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 developments which can be applied to many application areas. All authors of individual chapters are leading researchers on the specific topics, assuring high quality and up-to-date contents.An Introduction to Imprecise Probabilities provides a comprehensive introduction to imprecise probabilities, including theory and applications reflecting the current state if the art. Each chapter is written by experts on the respective topics, including: Sets of desirable gambles; Coherent lower (conditional) previsions; Special cases and links to literature; Decision making; Graphical models; Classification; Reliability and risk assessment; Statistical inference; Structural judgments; Aspects of implementation (including elicitation and computation); Models in finance; Game-theoretic probability; Stochastic processes (including Markov chains); Engineering applications.Essential reading for researchers in academia, research institutes and other organizations, as well as practitioners engaged in areas such as risk analysis and engineering.

PrefaceIntroductionAcknowledgementsOutline of this Book and Guide to ReadersContributors1 Desirability1.1 Introduction1.2 Reasoning about and with Sets of Desirable Gambles1.2.1 Rationality Criteria1.2.2 Assessments Avoiding Partial or Sure Loss1.2.3 Coherent Sets of Desirable Gambles1.2.4 Natural Extension1.2.5 Desirability Relative to Subspaces with Arbitrary Vector Orderings1.3 Deriving & Combining Sets of Desirable Gambles1.3.1 Gamble Space Transformations1.3.2 Derived Coherent Sets of Desirable Gambles1.3.3 Conditional Sets of Desirable Gambles1.3.4 Marginal Sets of Desirable Gambles1.3.5 Combining Sets of Desirable Gambles1.4 Partial Preference Orders1.4.1 Strict Preference1.4.2 Nonstrict Preference1.4.3 Nonstrict Preferences Implied by Strict Ones1.4.4 Strict Preferences Implied by Nonstrict Ones1.5 Maximally Committal Sets of Strictly Desirable Gambles1.6 Relationships with Other, Nonequivalent Models1.6.1 Linear Previsions1.6.2 Credal Sets1.6.3 To Lower and Upper Previsions1.6.4 Simplified Variants of Desirability1.6.5 From Lower Previsions1.6.6 Conditional Lower Previsions1.7 Further Reading2 Lower Previsions2.1 Introduction2.2 Coherent Lower Previsions2.2.1 Avoiding Sure Loss and Coherence2.2.2 Linear Previsions2.2.3 Sets of Desirable Gambles2.2.4 Natural Extension2.3 Conditional Lower Previsions2.3.1 Coherence of a Finite Number of Conditional Lower Previsions2.3.2 Natural Extension of Conditional Lower Previsions2.3.3 Coherence of an Unconditional and a Conditional Lower Prevision2.3.4 Updating with the Regular Extension2.4 Further Reading2.4.1 The Work of Williams2.4.2 The Work of Kuznetsov2.4.3 The Work of Weichselberger3 Structural Judgements3.1 Introduction3.2 Irrelevance and Independence3.2.1 Epistemic Irrelevance3.2.2 Epistemic Independence3.2.3 Envelopes of Independent Precise Models3.2.4 Strong Independence3.2.5 The Formalist Approach to Independence3.3 Invariance3.3.1 Weak Invariance3.3.2 Strong Invariance3.4 Exchangeability.3.4.1 Representation Theorem for Finite Sequences3.4.2 Exchangeable Natural Extension3.4.3 Exchangeable Sequences3.5 Further Reading3.5.1 Independence.3.5.2 Invariance3.5.3 Exchangeability4 Special Cases4.1 Introduction4.2 Capacities and n-monotonicity4.3 2-monotone Capacities4.4 Probability Intervals