Introduction to imprecise probabilities

"In recent years, the theory has become widely accepted and has been further developed, but a detailed introduction is needed in order to make the material available and accessible to a wide audience. This will be the first book providing such an introduction, covering core theory and recent de...

Descripción completa

Detalles Bibliográficos
Otros Autores:	Augustin, Thomas, editor (editor)
Formato:	Libro electrónico
Idioma:	Inglés
Publicado:	West Sussex, England : John Wiley & Sons 2014.
Edición:	1st edition
Colección:	Wiley series in probability and statistics.
Materias:	Probabilities.
Ver en Biblioteca Universitat Ramon Llull:	https://discovery.url.edu/permalink/34CSUC_URL/1im36ta/alma991009631568006719

Tabla de Contenidos:

Cover
Title Page
Copyright
Contents
Introduction
A brief outline of this book
Guide to the reader
Contributors
Acknowledgements
Chapter 1 Desirability
1.1 Introduction
1.2 Reasoning about and with sets of desirable gambles
1.2.1 Rationality criteria
1.2.2 Assessments avoiding partial or sure loss
1.2.3 Coherent sets of desirable gambles
1.2.4 Natural extension
1.2.5 Desirability relative to subspaces with arbitrary vector orderings
1.3 Deriving and combining sets of desirable gambles
1.3.1 Gamble space transformations
1.3.2 Derived coherent sets of desirable gambles
1.3.3 Conditional sets of desirable gambles
1.3.4 Marginal sets of desirable gambles
1.3.5 Combining sets of desirable gambles
1.4 Partial preference orders
1.4.1 Strict preference
1.4.2 Nonstrict preference
1.4.3 Nonstrict preferences implied by strict ones
1.4.4 Strict preferences implied by nonstrict ones
1.5 Maximally committal sets of strictly desirable gambles
1.6 Relationships with other, nonequivalent models
1.6.1 Linear previsions
1.6.2 Credal sets
1.6.3 To lower and upper previsions
1.6.4 Simplified variants of desirability
1.6.5 From lower previsions
1.6.6 Conditional lower previsions
1.7 Further reading
Acknowledgements
Chapter 2 Lower previsions
2.1 Introduction
2.2 Coherent lower previsions
2.2.1 Avoiding sure loss and coherence
2.2.2 Linear previsions
2.2.3 Sets of desirable gambles
2.2.4 Natural extension
2.3 Conditional lower previsions
2.3.1 Coherence of a finite number of conditional lower previsions
2.3.2 Natural extension of conditional lower previsions
2.3.3 Coherence of an unconditional and a conditional lower prevision
2.3.4 Updating with the regular extension
2.4 Further reading
2.4.1 The work of Williams.
2.4.2 The work of Kuznetsov
2.4.3 The work of Weichselberger
Acknowledgements
Chapter 3 Structural judgements
3.1 Introduction
3.2 Irrelevance and independence
3.2.1 Epistemic irrelevance
3.2.2 Epistemic independence
3.2.3 Envelopes of independent precise models
3.2.4 Strong independence
3.2.5 The formalist approach to independence
3.3 Invariance
3.3.1 Weak invariance
3.3.2 Strong invariance
3.4 Exchangeability
3.4.1 Representation theorem for finite sequences
3.4.2 Exchangeable natural extension
3.4.3 Exchangeable sequences
3.5 Further reading
3.5.1 Independence
3.5.2 Invariance
3.5.3 Exchangeability
Acknowledgements
Chapter 4 Special cases
4.1 Introduction
4.2 Capacities and n-monotonicity
4.3 2-monotone capacities
4.4 Probability intervals on singletons
4.5 ∞-monotone capacities
4.5.1 Constructing ∞-monotone capacities
4.5.2 Simple support functions
4.5.3 Further elements
4.6 Possibility distributions, p-boxes, clouds and related models
4.6.1 Possibility distributions
4.6.2 Fuzzy intervals
4.6.3 Clouds
4.6.4 p-boxes
4.7 Neighbourhood models
4.7.1 Pari-mutuel
4.7.2 Odds-ratio
4.7.3 Linear-vacuous
4.7.4 Relations between neighbourhood models
4.8 Summary
Chapter 5 Other uncertainty theories based on capacities
5.1 Imprecise probability = modal logic + probability
5.1.1 Boolean possibility theory and modal logic
5.1.2 A unifying framework for capacity based uncertainty theories
5.2 From imprecise probabilities to belief functions and possibility theory
5.2.1 Random disjunctive sets
5.2.1.1 Mass functions
5.2.1.2 Belief and plausibility functions
5.2.1.3 Belief functions and lower probabilities
5.2.2 Numerical possibility theory
5.2.2.1 Possibility distributions.
5.2.2.2 Possibility and necessity measures
5.2.3 Overall picture
5.3 Discrepancies between uncertainty theories
5.3.1 Objectivist vs. Subjectivist standpoints
5.3.2 Discrepancies in conditioning
5.3.3 Discrepancies in notions of independence
5.3.4 Discrepancies in fusion operations
5.4 Further reading
Chapter 6 Game-theoretic probability
6.1 Introduction
6.2 A law of large numbers
6.3 A general forecasting protocol
6.4 The axiom of continuity
6.5 Doob's argument
6.6 Limit theorems of probability
6.7 Lévy's zero-one law
6.8 The axiom of continuity revisited
