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Previous article Next article Non-Parametric Estimation of a Multivariate Probability DensityV. A. EpanechnikovV. A. Epanechnikovhttps://doi.org/10.1137/1114019PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Emanuel Parzen, On estimation of a probability density function and mode, Ann. Math. Statist., 33 (1962), 1065–1076 MR0143282 0116.11302 CrossrefGoogle Scholar[2] Murray Rosenblatt, Remarks on some nonparametric estimates of a density function, Ann. Math. Statist., 27 (1956), 832–837 MR0079873 0073.14602 CrossrefGoogle Scholar[3] G. M. Manija, Remarks on non-parametric estimates of a two-dimensional density function, Soobšč. Akad. Nauk Gruzin. SSR, 27 (1961), 385–390 MR0143303 Google Scholar[4] E. A. Nadaraya, Estimation of a bivariate probability density, Soobshch. Akad. Nauk Gruz. SSR, 36 (1964), 267–268 Google Scholar[5] R. E. Bellman, , I. Glicksberg and , O. A. Gross, Some aspects of the mathematical theory of control processes, Rand Corporation, Santa Monica, Calif., Rep. No. R-313, 1958rm xix+244 MR0094281 0086.11703 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Kernel-based learning of birth process from evolving spatiotemporal RFS data stream in SMC CPHD filter for multi-target trackingSignal Processing, Vol. 203 Cross Ref Assessing spatial connectivity effects on daily streamflow forecasting using Bayesian-based graph neural networkScience of The Total Environment, Vol. 855 Cross Ref Joint Source-Channel Decoding of Polar Codes for HEVC-Based Video StreamingACM Transactions on Multimedia Computing, Communications, and Applications, Vol. 18, No. 4 Cross Ref A novel structure adaptive new information priority discrete grey prediction model and its application in renewable energy generation forecastingApplied Energy, Vol. 325 Cross Ref A novel structure adaptive fractional discrete grey forecasting model and its application in China’s crude oil production predictionExpert Systems with Applications, Vol. 207 Cross Ref Age-varying effects of repeated emergency department presentations for children in Canada6 May 2022 | Journal of Health Services Research & Policy, Vol. 27, No. 4 Cross Ref Probabilistic adaptive power pinch analysis for islanded hybrid energy storage systemsJournal of Energy Storage, Vol. 54 Cross Ref Should I stay or should I fly? 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Preface.Acknowledgments.1 Introduction.1.1 Why Spatial Data in Public Health?1.2 Why Statistical Methods for Spatial Data?1.3 Intersection of Three Fields of Study.1.4 Organization of the Book.2 Analyzing Public Health Data.2.1 Observational vs. Experimental Data.2.2 Risk and Rates.2.2.1 Incidence and Prevalence.2.2.2 Risk.2.2.3 Estimating Risk: Rates and Proportions.2.2.4 Relative and Attributable Risks.2.3 Making Rates Comparable: Standardized Rates.2.3.1 Direct Standardization.2.3.2 Indirect Standardization.2.3.3 Direct or Indirect?2.3.4 Standardizing to What Standard?2.3.5 Cautions with Standardized Rates.2.4 Basic Epidemiological Study Designs.2.4.1 Prospective Cohort Studies.2.4.2 Retrospective Case-Control Studies.2.4.3 Other Types of Epidemiological Studies.2.5 Basic Analytic Tool: The Odds Ratio.2.6 Modeling Counts and Rates.2.6.1 Generalized Linear Models.2.6.2 Logistic Regression.2.6.3 Poisson Regression.2.7 Challenges in the Analysis of Observational Data.2.7.1 Bias.2.7.2 Confounding.2.7.3 Effect Modification.2.7.4 Ecological Inference and the Ecological Fallacy.2.8 Additional Topics and Further Reading.2.9 Exercises.3 Spatial