Lower Previsions.
Title:
Lower Previsions.
Author:
Troffaes, Matthias C. M.
ISBN:
9781118762646
Personal Author:
Edition:
1st ed.
Physical Description:
1 online resource (449 pages)
Series:
Wiley Series in Probability and Statistics
Contents:
Cover -- Title Page -- Copyright -- Contents -- Preface -- Acknowledgements -- Chapter 1 Preliminary notions and definitions -- 1.1 Sets of numbers -- 1.2 Gambles -- 1.3 Subsets and their indicators -- 1.4 Collections of events -- 1.5 Directed sets and Moore-Smith limits -- 1.6 Uniform convergence of bounded gambles -- 1.7 Set functions, charges and measures -- 1.8 Measurability and simple gambles -- 1.9 Real functionals -- 1.10 A useful lemma -- Part I Lower Previsions On Bounded Gambles -- Chapter 2 Introduction -- Chapter 3 Sets of acceptable bounded gambles -- 3.1 Random variables -- 3.2 Belief and behaviour -- 3.3 Bounded gambles -- 3.4 Sets of acceptable bounded gambles -- 3.4.1 Rationality criteria -- 3.4.2 Inference -- Chapter 4 Lower previsions -- 4.1 Lower and upper previsions -- 4.1.1 From sets of acceptable bounded gambles to lower previsions -- 4.1.2 Lower and upper previsions directly -- 4.2 Consistency for lower previsions -- 4.2.1 Definition and justification -- 4.2.2 A more direct justification for the avoiding sure loss condition -- 4.2.3 Avoiding sure loss and avoiding partial loss -- 4.2.4 Illustrating the avoiding sure loss condition -- 4.2.5 Consequences of avoiding sure loss -- 4.3 Coherence for lower previsions -- 4.3.1 Definition and justification -- 4.3.2 A more direct justification for the coherence condition -- 4.3.3 Illustrating the coherence condition -- 4.3.4 Linear previsions -- 4.4 Properties of coherent lower previsions -- 4.4.1 Interesting consequences of coherence -- 4.4.2 Coherence and conjugacy -- 4.4.3 Easier ways to prove coherence -- 4.4.4 Coherence and monotone convergence -- 4.4.5 Coherence and a seminorm -- 4.5 The natural extension of a lower prevision -- 4.5.1 Natural extension as least-committal extension -- 4.5.2 Natural extension and equivalence.
4.5.3 Natural extension to a specific domain -- 4.5.4 Transitivity of natural extension -- 4.5.5 Natural extension and avoiding sure loss -- 4.5.6 Simpler ways of calculating the natural extension -- 4.6 Alternative characterisations for avoiding sure loss, coherence, and natural extension -- 4.7 Topological considerations -- Chapter 5 Special coherent lower previsions -- 5.1 Linear previsions on finite spaces -- 5.2 Coherent lower previsions on finite spaces -- 5.3 Limits as linear previsions -- 5.4 Vacuous lower previsions -- 5.5 {0,1}-valued lower probabilities -- 5.5.1 Coherence and natural extension -- 5.5.2 The link with classical propositional logic -- 5.5.3 The link with limits inferior -- 5.5.4 Monotone convergence -- 5.5.5 Lower oscillations and neighbourhood filters -- 5.5.6 Extending a lower prevision defined on all continuous bounded gambles -- Chapter 6 n-Monotone lower previsions -- 6.1 n-Monotonicity -- 6.2 n-Monotonicity and coherence -- 6.2.1 A few observations -- 6.2.2 Results for lower probabilities -- 6.3 Representation results -- Chapter 7 Special n-monotone coherent lower previsions -- 7.1 Lower and upper mass functions -- 7.2 Minimum preserving lower previsions -- 7.2.1 Definition and properties -- 7.2.2 Vacuous lower previsions -- 7.3 Belief functions -- 7.4 Lower previsions associated with proper filters -- 7.5 Induced lower previsions -- 7.5.1 Motivation -- 7.5.2 Induced lower previsions -- 7.5.3 Properties of induced lower previsions -- 7.6 Special cases of induced lower previsions -- 7.6.1 Belief functions -- 7.6.2 Refining the set of possible values for a random variable -- 7.7 Assessments on chains of sets -- 7.8 Possibility and necessity measures -- 7.9 Distribution functions and probability boxes -- 7.9.1 Distribution functions -- 7.9.2 Probability boxes.
