
gambles to lower previsions
4.1.2 Lower and upper previsions directly 40 (1)
4.2 Consistency for lower previsions 41 (5)
4.2.1 Definition and justification 41 (3)
4.2.2 A more direct justification for the 44 (1)
avoiding sure loss condition
4.2.3 Avoiding sure loss and avoiding 45 (1)
partial loss
4.2.4 Illustrating the avoiding sure loss 45 (1)
condition
4.2.5 Consequences of avoiding sure loss 46 (1)
4.3 Coherence for lower previsions 46 (7)
4.3.1 Definition and justification 46 (4)
4.3.2 A more direct justification for the 50 (1)
Opherence condition
4.3.3 Illustrating the coherence condition 51 (1)
4.3.4 Linear previsions 51 (2)
4.4 Properties of coherent lower previsions 53 (12)
4.4.1 Interesting consequences of 53 (3)
coherence
4.4.2 Coherence and conjugacy 56 (1)
4.4.3 Easier ways to prove coherence 56 (7)
4.4.4 Coherence and monotone convergence 63 (1)
4.4.5 Coherence and a seminorm 64 (1)
4.5 The natural extension of a lower 65 (5)
prevision
4.5.1 Natural extension as 65 (1)
least-committal extension
4.5.2 Natural extension and equivalence 66 (1)
4.5.3 Natural extension to a specific 66 (1)
domain
4.5.4 Transitivity of natural extension 67 (1)
4.5.5 Natural extension and avoiding sure 67 (2)
loss
4.5.6 Simpler ways of calculating the 69 (1)
natural extension
4.6 Alternative characterisations for 70 (4)
avoiding sure loss, coherence, and natural
extension
4.7 Topological considerations 74 (2)
5 Special coherent lower previsions 76 (25)
5.1 Linear previsions on finite spaces 77 (1)
5.2 Coherent lower previsions on finite 78 (2)
spaces
5.3 Limits as linear previsions 80 (1)
5.4 Vacuous lower previsions 81 (1)
5.5 (0, 1)-valued lower probabilities 82 (19)
5.5.1 Coherence and natural extension 82 (6)
5.5.2 The link with classical 88 (2)
propositional logic
5.5.3 The link with limits inferior 90 (1)
5.5.4 Monotone convergence 91 (2)
5.5.5 Lower oscillations and 93 (5)
neighbourhood filters
5.5.6 Extending a lower prevision defined 98 (3)
on all continuous bounded gambles
6 n-Monotone lower previsions 101 (21)
6.1 n-Monotonicity 102 (5)
6.2 n-Monotonicity and coherence 107 (6)
6.2.1 A few observations 107 (2)
6.2.2 Results for lower probabilities 109 (4)
6.3 Representation results 113 (9)
7 Special n-monotone coherent lower previsions 122 (29)
7.1 Lower and upper mass functions 123 (4)
7.2 Minimum preserving lower previsions 127 (1)
7.2.1 Definition and properties 127 (1)
7.2.2 Vacuous lower previsions 128 (1)
7.3 Belief functions 128 (1)
7.4 Lower previsions associated with proper 129 (2)
filters
7.5 Induced lower previsions 131 (7)
7.5.1 Motivation 131 (2)
7.5.2 Induced lower previsions 133 (1)
7.5.3 Properties of induced lower 134 (4)
previsions
7.6 Special cases of induced lower 138 (4)
previsions
7.6.1 Belief functions 139 (1)
7.6.2 Refining the set of possible values 139 (3)
for a random variable
7.7 Assessments on chains of sets 142 (1)
7.8 Possibility and necessity measures 143 (4)
7.9 Distribution functions and probability 147 (4)
boxes
7.9.1 Distribution functions 147 (2)
7.9.2 Probability boxes 149 (2)
8 Linear previsions, integration and duality 151 (30)
8.1 Linear extension and integration 153 (6)
8.2 Integration of probability charges 159 (4)
8.3 Inner and outer set function, 163 (3)
completion and other extensions
8.4 Linear previsions and probability 166 (2)
charges
8.5 The S-integral 168 (3)
8.6 The Lebesgue integral 171 (1)
8.7 The Dunford integral 172 (5)
8.8 Consequences of duality 177 (4)
9 Examples of linear extension 181 (10)
9.1 Distribution functions 181 (1)
9.2 Limits inferior 182 (1)
9.3 Lower and upper oscillations 183 (1)
9.4 Linear extension of a probability 183 (4)
measure
9.5 Extending a linear prevision from 187 (1)
continuous bounded gambles
9.6 Induced lower previsions and random sets 188 (3)
10 Lower previsions and symmetry 191 (23)
10.1 Invariance for lower previsions 192 (8)
10.1.1 Definition 192 (2)
10.1.2 Existence of invariant lower 194 (1)
previsions
10.1.3 Existence of strongly invariant 195 (5)
lower previsions
10.2 An important special case 200 (5)
10.3 Interesting examples 205 (9)
10.3.1 Permutation invariance on finite 205 (3)
spaces
10.3.2 Shift invariance and Banach limits 208 (2)
10.3.3 Stationary random processes 210 (4)
11 Extreme lower previsions 214 (17)
11.1 Preliminary results concerning real 215 (2)
functionals
11.2 Inequality preserving functionals 217 (3)
11.2.1 Definition 217 (1)
11.2.2 Linear functionals 217 (1)
11.2.3 Monotone functionals 218 (1)
11.2.4 n-Monotone functionals 218 (1)
11.2.5 Coherent lower previsions 219 (1)
11.2.6 Combinations 220 (1)
11.3 Properties of inequality preserving 220 (1)
functionals
11.4 Infinite non-negative linear 221 (3)
combinations of inequality preserving
functionals
11.4.1 Definition 221 (1)
11.4.2 Examples 222 (1)