Data.3.1 Components of Spatial Data.3.2 An Odyssey into Geodesy.3.2.1 Measuring Location: Geographical Coordinates.3.2.2 Flattening the Globe: Map Projections and Coordinate Systems.3.2.3 Mathematics of Location: Vector and Polygon Geometry.3.3 Sources of Spatial Data.3.3.1 Health Data.3.3.2 Census-Related Data.3.3.3 Geocoding.3.3.4 Digital Cartographic Data.3.3.5 Environmental and Natural Resource Data.3.3.6 Remotely Sensed Data.3.3.7 Digitizing.3.3.8 Collect Your Own!3.4 Geographic Information Systems.3.4.1 Vector and Raster GISs.3.4.2 Basic GIS Operations.3.4.3 Spatial Analysis within GIS.3.5 Problems with Spatial Data and GIS.3.5.1 Inaccurate and Incomplete Databases.3.5.2 Confidentiality.3.5.3 Use of ZIP Codes.3.5.4 Geocoding Issues.3.5.5 Location Uncertainty.4 Visualizing Spatial Data.4.1 Cartography: The Art and Science of Mapmaking.4.2 Types of Statistical Maps.MAP STUDY: Very Low Birth Weights in Georgia Health Care District 9.4.2.1 Maps for Point Features.4.2.2 Maps for Areal Features.4.3 Symbolization.4.3.1 Map Generalization.4.3.2 Visual Variables.4.3.3 Color.4.4 Mapping Smoothed Rates and Probabilities.4.4.1 Locally Weighted Averages.4.4.2 Nonparametric Regression.4.4.3 Empirical Bayes Smoothing.4.4.4 Probability Mapping.4.4.5 Practical Notes and Recommendations.CASE STUDY: Smoothing New York Leukemia Data.4.5 Modifiable Areal Unit Problem.4.6 Additional Topics and Further Reading.4.6.1 Visualization.4.6.2 Additional Types of Maps.4.6.3 Exploratory Spatial Data Analysis.4.6.4 Other Smoothing Approaches.4.6.5 Edge Effects.4.7 Exercises.5 Analysis of Spatial Point Patterns.5.1 Types of Patterns.5.2 Spatial Point Processes.5.2.1 Stationarity and Isotropy.5.2.2 Spatial Poisson Processes and CSR.5.2.3 Hypothesis Tests of CSR via Monte Carlo Methods.5.2.4 Heterogeneous Poisson Processes.5.2.5 Estimating Intensity Functions.DATA BREAK: Early Medieval Grave Sites.5.3 K Function.5.3.1 Estimating the K Function.5.3.2 Diagnostic Plots Based on the K Function.5.3.3 Monte Carlo Assessments of CSR Based on the K Function.DATA BREAK: Early Medieval Grave Sites.5.3.4 Roles of First- and Second-Order Properties.5.4 Other Spatial Point Processes.5.4.1 Poisson Cluster Processes.5.4.2 Contagion/Inhibition Processes.5.4.3 Cox Processes.5.4.4 Distinguishing Processes.5.5 Additional Topics and Further Reading.5.6 Exercises.6 Spatial Clusters of Health Events: Point Data for Cases and Controls.6.1 What Do We Have? Data Types and Related Issues.6.2 What Do We Want? Null and Alternative Hypotheses.6.3 Categorization of Methods.6.4 Comparing Point Process Summaries.6.4.1 Goals.6.4.2 Assumptions and Typical Output.6.4.3 Method: Ratio of Kernel Intensity Estimates.DATA BREAK: Early Medieval Grave Sites.6.4.4 Method: Difference between K Functions.DATA BREAK: Early Medieval Grave Sites.6.5 Scanning Local Rates.6.5.1 Goals.6.5.2 Assumptions and Typical Output.6.5.3 Method: Geographical Analysis Machine.6.5.4 Method: Overlapping Local Case Proportions.DATA BREAK: Early Medieval Grave Sites.6.5.5 Method: Spatial Scan Statistics.DATA BREAK: Early Medieval Grave Sites.6.6 Nearest-Neighbor Statistics.6.6.1 Goals.6.6.2 Assumptions and Typical Output.6.6.3 Method: q Nearest Neighbors of Cases.CASE STUDY: San Diego Asthma.6.7 Further Reading.6.8 Exercises.7 Spatial Clustering of Health Events: Regional Count Data.7.1 What Do We Have and What Do We Want?7.1.1 Data Structure.7.1.2 Null Hypotheses.7.1.3 Alternative Hypotheses.7.2 Categorization of Methods.7.3 Scanning Local Rates.7.3.1 Goals.7.3.2 Assumptions.7.3.3 Method: Overlapping