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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…mehr
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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.
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.
Produktdetails
- Produktdetails
- Wiley Series in Probability and Statistics
- Verlag: Wiley & Sons
- 1. Auflage
- Seitenzahl: 448
- Erscheinungstermin: Mai 2014
- Englisch
- Abmessung: 234mm x 161mm x 30mm
- Gewicht: 688g
- ISBN-13: 9780470723777
- ISBN-10: 0470723777
- Artikelnr.: 27787700
- Wiley Series in Probability and Statistics
- Verlag: Wiley & Sons
- 1. Auflage
- Seitenzahl: 448
- Erscheinungstermin: Mai 2014
- Englisch
- Abmessung: 234mm x 161mm x 30mm
- Gewicht: 688g
- ISBN-13: 9780470723777
- ISBN-10: 0470723777
- Artikelnr.: 27787700
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
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
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
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