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.
Chapter
1.8 Measurability and simple gambles
Part I Lower Previsions On Bounded Gambles
Chapter 3 Sets of acceptable bounded gambles
3.4 Sets of acceptable bounded gambles
3.4.1 Rationality criteria
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
4.4 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
6.2 n-Monotonicity and coherence
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
7.4 Lower previsions associated with proper filters
7.5 Induced lower previsions
7.5.2 Induced lower previsions
7.5.3 Properties of induced lower previsions
7.6 Special cases of induced lower previsions
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
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
8.6 The Lebesgue integral
8.8 Consequences of duality
Chapter 9 Examples of linear extension
9.1 Distribution functions
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
10.1.2 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
11.2.2 Linear functionals
11.2.3 Monotone functionals
11.2.4 n-Monotone functionals
11.2.5 Coherent lower previsions
11.3 Properties of inequality preserving functionals
11.4 Infinite non-negative linear combinations of inequality preserving functionals
11.5 Representation results
11.6 Lower previsions associated with proper filters
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 13 Conditional lower previsions
13.2 Sets of acceptable gambles
13.2.1 Rationality criteria
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
13.6 Properties of coherent conditional lower previsions
13.6.1 Interesting consequences of coherence
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
13.10 Extending a lower prevision from bounded gambles to conditional gambles
13.10.2 Linear previsions and probability charges
13.10.3 Vacuous lower previsions
13.10.4 Lower previsions associated with proper filters
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
14.5.1 Linear previsions and probability charges
14.5.2 Vacuous lower previsions
14.5.3 Lower previsions associated with proper filters
14.5.6 Possibility measures
Chapter 15 Lower previsions for previsible gambles
15.1 Convergence in probability
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
15.9 Lower envelope theorem
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
15.10.7 Possibility measures
Appendix A Linear spaces, linear lattices and convexity
Appendix B Notions and results from topology
B.4 Topological linear spaces
Appendix C The Choquet integral
C.1.1 The improper Riemann integral of a non-increasing function
C.2 Definition of the Choquet integral
C.3 Basic properties of the Choquet integral
C.4 A simple but useful equality
C.5 A simplified version of Greco's representation theorem
Appendix D The extended real calculus
Wiley Series in Probability and Statistics
Lower Previsions
( Wiley Series in Probability and Statistics )
Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.