Local Rates.DATA BREAK: New York Leukemia Data.7.3.4 Method: Turnbull et al.'s CEPP.7.3.5 Method: Besag and Newell Approach.7.3.6 Method: Spatial Scan Statistics.7.4 Global Indexes of Spatial Autocorrelation.7.4.1 Goals.7.4.2 Assumptions and Typical Output.7.4.3 Method: Moran's I .7.4.4 Method: Geary's c.7.5 Local Indicators of Spatial Association.7.5.1 Goals.7.5.2 Assumptions and Typical Output.7.5.3 Method: Local Moran's I.7.6 Goodness-of-Fit Statistics.7.6.1 Goals.7.6.2 Assumptions and Typical Output.7.6.3 Method: Pearson's chi2.7.6.4 Method: Tango's Index.7.6.5 Method: Focused Score Tests of Trend.7.7 Statistical Power and Related Considerations.7.7.1 Power Depends on the Alternative Hypothesis.7.7.2 Power Depends on the Data Structure.7.7.3 Theoretical Assessment of Power.7.7.4 Monte Carlo Assessment of Power.7.7.5 Benchmark Data and Conditional Power Assessments.7.8 Additional Topics and Further Reading.7.8.1 Related Research Regarding Indexes of Spatial Association.7.8.2 Additional Approaches for Detecting Clusters and/or Clustering.7.8.3 Space-Time Clustering and Disease Surveillance.7.9 Exercises.8 Spatial Exposure Data.8.1 Random Fields and Stationarity.8.2 Semivariograms.8.2.1 Relationship to Covariance Function and Correlogram.8.2.2 Parametric Isotropic Semivariogram Models.8.2.3 Estimating the Semivariogram.DATA BREAK: Smoky Mountain pH Data.8.2.4 Fitting Semivariogram Models.8.2.5 Anisotropic Semivariogram Modeling.8.3 Interpolation and Spatial Prediction.8.3.1 Inverse-Distance Interpolation.8.3.2 Kriging.CASE STUDY: Hazardous Waste Site Remediation.8.4 Additional Topics and Further Reading.8.4.1 Erratic Experimental Semivariograms.8.4.2 Sampling Distribution of the Classical Semivariogram Estimator.8.4.3 Nonparametric Semivariogram Models.8.4.4 Kriging Non-Gaussian Data.8.4.5 Geostatistical Simulation.8.4.6 Use of Non-Euclidean Distances in Geostatistics.8.4.7 Spatial Sampling and Network Design.8.5 Exercises.9 Linking Spatial Exposure Data to Health Events.9.1 Linear Regression Models for Independent Data.9.1.1 Estimation and Inference.9.1.2 Interpretation and Use with Spatial Data.DATA BREAK: Raccoon Rabies in Connecticut.9.2 Linear Regression Models for Spatially Autocorrelated Data.9.2.1 Estimation and Inference.9.2.2 Interpretation and Use with Spatial Data.9.2.3 Predicting New Observations: Universal Kriging.DATA BREAK: New York Leukemia Data.9.3 Spatial Autoregressive Models.9.3.1 Simultaneous Autoregressive Models.9.3.2 Conditional Autoregressive Models.9.3.3 Concluding Remarks on Conditional Autoregressions.9.3.4 Concluding Remarks on Spatial Autoregressions.9.4 Generalized Linear Models.9.4.1 Fixed Effects and the Marginal Specification.9.4.2 Mixed Models and Conditional Specification.9.4.3 Estimation in Spatial GLMs and GLMMs.DATA BREAK: Modeling Lip Cancer Morbidity in Scotland.9.4.4 Additional Considerations in Spatial GLMs.CASE STUDY: Very Low Birth Weights in Georgia Health Care District 9.9.5 Bayesian Models for Disease Mapping.9.5.1 Hierarchical Structure.9.5.2 Estimation and Inference.9.5.3 Interpretation and Use with Spatial Data.9.6 Parting Thoughts.9.7 Additional Topics and Further Reading.9.7.1 General References.9.7.2 Restricted Maximum Likelihood Estimation.9.7.3 Residual Analysis with Spatially Correlated Error Terms.9.7.4 Two-Parameter Autoregressive Models.9.7.5 Non-Gaussian Spatial Autoregressive Models.9.7.6 Classical/Bayesian GLMMs.9.7.7 Prediction with GLMs.9.7.8 Bayesian Hierarchical Models for Spatial Data.9.8 Exercises.References.Author Index.Subject Index.