on Singletons4.5 1-monotone Capacities4.5.1 Constructing 1-monotone Capacities4.5.2 Simple Support Functions4.5.3 Further Elements4.6 Possibility Distributions, p-boxes, Clouds and Related Models.4.6.1 Possibility Distributions4.6.2 Fuzzy Intervals4.6.3 Clouds4.6.4 p-boxes.4.7 Neighbourhood Models4.7.1 Pari-mutuel4.7.2 Odds-ratio4.7.3 Linear-vacuous4.7.4 Relations between Neighbourhood Models4.8 Summary5 Other Uncertainty Theories Based on Capacities5.1 Imprecise Probability = Modal Logic + Probability5.1.1 Boolean Possibility Theory and Modal Logic5.1.2 A Unifying Framework for Capacity Based Uncertainty Theories5.2 From Imprecise Probabilities to Belief Functions and Possibility Theory5.2.1 Random Disjunctive Sets5.2.2 Numerical Possibility Theory5.2.3 Overall Picture5.3 Discrepancies between Uncertainty Theories5.3.1 Objectivist vs. Subjectivist Standpoints5.3.2 Discrepancies in Conditioning5.3.3 Discrepancies in Notions of Independence5.3.4 Discrepancies in Fusion Operations5.4 Further Reading6 Game-Theoretic Probability6.1 Introduction6.2 A Law of Large Numbers6.3 A General Forecasting Protocol6.4 The Axiom of Continuity6.5 Doob's Argument6.6 Limit Theorems of Probability6.7 Lévy's Zero-One Law.6.8 The Axiom of Continuity Revisited6.9 Further Reading7 Statistical Inference7.1 Background and Introduction7.1.1 What is Statistical Inference?7.1.2 (Parametric) Statistical Models and i.i.d. Samples7.1.3 Basic Tasks and Procedures of Statistical Inference7.1.4 Some Methodological Distinctions7.1.5 Examples: Multinomial and Normal Distribution7.2 Imprecision in Statistics, some General Sources and Motives7.2.1 Model and Data Imprecision; Sensitivity Analysis and Ontological Views on Imprecision7.2.2 The Robustness Shock, Sensitivity Analysis7.2.3 Imprecision as a Modelling Tool to Express the Quality of Partial Knowledge7.2.4 The Law of Decreasing Credibility7.2.5 Imprecise Sampling Models: Typical Models and Motives7.3 Some Basic Concepts of Statistical Models Relying on Imprecise Probabilities7.3.1 Most Common Classes of Models and Notation7.3.2 Imprecise Parametric Statistical Models and Corresponding i.i.d. Samples.7.4 Generalized Bayesian Inference7.4.1 Some Selected Results from Traditional Bayesian Statistics.7.4.2 Sets of Precise Prior Distributions, Robust Bayesian Inference and the Generalized Bayes Rule7.4.3 A Closer Exemplary Look at a Popular Class of Models: The IDM and Other Models Based on Sets of Conjugate Priors in Exponential Families.7.4.4 Some Further Comments and a Brief Look at Other Models for Generalized Bayesian Inference7.5 Frequentist Statistics with Imprecise Probabilities7.5.1 The Non-robustness of Classical Frequentist Methods.7.5.2 (Frequentist) Hypothesis Testing under Imprecise Probability: Huber-Strassen Theory and Extensions7.5.3 Towards a Frequentist Estimation Theory under Imprecise Probabilities-- Some Basic Criteria and First Results7.5.4 A Brief Outlook on Frequentist Methods7.6 Nonparametric Predictive Inference (NPI)7.6.1 Overview7.6.2 Applications and Challenges7.7 A Brief Sketch of Some Further Approaches and Aspects7.8 Data Imprecision, Partial Identification7.8.1 Data Imprecision7.8.2 Cautious Data Completion7.8.3 Partial Identification and Observationally Equivalent Models7.8.4 A Brief Outlook on Some Further Aspects7.9 Some General Further Reading7.10 Some General