Lower previsions / Matthias C.M. Troffaes, Gert de Cooman.

Format:

Book

Author/Creator:

Troffaes, Matthias C. M., 1977- author.

Cooman, Gert de, author.

Series:

Wiley series in probability and statistics.

Wiley Series in Probability and Statistics

Language:

English

Subjects (All):

Probabilities.

Statistical decision.

Games of chance (Mathematics).

Physical Description:

1 online resource (449 p.)

Edition:

1st ed.

Place of Publication:

Chichester, England : Wiley, 2014.

Language Note:

English

Summary:

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)

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 gambles3.4.1 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 justification4.3.2 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 extension4.5.5 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 inferior5.5.4 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

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

1-118-76113-8

1-118-76262-2

1-118-76264-9

OCLC:

869552650