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

Preface. Part I. 1 Introduction - uncertainty and risk in geotechnical engineering. 1.1 Offshore platforms. 1.2 Pit mine slopes. 1.3 Balancing risk and reliability in a geotechnical design. 1.4 Historical development of reliability methods in civil engineering. 1.5 Some terminological and philosophical issues. 1.6 The organization of this book. 1.7 A comment on notation and nomenclature. 2 Uncertainty. 2.1 Randomness, uncertainty, and the world. 2.2 Modeling uncertainties in risk and reliability analysis. 2.3 Probability. 3 Probability. 3.1 Histograms and frequency diagrams. 3.2 Summary statistics. 3.3 Probability theory. 3.4 Random variables. 3.5 Random process models. 3.6 Fitting mathematical pdf models to data. 3.7 Covariance among variables. 4 Inference. 4.1 Frequentist theory. 4.2 Bayesian theory. 4.3 Prior probabilities. 4.4 Inferences from sampling. 4.5 Regression analysis. 4.6 Hypothesis tests. 4.7 Choice among models. 5 Risk, decisions and judgment. 5.1 Risk. 5.2 Optimizing decisions. 5.3 Non-optimizing decisions. 5.4 Engineering judgment. Part II. 6 Site characterization. 6.1 Developments in site characterization. 6.2 Analytical approaches to site characterization. 6.3 Modeling site characterization activities. 6.4 Some pitfalls of intuitive data evaluation. 6.5 Organization of Part II. 7 Classification and mapping. 7.1 Mapping discrete variables. 7.2 Classification. 7.3 Discriminant analysis. 7.4 Mapping. 7.5 Carrying out a discriminant or logistic analysis. 8 Soil variability. 8.1 Soil properties. 8.2 Index tests and classification of soils. 8.3 Consolidation properties. 8.4 Permeability. 8.5 Strength properties. 8.6 Distributional properties. 8.7 Measurement error. 9 Spatial variability within homogeneous deposits. 9.1 Trends and variations about trends. 9.2 Residual variations. 9.3 Estimating autocorrelation and autocovariance. 9.4 Variograms and geostatistics. Appendix: algorithm for maximizing log-likelihood of autocovariance. 10 Random field theory. 10.1 Stationary processes. 10.2 Mathematical properties of autocovariance functions. 10.3 Multivariate (vector) random fields. 10.4 Gaussian random fields. 10.5 Functions of random fields. 11 Spatial sampling. 11.1 Concepts of sampling. 11.2 Common spatial sampling plans. 11.3 Interpolating random fields. 11.4 Sampling for autocorrelation. 12 Search theory. 12.1 Brief history of search theory. 12.2 Logic of a search process. 12.3 Single stage search. 12.4 Grid search. 12.5 Inferring target characteristics. 12.6 Optimal search. 12.7 Sequential search. Part III. 13 Reliability analysis and error propagation. 13.1 Loads, resistances and reliability. 13.2 Results for different distributions of the performance function. 13.3 Steps and approximations in reliability analysis. 13.4 Error propagation - statistical moments of the performance function. 13.5 Solution techniques for practical cases. 13.6 A simple conceptual model of practical significance. 14 First order second moment (FOSM) methods. 14.1 The James Bay dikes. 14.2 Uncertainty in geotechnical parameters. 14.3 FOSM calculations. 14.4 Extrapolations and consequences. 14.5 Conclusions from the James Bay study. 14.6 Final comments. 15 Point estimate methods. 15.1 Mathematical background. 15.2 Rosenblueth's cases and notation. 15.3 Numerical results for simple cases. 15.4 Relation to orthogonal polynomial quadrature. 15.5 Relation with 'Gauss points' in the finite element method. 15.6 Limitations of orthogonal polynomial quadrature. 15.7 Accuracy, or when to use the point-estimate method. 15.8 The problem of the number of computation points. 15.9 Final comments and conclusions. 16 The Hasofer-Lind approach (FORM). 16.1 Justification for improvement - vertical cut in cohesive soil. 16.2 The Hasofer-Lind formulation. 16.3 Linear or non-linear failure criteria and uncorrelated variables. 