Challenges8 Decision Making8.1 Non-Sequential Decision Problems8.1.1 Choosing From a Set of Gambles8.1.2 Choice Functions for Coherent Lower Previsions8.2 Sequential Decision Problems8.2.1 Static Sequential Solutions: Normal Form8.2.2 Dynamic Sequential Solutions: Extensive Form8.3 Examples and Applications8.3.1 Ellsberg's Paradox8.3.2 Robust Bayesian Statistics9 Probabilistic Graphical Models9.1 Introduction9.2 Credal Sets9.2.1 Definition and Relation with Lower Previsions9.2.2 Marginalisation and Conditioning9.2.3 Composition.9.3 Independence9.4 Credal Networks9.4.1 Non-Separately Specified Credal Networks9.5 Computing with Credal Networks9.5.1 Credal Networks Updating9.5.2 Modelling and Updating with Missing Data9.5.3 Algorithms for Credal Networks Updating9.5.4 Inference on Credal Networks as a Multilinear Programming Task9.6 Further Reading10 Classification10.1 Introduction10.2 Naive Bayes10.3 Naive Credal Classifier (NCC)10.4 Extensions and Developments of the Naive Credal Classifier10.4.1 Lazy Naive Credal Classifier10.4.2 Credal Model Averaging10.4.3 Profile-likelihood Classifiers10.4.4 Tree-Augmented Networks (TAN)10.5 Tree-based Credal Classifiers10.5.1 Uncertainty Measures on Credal Sets. The Maximum Entropy Function.10.5.2 Obtaining Conditional Probability Intervals with the Imprecise Dirichlet Model10.5.3 Classification Procedure10.6 Metrics, Experiments and Software10.6.1 Software.10.6.2 Experiments.11 Stochastic Processes11.1 The Classical Characterization of Stochastic Processes11.1.1 Basic Definitions11.1.2 Precise Markov Chains11.2 Event-driven Random Processes11.3 Imprecise Markov Chains11.3.1 From Precise to Imprecise Markov Chains11.3.2 Imprecise Markov Models under Epistemic Irrelevance.11.3.3 Imprecise Markov Models Under Strong Independence.11.3.4 When Does the Interpretation of Independence (not) Matter?11.4 Limit Behaviour of Imprecise Markov Chains11.4.1 Metric Properties of Imprecise Probability Models11.4.2 The Perron-Frobenius Theorem11.4.3 Invariant Distributions11.4.4 Coefficients of Ergodicity11.4.5 Coefficients of Ergodicity for Imprecise Markov Chains.11.5 Further Reading12 Financial Risk Measurement12.1 Introduction12.2 Imprecise Previsions and Betting12.3 Imprecise Previsions and Risk Measurement12.3.1 Risk Measures as Imprecise Previsions12.3.2 Coherent Risk Measures12.3.3 Convex Risk Measures (and Previsions)12.4 Further Reading13 Engineering13.1 Introduction13.2 Probabilistic Dimensioning in a Simple Example13.3 Random Set Modelling of the Output Variability13.4 Sensitivity Analysis13.5 Hybrid Models.13.6 Reliability Analysis and Decision Making in Engineering13.7 Further Reading14 Reliability and Risk14.1 Introduction14.2 Stress-strength Reliability14.3 Statistical Inference in Reliability and Risk14.4 NPI in Reliablity and Risk14.5 Discussion and Research Challenges15 Elicitation15.1 Methods and Issues15.2 Evaluating Imprecise Probability Judgements15.3 Factors Affecting Elicitation15.4 Further Reading16 Computation16.1 Introduction16.2 Natural Extension16.2.1 Conditional Lower Previsions with Arbitrary Domains.16.2.2 The Walley-Pelessoni-Vicig Algorithm16.2.3 Choquet Integration16.2.4 Möbius Inverse16.2.5 Linear-Vacuous Mixture16.3 Decision Making16.3.1 Maximin, Maximax, and Hurwicz16.3.2 Maximality16.3.3 E-Admissibility16.3.4 Interval DominanceReferencesAuthor indexSubject index