Chapter 8 Linear previsions, integration and duality -- 8.1 Linear extension and integration -- 8.2 Integration of probability charges -- 8.3 Inner and outer set function, completion and other extensions -- 8.4 Linear previsions and probability charges -- 8.5 The S-integral -- 8.6 The Lebesgue integral -- 8.7 The Dunford integral -- 8.8 Consequences of duality -- Chapter 9 Examples of linear extension -- 9.1 Distribution functions -- 9.2 Limits inferior -- 9.3 Lower and upper oscillations -- 9.4 Linear extension of a probability measure -- 9.5 Extending a linear prevision from continuous bounded gambles -- 9.6 Induced lower previsions and random sets -- Chapter 10 Lower previsions and symmetry -- 10.1 Invariance for lower previsions -- 10.1.1 Definition -- 10.1.2 Existence of invariant lower previsions -- 10.1.3 Existence of strongly invariant lower previsions -- 10.2 An important special case -- 10.3 Interesting examples -- 10.3.1 Permutation invariance on finite spaces -- 10.3.2 Shift invariance and Banach limits -- 10.3.3 Stationary random processes -- Chapter 11 Extreme lower previsions -- 11.1 Preliminary results concerning real functionals -- 11.2 Inequality preserving functionals -- 11.2.1 Definition -- 11.2.2 Linear functionals -- 11.2.3 Monotone functionals -- 11.2.4 n-Monotone functionals -- 11.2.5 Coherent lower previsions -- 11.2.6 Combinations -- 11.3 Properties of inequality preserving functionals -- 11.4 Infinite non-negative linear combinations of inequality preserving functionals -- 11.4.1 Definition -- 11.4.2 Examples -- 11.4.3 Main result -- 11.5 Representation results -- 11.6 Lower previsions associated with proper filters -- 11.6.1 Belief functions -- 11.6.2 Possibility measures.
11.6.3 Extending a linear prevision defined on all continuous bounded gambles -- 11.6.4 The connection with induced lower previsions -- 11.7 Strongly invariant coherent lower previsions -- Part II Extending the Theory to Unbounded Gambles -- Chapter 12 Introduction -- Chapter 13 Conditional lower previsions -- 13.1 Gambles -- 13.2 Sets of acceptable gambles -- 13.2.1 Rationality criteria -- 13.2.2 Inference -- 13.3 Conditional lower previsions -- 13.3.1 Going from sets of acceptable gambles to conditional lower previsions -- 13.3.2 Conditional lower previsions directly -- 13.4 Consistency for conditional lower previsions -- 13.4.1 Definition and justification -- 13.4.2 Avoiding sure loss and avoiding partial loss -- 13.4.3 Compatibility with the definition for lower previsions on bounded gambles -- 13.4.4 Comparison with avoiding sure loss for lower previsions on bounded gambles -- 13.5 Coherence for conditional lower previsions -- 13.5.1 Definition and justification -- 13.5.2 Compatibility with the definition for lower previsions on bounded gambles -- 13.5.3 Comparison with coherence for lower previsions on bounded gambles -- 13.5.4 Linear previsions -- 13.6 Properties of coherent conditional lower previsions -- 13.6.1 Interesting consequences of coherence -- 13.6.2 Trivial extension -- 13.6.3 Easier ways to prove coherence -- 13.6.4 Separate coherence -- 13.7 The natural extension of a conditional lower prevision -- 13.7.1 Natural extension as least-committal extension -- 13.7.2 Natural extension and equivalence -- 13.7.3 Natural extension to a specific domain and the transitivity of natural extension -- 13.7.4 Natural extension and avoiding sure loss -- 13.7.5 Simpler ways of calculating the natural extension.