16.4 Higher order reliability. 16.5 Correlated variables. 16.6 Non-normal variables. 17 Monte Carlo simulation methods. 17.1 Basic considerations. 17.2 Computer programming considerations. 17.3 Simulation of random processes. 17.4 Variance reduction methods. 17.5 Summary. 18 Load and resistance factor design. 18.1 Limit state design and code development. 18.2 Load and resistance factor design. 18.3 Foundation design based on LRFD. 18.4 Concluding remarks. 19 Stochastic finite elements. 19.1 Elementary finite element issues. 19.2 Correlated properties. 19.3 Explicit formulation. 19.4 Monte Carlo study of differential settlement. 19.5 Summary and conclusions. Part IV. 20 Event tree analysis. 20.1 Systems failure. 20.2 Influence diagrams. 20.3 Constructing event trees. 20.4 Branch probabilities. 20.5 Levee example revisited. 21 Expert opinion. 21.1 Expert opinion in geotechnical practice. 21.2 How do people estimate subjective probabilities? 21.3 How well do people estimate subjective probabilities? 21.4 Can people learn to be well-calibrated? 21.5 Protocol for assessing subjective probabilities. 21.6 Conducting a process to elicit quantified judgment. 21.7 Practical suggestions and techniques. 21.8 Summary. 22 System reliability assessment. 22.1 Concepts of system reliability. 22.2 Dependencies among component failures. 22.3 Event tree representations. 22.4 Fault tree representations. 22.5 Simulation approach to system reliability. 22.6 Combined approaches. 22.7 Summary. Appendix A: A primer on probability theory. A.1 Notation and axioms. A.2 Elementary results. A.3 Total probability and Bayes' theorem. A.4 Discrete distributions. A.5 Continuous distributions. A.6 Multiple variables. A.7 Functions of random variables. References. Index.

Foreword.Preface.PART ONE. SURVEY OF PROBABILITY THEORY.Chapter 1. Introduction.Chapter 2. Experiments, Sample Spaces, and Probability.2.1 Experiments and Sample Spaces.2.2 Set Theory.2.3 Events and Probability.2.4 Conditional Probability.2.5 Binomial Coefficients.Exercises.Chapter 3. Random Variables, Random Vectors, and Distributions Functions.3.1 Random Variables and Their Distributions.3.2 Multivariate Distributions.3.3 Sums and Integrals.3.4 Marginal Distributions and Independence.3.5 Vectors and Matrices.3.6 Expectations, Moments, and Characteristic Functions.3.7 Transformations of Random Variables.3.8 Conditional Distributions.Exercises.Chapter 4. Some Special Univariate Distributions.4.1 Introduction.4.2 The Bernoulli Distributions.4.3 The Binomial Distribution.4.4 The Poisson Distribution.4.5 The Negative Binomial Distribution.4.6 The Hypergeometric Distribution.4.7 The Normal Distribution.4.8 The Gamma Distribution.4.9 The Beta Distribution.4.10 The Uniform Distribution.4.11 The Pareto Distribution.4.12 The t Distribution.4.13 The F Distribution.Exercises.Chapter 5. Some Special Multivariate Distributions.5.1 Introduction.5.2 The Multinomial Distribution.5.3 The Dirichlet Distribution.5.4 The Multivariate Normal Distribution.5.5 The Wishart Distribution.5.6 The Multivariate t Distribution.5.7 The Bilateral Bivariate Pareto Distribution.Exercises.PART TWO. SUBJECTIVE PROBABILITY AND UTILITY.Chapter 6. Subjective Probability.6.1 Introduction.6.2 Relative Likelihood.6.3 The Auxiliary Experiment.6.4 Construction of the Probability Distribution.6.5 Verification of the Properties of a Probability Distribution.6.6 Conditional Likelihoods.Exercises.Chapter 7. Utility.7.1 Preferences Among Rewards.7.2 Preferences Among Probability Distributions.7.3 The Definitions of a Utility Function.7.4 Some Properties of Utility Functions.7.5 The Utility of Monetary Rewards.7.6 Convex and Concave Utility Functions.7.7 The Anxiomatic Development of Utility.7.8 Construction of the Utility Function.7.9 Verification of the Properties of a Utility Function.7.10 Extension of the Properties of a Utility Function to the