Introduction to Imprecise Probabilities

Thomas Augustin, Department of Statistics, University of Munich, Germany. Frank Coolen, Department of Mathematical Sciences, Durham University, UK. Gert de Cooman, Research Professor in Uncertainty Modelling and Systems Science, Ghent University, Belgium. Matthias Troffaes, Department of Mathematical Sciences, Durham University, UK.

• Introduction xiiiA brief outline of this book xvGuide to the reader xviiContributors xxiAcknowledgements xxvii1 Desirability 1Erik Quaeghebeur1.1 Introduction 11.2 Reasoning about and with sets of desirable gambles 21.2.1 Rationality criteria 21.2.2 Assessments avoiding partial or sure loss 31.2.3 Coherent sets of desirable gambles 41.2.4 Natural extension 51.2.5 Desirability relative to subspaces with arbitrary vector orderings 51.3 Deriving and combining sets of desirable gambles 61.3.1 Gamble space transformations 61.3.2 Derived coherent sets of desirable gambles 71.3.3 Conditional sets of desirable gambles 81.3.4 Marginal sets of desirable gambles 81.3.5 Combining sets of desirable gambles 91.4 Partial preference orders 111.4.1 Strict preference 121.4.2 Nonstrict preference 121.4.3 Nonstrict preferences implied by strict ones 141.4.4 Strict preferences implied by nonstrict ones 151.5 Maximally committal sets of strictly desirable gambles 161.6 Relationships with other, nonequivalent models 181.6.1 Linear previsions 181.6.2 Credal sets 191.6.3 To lower and upper previsions 211.6.4 Simplified variants of desirability 221.6.5 From lower previsions 231.6.6 Conditional lower previsions 251.7 Further reading 26Acknowledgements 272 Lower previsions 28Enrique Miranda and Gert de Cooman2.1 Introduction 282.2 Coherent lower previsions 292.2.1 Avoiding sure loss and coherence 312.2.2 Linear previsions 352.2.3 Sets of desirable gambles 392.2.4 Natural extension 402.3 Conditional lower previsions 422.3.1 Coherence of a finite number of conditional lower previsions 452.3.2 Natural extension of conditional lower previsions 472.3.3 Coherence of an unconditional and a conditional lower prevision 492.3.4 Updating with the regular extension 522.4 Further reading 532.4.1 The work of Williams 532.4.2 The work of Kuznetsov 542.4.3 The work of Weichselberger 54Acknowledgements 553 Structural judgements 56Enrique Miranda and Gert de Cooman3.1 Introduction 563.2 Irrelevance and independence 573.2.1 Epistemic irrelevance 593.2.2 Epistemic independence 613.2.3 Envelopes of independent precise models 633.2.4 Strong independence 653.2.5 The formalist approach to independence 663.3 Invariance 673.3.1 Weak invariance 683.3.2 Strong invariance 693.4 Exchangeability 713.4.1 Representation theorem for finite sequences 723.4.2 Exchangeable natural extension 743.4.3 Exchangeable sequences 753.5 Further reading 773.5.1 Independence 773.5.2 Invariance 773.5.3 Exchangeability 77Acknowledgements 784 Special cases 79Sébastien Destercke and Didier Dubois4.1 Introduction 794.2 Capacities and n-monotonicity 804.3 2-monotone capacities 814.4 Probability intervals on singletons 824.5 ∞-monotone capacities 824.5.1 Constructing ∞-monotone capacities 834.5.2 Simple support functions 834.5.3 Further elements 844.6 Possibility distributions, p-boxes, clouds and related models 844.6.1 Possibility distributions 844.6.2 Fuzzy intervals 864.6.3 Clouds 874.6.4 p-boxes 884.7 Neighbourhood models 894.7.1 Pari-mutuel 894.7.2 Odds-ratio 904.7.3 Linear-vacuous 904.7.4 Relations between neighbourhood models 914.8 Summary 915 Other uncertainty theories based on capacities 93Sébastien Destercke and Didier Dubois5.1 Imprecise probability = modal logic + probability 955.1.1 Boolean possibility theory and modal logic 955.1.2 A unifying framework for capacity based uncertainty theories 975.2 From imprecise probabilities