6.9 Further reading
Acknowledgements
Chapter 7 Statistical inference
7.1 Background and introduction
7.1.1 What is statistical inference?
7.1.2 (Parametric) statistical models and i.i.d. samples
7.1.3 Basic tasks and procedures of statistical inference
7.1.4 Some methodological distinctions
7.1.5 Examples: Multinomial and normal distribution
7.2 Imprecision in statistics, some general sources and motives
7.2.1 Model and data imprecision
sensitivity analysis and ontological views on imprecision
7.2.2 The robustness shock, sensitivity analysis
7.2.3 Imprecision as a modelling tool to express the quality of partial knowledge
7.2.4 The law of decreasing credibility
7.2.5 Imprecise sampling models: Typical models and motives
7.3 Some basic concepts of statistical models relying on imprecise probabilities
7.3.1 Most common classes of models and notation
7.3.2 Imprecise parametric statistical models and corresponding i.i.d. samples
7.4 Generalized Bayesian inference
7.4.1 Some selected results from traditional Bayesian statistics
7.4.1.1 Conjugate families of distributions
7.4.2 Sets of precise prior distributions, robust Bayesian inference and the generalized Bayes rule.
7.4.2.1 Robust Bayes and imprecise probabilities
7.4.2.2 Some computational aspects
7.4.3 A closer exemplary look at a popular class of models: The IDM and other models based on sets of conjugate priors in exponential families
7.4.3.1 The general model
7.4.3.2 The IDM and other prior near-ignorance models
7.4.3.3 Substantial prior information and sensitivity to prior-data conflict
7.4.4 Some further comments and a brief look at other models for generalized Bayesian inference
7.5 Frequentist statistics with imprecise probabilities
7.5.1 The nonrobustness of classical frequentist methods
7.5.2 (Frequentist) hypothesis testing under imprecise probability: Huber-Strassen theory and extensions
7.5.2.1 A generalized Neyman-Pearson setting: Basic concepts
7.5.2.2 Globally least favourable pairs and Huber-Strassen theory
7.5.3 Towards a frequentist estimation theory under imprecise probabilities-some basic criteria and first results
7.5.3.1 Basic traditional criteria
7.5.3.2 Generalized unbiasedness
7.5.3.3 Minimum distance estimation
7.5.3.4 Asymptotics
7.5.4 A brief outlook on frequentist methods
7.6 Nonparametric predictive inference
7.6.1 Overview
7.6.2 Applications and challenges
7.7 A brief sketch of some further approaches and aspects
7.8 Data imprecision, partial identification
7.8.1 Data imprecision
7.8.1.1 The basic setting
7.8.1.2 Basic coarsening model
7.8.1.3 Coarse data in traditional statistics
7.8.2 Cautious data completion
7.8.3 Partial identification and observationally equivalent models
7.8.3.1 (Partial) Identifiability, identification regions
7.8.3.2 Partial identification and misclassification
7.8.4 A brief outlook on some further aspects
7.8.4.1 Partial identification and other concepts of imprecise probabilities.
7.8.4.2 A brief outlook on further developments in the literature
7.9 Some general further reading
7.10 Some general challenges
Acknowledgements
Chapter 8 Decision making
8.1 Non-sequential decision problems
8.1.1 Choosing from a set of gambles
8.1.2 Choice functions for coherent lower previsions
8.2 Sequential decision problems
8.2.1 Static sequential solutions: Normal form
8.2.2 Dynamic sequential solutions: Extensive form
8.3 Examples and applications
8.3.1 Ellsberg's paradox
8.3.2 Robust Bayesian statistics
Chapter 9 Probabilistic graphical models
9.1 Introduction
9.2 Credal sets
9.2.1 Definition and relation with lower previsions
9.2.2 Marginalization and conditioning
9.2.3 Composition
9.3 Independence
9.4 Credal networks
9.4.1 Nonseparately specified credal networks
9.5 Computing with credal networks
9.5.1 Credal networks updating
9.5.2 Modelling and updating with missing data
9.5.3 Algorithms for credal networks updating
9.5.4 Inference on credal networks as a multilinear programming task
9.6 Further reading
Acknowledgements
Chapter 10 Classification
10.1 Introduction
10.2 Naive Bayes
10.2.1 Derivation of naive Bayes
10.3 Naive credal classifier (NCC)
10.3.1 Checking Credal-dominance
10.3.2 Particular behaviours of NCC
10.3.3 NCC2: Conservative treatment of missing data
10.4 Extensions and developments of the naive credal classifier
10.4.1 Lazy naive credal classifier
10.4.2 Credal model averaging
10.4.3 Profile-likelihood classifiers
10.4.4 Tree-augmented networks (TAN)
10.5 Tree-based credal classifiers
10.5.1 Uncertainty measures on credal sets: The maximum entropy function
10.5.2 Obtaining conditional probability intervals with the imprecise Dirichlet model
10.5.3 Classification procedure.
10.6 Metrics, experiments and software.

Introduction to imprecise probabilities

Ejemplares similares