Frank P. A. Coolen

Introduction to imprecise probabilities

Preface Introduction Acknowledgements Outline of this Book and Guide to Readers Contributors 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 & 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 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 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 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 1-monotone Capacities 4.5.1 Constructing 1-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 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 5.2.2 Numerical Possibility Theory 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 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 Levy s Zero-One Law. 6.8 The Axiom of Continuity Revisited 6.9 Further Reading 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. 7.4.2 Sets of Precise Prior Distributions, Robust Bayesian Inference and the Generalized Bayes Rule 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. 7.4.4 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 Non-robustness of Classical Frequentist Methods. 7.5.2 (Frequentist) Hypothesis Testing under Imprecise Probability: Huber-Strassen Theory and Extensions 7.5.3 Towards a Frequentist Estimation Theory under Imprecise Probabilities Some Basic Criteria and First Results 7.5.4 A Brief Outlook on Frequentist Methods 7.6 Nonparametric Predictive Inference (NPI) 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 7.8.2 Cautious Data Completion 7.8.3 Partial Identification and Observationally Equivalent Models 7.8.4 A Brief Outlook on Some Further Aspects 7.9 Some General Further Reading 7.10 Some General Challenges 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 9 Probabilistic Graphical Models 9.1 Introduction 9.2 Credal Sets 9.2.1 Definition and Relation with Lower Previsions 9.2.2 Marginalisation and Conditioning 9.2.3 Composition. 9.3 Independence 9.4 Credal Networks 9.4.1 Non-Separately 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 10 Classification 10.1 Introduction 10.2 Naive Bayes 10.3 Naive Credal Classifier (NCC) 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 10.6.1 Software. 10.6.2 Experiments. 11 Stochastic Processes 11.1 The Classical Characterization of Stochastic Processes 11.1.1 Basic Definitions 11.1.2 Precise Markov Chains 11.2 Event-driven Random Processes 11.3 Imprecise Markov Chains 11.3.1 From Precise to Imprecise Markov Chains 11.3.2 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 Chains 11.4.1 Metric Properties of Imprecise Probability Models 11.4.2 The Perron-Frobenius Theorem 11.4.3 Invariant Distributions 11.4.4 Coefficients of Ergodicity 11.4.5 Coefficients of Ergodicity for Imprecise Markov Chains. 11.5 Further Reading 12 Financial Risk Measurement 12.1 Introduction 12.2 Imprecise Previsions and Betting 12.3 Imprecise Previsions and Risk Measurement 12.3.1 Risk Measures as Imprecise Previsions 12.3.2 Coherent Risk Measures 12.3.3 Convex Risk Measures (and Previsions) 12.4 Further Reading 13 Engineering 13.1 Introduction 13.2 Probabilistic Dimensioning in a Simple Example 13.3 Random Set Modelling of the Output Variability 13.4 Sensitivity Analysis 13.5 Hybrid Models. 13.6 Reliability Analysis and Decision Making in Engineering 13.7 Further Reading 14 Reliability and Risk 14.1 Introduction 14.2 Stress-strength Reliability 14.3 Statistical Inference in Reliability and Risk 14.4 NPI in Reliablity and Risk 14.5 Discussion and Research Challenges 15 Elicitation 15.1 Methods and Issues 15.2 Evaluating Imprecise Probability Judgements 15.3 Factors Affecting Elicitation 15.4 Further Reading 16 Computation 16.1 Introduction 16.2 Natural Extension 16.2.1 Conditional Lower Previsions with Arbitrary Domains. 16.2.2 The Walley-Pelessoni-Vicig Algorithm 16.2.3 Choquet Integration 16.2.4 Mobius Inverse 16.2.5 Linear-Vacuous Mixture 16.3 Decision Making 16.3.1 Maximin, Maximax, and Hurwicz 16.3.2 Maximality 16.3.3 E-Admissibility 16.3.4 Interval Dominance References Author index Subject index

On Nonparametric Predictive Inference and Objective Bayesianism

This paper consists of three main parts. First, we give an introduction to Hill’s assumption A(n) and to theory of interval probability, and an overview of recently developed theory and methods for nonparametric predictive inference (NPI), which is based on A(n) and uses interval probability to quantify uncertainty. Thereafter, we illustrate NPI by introducing a variation to the assumption A(n), suitable for inference based on circular data, with applications to several data sets from the literature. This includes attention to comparison of two groups of circular data, and to grouped data. We briefly discuss such inference for multiple future observations. We end the paper with a discussion of NPI and objective Bayesianism.

Imprecise Reliability: An Introductory Overview

A lot of methods and models in classical reliability theory assume that all probabilities are precise, that is, that every probability involved is perfectly determinable. Moreover, it is usually assumed that there exists some complete probabilistic information about the system and component reliability behavior.