11.4.3 Main result 223 (1)
11.5 Representation results 224 (1)
11.6 Lower previsions associated with 225 (3)
proper filters
11.6.1 Belief functions 225 (1)
11.6.2 Possibility measures 226 (1)
11.6.3 Extending a linear prevision 226 (1)
defined on all continuous bounded gambles
11.6.4 The connection with induced lower 227 (1)
previsions
11.7 Strongly invariant coherent lower 228 (3)
previsions
Part II Extending The Theory To Unbounded 231 (137)
Gambles
12 Introduction 233 (2)
13 Conditional lower previsions 235 (69)
13.1 Gambles 236 (1)
13.2 Sets of acceptable gambles 236 (4)
13.2.1 Rationality criteria 236 (2)
13.2.2 Inference 238 (2)
13.3 Conditional lower previsions 240 (14)
13.3.1 Going from sets of acceptable 240 (12)
gambles to conditional lower previsions
13.3.2 Conditional lower previsions 252 (2)
directly
13.4 Consistency for conditional lower 254 (5)
previsions
13.4.1 Definition and justification 254 (3)
13.4.2 Avoiding sure loss and avoiding 257 (1)
partial loss
13.4.3 Compatibility with the definition 258 (1)
for lower previsions on bounded gambles
13.4.4 Comparison with avoiding sure loss 258 (1)
for lower previsions on bounded gambles
13.5 Coherence for conditional lower 259 (7)
previsions
13.5.1 Definition and justification 259 (5)
13.5.2 Compatibility with the definition 264 (1)
for lower previsions on bounded gambles
13.5.3 Comparison with coherence for 264 (1)
lower previsions on bounded gambles
13.5.4 Linear previsions 264 (2)
13.6 Properties of coherent conditional 266 (13)
lower previsions
13.6.1 Interesting consequences of 266 (3)
coherence
13.6.2 Trivial extension 269 (1)
13.6.3 Easier ways to prove coherence 270 (8)
13.6.4 Separate coherence 278 (1)
13.7 The natural extension of a conditional 279 (8)
lower prevision
13.7.1 Natural extension as 280 (1)
least-committal extension
13.7.2 Natural extension and equivalence 281 (1)
13.7.3 Natural extension to a specific 282 (1)
domain and the transitivity of natural
extension
13.7.4 Natural extension and avoiding 283 (2)
sure loss
13.7.5 Simpler ways of calculating the 285 (1)
natural extension
13.7.6 Compatibility with the definition 286 (1)
for lower previsions on bounded gambles
13.8 Alternative characterisations for 287 (1)
avoiding sure loss, coherence and natural
extension
13.9 Marginal extension 288 (7)
13.10 Extending a lower prevision from 295 (6)
bounded gambles to conditional gambles
13.10.1 General case 295 (2)
13.10.2 Linear previsions and probability 297 (1)
charges
13.10.3 Vacuous lower previsions 298 (2)
13.10.4 Lower previsions associated with 300 (1)
proper filters
13.10.5 Limits inferior 300 (1)
13.11 The need for infinity? 301 (3)
14 Lower previsions for essentially bounded 304 (23)
gambles
14.1 Null sets and null gambles 305 (5)
14.2 Null bounded gambles 310 (1)
14.3 Essentially bounded gambles 311 (5)
14.4 Extension of lower and upper 316 (6)
previsions to essentially bounded gambles
14.5 Examples 322 (5)
14.5.1 Linear previsions and probability 322 (1)
charges
14.5.2 Vacuous lower previsions 323 (1)
14.5.3 Lower previsions associated with 323 (1)
proper filters
14.5.4 Limits inferior 324 (1)
14.5.5 Belief functions 325 (1)
14.5.6 Possibility measures 325 (2)
15 Lower previsions for previsible gambles 327 (44)
15.1 Convergence in probability 328 (3)
15.2 Previsibility 331 (9)
15.3 Measurability 340 (3)
15.4 Lebesgue's dominated convergence 343 (5)
theorem
15.5 Previsibility by cuts 348 (2)
15.6 A sufficient condition for 350 (2)
previsibility
15.7 Previsibility for 2-monotone lower 352 (3)
previsions
15.8 Convex combinations 355 (1)
15.9 Lower envelope theorem 355 (3)
15.10 Examples 358 (18)
15.10.1 Linear previsions and probability 358 (1)
charges
15.10.2 Probability density functions: 359 (1)
The normal density
15.10.3 Vacuous lower previsions 360 (1)
15.10.4 Lower previsions associated with 361 (1)
proper filters
15.10.5 Limits inferior 361 (1)
15.10.6 Belief functions 362 (1)
15.10.7 Possibility measures 362 (3)
15.10.8 Estimation 365 (3)
Appendix A Linear spaces, linear lattices and 368 (3)
convexity
Appendix B Notions and results from topology 371 (5)
B.1 Basic definitions 371 (1)
B.2 Metric spaces 372 (1)
B.3 Continuity 373 (1)
B.4 Topological linear spaces 374 (1)
B.5 Extreme points 374 (2)
Appendix C The Choquet integral 376 (15)
C.1 Preliminaries 376 (2)
C.1.1 The improper Riemann integral of a 376 (2)
non-increasing function
C.1.2 Comonotonicity 378 (1)
C.2 Definition of the Choquet integral 378 (1)
C.3 Basic properties of the Choquet integral 379 (8)
C.4 A simple but useful equality 387 (2)
C.5 A simplified version of Greco's 389 (2)
representation theorem
Appendix D The extended real calculus 391 (5)
D.1 Definitions 391 (1)
D.2 Properties 392 (4)
Appendix E Symbols and notation 396 (2)
References 398 (9)
Index 407