to belief functions and possibility theory 975.2.1 Random disjunctive sets 985.2.2 Numerical possibility theory 1005.2.3 Overall picture 1025.3 Discrepancies between uncertainty theories 1025.3.1 Objectivist vs. Subjectivist standpoints 1035.3.2 Discrepancies in conditioning 1045.3.3 Discrepancies in notions of independence 1075.3.4 Discrepancies in fusion operations 1095.4 Further reading 1126 Game-theoretic probability 114Vladimir Vovk and Glenn Shafer6.1 Introduction 1146.2 A law of large numbers 1156.3 A general forecasting protocol 1186.4 The axiom of continuity 1226.5 Doob’s argument 1246.6 Limit theorems of probability 1276.7 Lévy’s zero-one law 1286.8 The axiom of continuity revisited 1296.9 Further reading 132Acknowledgements 1347 Statistical inference 135Thomas Augustin, Gero Walter, and Frank P. A. Coolen7.1 Background and introduction 1367.1.1 What is statistical inference? 1367.1.2 (Parametric) statistical models and i.i.d. samples 1377.1.3 Basic tasks and procedures of statistical inference 1397.1.4 Some methodological distinctions 1407.1.5 Examples: Multinomial and normal distribution 1417.2 Imprecision in statistics, some general sources and motives 1437.2.1 Model and data imprecision; sensitivity analysis and ontological views on imprecision 1437.2.2 The robustness shock, sensitivity analysis 1447.2.3 Imprecision as a modelling tool to express the quality of partial knowledge 1457.2.4 The law of decreasing credibility 1457.2.5 Imprecise sampling models: Typical models and motives 1467.3 Some basic concepts of statistical models relying on imprecise probabilities 1477.3.1 Most common classes of models and notation 1477.3.2 Imprecise parametric statistical models and corresponding i.i.d. samples 1487.4 Generalized Bayesian inference 1497.4.1 Some selected results from traditional Bayesian statistics 1507.4.2 Sets of precise prior distributions, robust Bayesian inference and the generalized Bayes rule 1547.4.3 A closer exemplary look at a popular class of models: The IDM and other models based on sets of conjugate priors in exponential families 1557.4.4 Some further comments and a brief look at other models for generalized Bayesian inference 1647.5 Frequentist statistics with imprecise probabilities 1657.5.1 The nonrobustness of classical frequentist methods 1667.5.2 (Frequentist) hypothesis testing under imprecise probability: Huber-Strassen theory and extensions 1697.5.3 Towards a frequentist estimation theory under imprecise probabilities – some basic criteria and first results 1717.5.4 A brief outlook on frequentist methods 1747.6 Nonparametric predictive inference 1757.6.1 Overview 1757.6.2 Applications and challenges 1777.7 A brief sketch of some further approaches and aspects 1787.8 Data imprecision, partial identification 1797.8.1 Data imprecision 1807.8.2 Cautious data completion 1817.8.3 Partial identification and observationally equivalent models 1837.8.4 A brief outlook on some further aspects 1867.9 Some general further reading 1877.10 Some general challenges 188Acknowledgements 1898 Decision making 190Nathan Huntley, Robert Hable, and Matthias C. M. Troffaes8.1 Non-sequential decision problems 1908.1.1 Choosing from a set of gambles 1918.1.2 Choice functions for coherent lower previsions 1928.2 Sequential decision problems 1978.2.1 Static sequential solutions: Normal form 1988.2.2 Dynamic sequential solutions: Extensive form 1998.3 Examples and applications 2028.3.1 Ellsberg’s paradox 2028.3.2 Robust Bayesian statistics 2059 Probabilistic graphical models 207Alessandro Antonucci, Cassio P. de Campos, and Marco Zaffalon9.1 Introduction 2079.2 Credal sets 2089.2.1 Definition and relation with lower previsions 2089.2.2 Marginalization and conditioning 2109.2.3 Composition 2129.3 Independence 2139.4 Credal