Bayesian graphical models for software testing

This paper describes a new approach to the problem of software testing. The approach is based on Bayesian graphical models and presents formal mechanisms for the logical structuring of the software testing problem, the probabilistic and statistical treatment of the uncertainties to be addressed, the test design and analysis process, and the incorporation and implication of test results. Once constructed, the models produced are dynamic representations of the software testing problem. They may be used to drive test design, answer what-if questions, and provide decision support to managers and testers. The models capture the knowledge of the software tester for further use. Experiences of the approach in case studies are briefly discussed.

Nonparametric predictive inference in reliability

Abstract We introduce a recently developed statistical approach, called nonparametric predictive inference (NPI), to reliability. Bounds for the survival function for a future observation are presented. We illustrate how NPI can deal with right-censored data, and discuss aspects of competing risks. We present possible applications of NPI for Bernoulli data, and we briefly outline applications of NPI for replacement decisions. The emphasis is on introduction and illustration of NPI in reliability contexts, detailed mathematical justifications are presented elsewhere.

A nonparametric predictive alternative to the Imprecise Dirichlet Model: The case of a known number of categories

Nonparametric predictive inference (NPI) is a general methodology to learn from data in the absence of prior knowledge and without adding unjustified assumptions. This paper develops NPI for multinomial data when the total number of possible categories for the data is known. We present the upper and lower probabilities for events involving the next observation and several of their properties. We also comment on differences between this NPI approach and corresponding inferences based on Walleys Imprecise Dirichlet Model.

Guidelines for corrective replacement based on low stochastic structure assumptions

This paper presents corrective replacement decisions, e.g. for machines in a production process or other technical systems. In an attempt to base decisions on observed failure times only, some guidelines are provided for replacing failed machines. The method does not provide an optimal strategy in all situations, indicating that sometimes more information or assumptions are needed. The optimal policy indicates how to act if the low assumptions model recommends action. If the model does not strongly indicate an action, more data need to be collected or more sophisticated modelling is needed. Further modelling would require additional assumptions or input from expert judgements, and could be an expensive exercise. A method that gives clear guidelines if the data are strongly indicative may save time and money. This paper presents the model in an elementary form and is intended as a first step towards modelling more realistic maintenance situations.

An imprecise Dirichlet model for Bayesian analysis of failure data including right-censored observations

Abstract This paper is intended to make researchers in reliability theory aware of a recently introduced Bayesian model with imprecise prior distributions for statistical inference on failure data, that can also be considered as a robust Bayesian model. The model consists of a multinomial distribution with Dirichlet priors, making the approach basically nonparametric. New results for the model are presented, related to right-censored observations, where estimation based on this model is closely related to the product-limit estimator, which is an important statistical method to deal with reliability or survival data including right-censored observations. As for the product-limit estimator, the model considered in this paper aims at not using any information other than that provided by observed data, but our model fits into the robust Bayesian context which has the advantage that all inferences can be based on probabilities or expectations, or bounds for probabilities or expectations. The model uses a finite partition of the time-axis, and as such it is also related to life-tables.

Analysis of a 2-phase model for optimization of condition-monitoring intervals

Condition monitoring is a maintenance strategy where decisions are made depending on either continuously or regularly measured equipment states. It reduces uncertainty with respect to actual states of equipment, and can thus avoid unnecessary repair or replacement. However, it involves capital expenditure and/or operational costs to perform measurements. This paper presents a basic model for the economic evaluation and optimization of the interval between successive condition measurements (also called inspections), where measurements are expensive and cannot be made continuously. It assumes that the technique can detect an intermediate state to failure for a failure mode of interest. The influence of competing risks is analyzed, leading to the conclusion that once the cost-effectiveness of the condition-monitoring has been established, competing risks need not be considered in determining the optimum condition monitoring interval. Inspection is cost-effective if the intermediate state has a: (1) nondecreasing hazard rate, and (2) shorter mean residence time than the good state (good-as-new condition), while costs of failure are high enough compared with inspection and repair costs in the intermediate state. Assuming that the distribution of the residence time in the second state is unimodal, estimation of the mean (or scale parameter) and standard deviation of this state, in many cases, provides enough information to make a good decision on the inspection interval. The most important model parameters are identified by sensitivity analyses; it is shown that the model can be simplified without seriously affecting optimal decision making. >

Generalizing the signature to systems with multiple types of components

The concept of signature was introduced to simplify quantification of reliability for coherent systems and networks consisting of a single type of components, and for comparison of such systems’ reliabilities. The signature describes the structure of the system and can be combined with order statistics of the component failure times to derive inferences on the reliability of a system and to compare multiple systems. However, the restriction to use for systems with a single type of component prevents its application to most practical systems. We discuss the difficulty of generalization of the signature to systems with multiple types of components. We present an alternative, called the survival signature, which has similar characteristics and is closely related to the signature. The survival signature provides a feasible generalization to systems with multiple types of components.