13.7.6 Compatibility with the definition for lower previsions on bounded gambles -- 13.8 Alternative characterisations for avoiding sure loss, coherence and natural extension -- 13.9 Marginal extension -- 13.10 Extending a lower prevision from bounded gambles to conditional gambles -- 13.10.1 General case -- 13.10.2 Linear previsions and probability charges -- 13.10.3 Vacuous lower previsions -- 13.10.4 Lower previsions associated with proper filters -- 13.10.5 Limits inferior -- 13.11 The need for infinity? -- Chapter 14 Lower previsions for essentially bounded gambles -- 14.1 Null sets and null gambles -- 14.2 Null bounded gambles -- 14.3 Essentially bounded gambles -- 14.4 Extension of lower and upper previsions to essentially bounded gambles -- 14.5 Examples -- 14.5.1 Linear previsions and probability charges -- 14.5.2 Vacuous lower previsions -- 14.5.3 Lower previsions associated with proper filters -- 14.5.4 Limits inferior -- 14.5.5 Belief functions -- 14.5.6 Possibility measures -- Chapter 15 Lower previsions for previsible gambles -- 15.1 Convergence in probability -- 15.2 Previsibility -- 15.3 Measurability -- 15.4 Lebesgue's dominated convergence theorem -- 15.5 Previsibility by cuts -- 15.6 A sufficient condition for previsibility -- 15.7 Previsibility for 2-monotone lower previsions -- 15.8 Convex combinations -- 15.9 Lower envelope theorem -- 15.10 Examples -- 15.10.1 Linear previsions and probability charges -- 15.10.2 Probability density functions: The normal density -- 15.10.3 Vacuous lower previsions -- 15.10.4 Lower previsions associated with proper filters -- 15.10.5 Limits inferior -- 15.10.6 Belief functions -- 15.10.7 Possibility measures -- 15.10.8 Estimation -- Appendix A Linear spaces, linear lattices and convexity -- Appendix B Notions and results from topology -- B.1 Basic definitions.
B.2 Metric spaces.
Abstract:
This book has two main purposes. On the one hand, it provides aconcise and systematic development of the theory of lower previsions,based on the concept of acceptability, in spirit of the work ofWilliams and Walley. On the other hand, it also extends this theory todeal with unbounded quantities, which abound in practicalapplications. Following Williams, we start out with sets of acceptable gambles. Fromthose, we derive rationality criteria---avoiding sure loss andcoherence---and inference methods---natural extension---for(unconditional) lower previsions. We then proceed to study variousaspects of the resulting theory, including the concept of expectation(linear previsions), limits, vacuous models, classical propositionallogic, lower oscillations, and monotone convergence. We discussn-monotonicity for lower previsions, and relate lower previsions withChoquet integration, belief functions, random sets, possibilitymeasures, various integrals, symmetry, and representation theoremsbased on the Bishop-De Leeuw theorem. Next, we extend the framework of sets of acceptable gambles to consideralso unbounded quantities. As before, we again derive rationalitycriteria and inference methods for lower previsions, this time alsoallowing for conditioning. We apply this theory to constructextensions of lower previsions from bounded random quantities to alarger set of random quantities, based on ideas borrowed from thetheory of Dunford integration. A first step is to extend a lower prevision to random quantities thatare bounded on the complement of a null set (essentially boundedrandom quantities). This extension is achieved by a natural extensionprocedure that can be motivated by a rationality axiom stating thatadding null random quantities does not affect acceptability. In a further step, we approximate unbounded random quantities by asequences of
bounded ones, and, in essence, we identify those forwhich the induced lower prevision limit does not depend on the detailsof the approximation. We call those random quantities 'previsible'. Westudy previsibility by cut sequences, and arrive at a simplesufficient condition. For the 2-monotone case, we establish a Choquetintegral representation for the extension. For the general case, weprove that the extension can always be written as an envelope ofDunford integrals. We end with some examples of the theory.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
Genre:
Added Author:
Electronic Access:
Click to View