networks 2159.4.1 Nonseparately specified credal networks 2179.5 Computing with credal networks 2209.5.1 Credal networks updating 2209.5.2 Modelling and updating with missing data 2219.5.3 Algorithms for credal networks updating 2239.5.4 Inference on credal networks as a multilinear programming task 2249.6 Further reading 226Acknowledgements 22910 Classification 230Giorgio Corani, Joaquín Abellán, Andrés Masegosa, Serafin Moral, and Marco Zaffalon10.1 Introduction 23010.2 Naive Bayes 23110.2.1 Derivation of naive Bayes 23210.3 Naive credal classifier (NCC) 23310.3.1 Checking Credal-dominance 23310.3.2 Particular behaviours of NCC 23510.3.3 NCC2: Conservative treatment of missing data 23610.4 Extensions and developments of the naive credal classifier 23710.4.1 Lazy naive credal classifier 23710.4.2 Credal model averaging 23810.4.3 Profile-likelihood classifiers 23910.4.4 Tree-augmented networks (TAN) 24010.5 Tree-based credal classifiers 24210.5.1 Uncertainty measures on credal sets: The maximum entropy function 24210.5.2 Obtaining conditional probability intervals with the imprecise Dirichlet model 24510.5.3 Classification procedure 24610.6 Metrics, experiments and software 24910.7 Scoring the conditional probability of the class 25110.7.1 Software 25110.7.2 Experiments 25110.7.3 Experiments comparing conditional probabilities of the class 253Acknowledgements 25711 Stochastic processes 258Filip Hermans and Damjan Škulj11.1 The classical characterization of stochastic processes 25811.1.1 Basic definitions 25811.1.2 Precise Markov chains 25911.2 Event-driven random processes 26111.3 Imprecise Markov chains 26311.3.1 From precise to imprecise Markov chains 26411.3.2 Imprecise Markov models under epistemic irrelevance 26511.3.3 Imprecise Markov models under strong independence 26811.3.4 When does the interpretation of independence (not) matter? 27011.4 Limit behaviour of imprecise Markov chains 27211.4.1 Metric properties of imprecise probability models 27211.4.2 The Perron-Frobenius theorem 27311.4.3 Invariant distributions 27411.4.4 Coefficients of ergodicity 27511.4.5 Coefficients of ergodicity for imprecise Markov chains 27511.5 Further reading 27712 Financial risk measurement 279Paolo Vicig12.1 Introduction 27912.2 Imprecise previsions and betting 28012.3 Imprecise previsions and risk measurement 28212.3.1 Risk measures as imprecise previsions 28312.3.2 Coherent risk measures 28412.3.3 Convex risk measures (and previsions) 28512.4 Further reading 28913 Engineering 291Michael Oberguggenberger13.1 Introduction 29113.2 Probabilistic dimensioning in a simple example 29513.3 Random set modelling of the output variability 29813.4 Sensitivity analysis 30013.5 Hybrid models 30113.6 Reliability analysis and decision making in engineering 30213.7 Further reading 30314 Reliability and risk 305Frank P. A. Coolen and Lev V. Utkin14.1 Introduction 30514.2 Stress-strength reliability 30614.3 Statistical inference in reliability and risk 31014.4 Nonparametric predictive inference in reliability and risk 31214.5 Discussion and research challenges 31715 Elicitation 318Michael Smithson15.1 Methods and issues 31815.2 Evaluating imprecise probability judgements 32215.3 Factors affecting elicitation 32415.4 Matching methods with purposes 32715.5 Further reading 32816 Computation 329Matthias C. M. Troffaes and Robert Hable16.1 Introduction 32916.2 Natural extension 32916.2.1 Conditional lower previsions with arbitrary domains 33016.2.2 The Walley–Pelessoni–Vicig algorithm 33116.2.3 Choquet integration 33216.2.4 Möbius inverse 33416.2.5 Linear-vacuous mixture 33416.3 Decision making 33516.3.1 Γ-maximin, Γ-maximax and Hurwicz 33516.3.2 Maximality 33516.3.3 E-admissibility 33616.3.4 Interval dominance 337References 338Author index 375Subject index 385