Class ?E.Exercises.PART THREE. STATISTICAL DECISION PROBLEMS.Chapter 8. Decision Problems.8.1 Elements of a Decision Problem.8.2 Bayes Risk and Bayes Decisions.8.3 Nonnegative Loss Functions.8.4 Concavity of the Bayes Risk.8.5 Randomization and Mixed Decisions.8.6 Convex Sets.8.7 Decision Problems in Which ~2 and D Are Finite.8.8 Decision Problems with Observations.8.9 Construction of Bayes Decision Functions.8.10 The Cost of Observation.8.11 Statistical Decision Problems in Which Both ? and D contains Two Points.8.12 Computation of the Posterior Distribution When the Observations Are Made in More Than One Stage.Exercises.Chapter 9. Conjugate Prior Distributions.9.1 Sufficient Statistics.9.2 Conjugate Families of Distributions.9.3 Construction of the Conjugate Family.9.4 Conjugate Families for Samples from Various Standard Distributions.9.5 Conjugate Families for Samples from a Normal Distribution.9.6 Sampling from a Normal Distribution with Unknown Mean and Unknown Precision.9.7 Sampling from a Uniform Distribution.9.8 A Conjugate Family for Multinomial Observations.9.9 Conjugate Families for Samples from a Multivariate Normal Distribution.9.10 Multivariate Normal Distributions with Unknown Mean Vector and Unknown Precision matrix.9.11 The Marginal Distribution of the Mean Vector.9.12 The Distribution of a Correlation.9.13 Precision Matrices Having an Unknown Factor.Exercises.Chapter 10. Limiting Posterior Distributions.10.1 Improper Prior Distributions.10.2 Improper Prior Distributions for Samples from a Normal Distribution.10.3 Improper Prior Distributions for Samples from a Multivariate Normal Distribution.10.4 Precise Measurement.10.5 Convergence of Posterior Distributions.10.6 Supercontinuity.10.7 Solutions of the Likelihood Equation.10.8 Convergence of Supercontinuous Functions.10.9 Limiting Properties of the Likelihood Function.10.10 Normal Approximation to the Posterior Distribution.10.11 Approximation for Vector Parameters.10.12 Posterior Ratios.Exercises.Chapter 11. Estimation, Testing Hypotheses, and linear Statistical Models.11.1 Estimation.11.2 Quadratic Loss.11.3 Loss Proportional to the Absolute Value of the Error.11.4 Estimation of a Vector.11.5 Problems of Testing Hypotheses.11.6 Testing a Simple Hypothesis About the Mean of a Normal Distribution.11.7 Testing Hypotheses about the Mean of a Normal Distribution.11.8 Deciding Whether a Parameter Is Smaller or larger Than a Specific Value.11.9 Deciding Whether the Mean of a Normal Distribution Is Smaller or larger Than a Specific Value.11.10 Linear Models.11.11 Testing Hypotheses in Linear Models.11.12 Investigating the Hypothesis That Certain Regression Coefficients Vanish.11.13 One-Way Analysis of Variance.Exercises.PART FOUR. SEQUENTIAL DECISIONS.Chapter 12. Sequential Sampling.12.1 Gains from Sequential Sampling.12.2 Sequential Decision Procedures.12.3 The Risk of a Sequential Decision Procedure.12.4 Backward Induction.12.5 Optimal Bounded Sequential Decision procedures.12.6 Illustrative Examples.12.7 Unbounded Sequential Decision Procedures.12.8 Regular Sequential Decision Procedures.12.9 Existence of an Optimal Procedure.12.10 Approximating an Optimal Procedure by Bounded Procedures.12.11 Regions for Continuing or Terminating Sampling.12.12 The Functional Equation.12.13 Approximations and Bounds for the Bayes Risk.12.14 The Sequential Probability-ratio Test.12.15 Characteristics of Sequential Probability-ratio Tests.12.16 Approximating the Expected Number of Observations.Exercises.Chapter 13. Optimal Stopping.13.1 Introduction.13.2 The Statistician's Reward.13.3 Choice of the Utility Function.13.4 Sampling Without Recall.13.5 Further Problems of Sampling with Recall and Sampling without Recall.13.6 Sampling without Recall from a Normal Distribution with Unknown Mean.13.7 Sampling with Recall from a Normal Distribution with Unknown Mean.13.8 Existence of Optimal Stopping Rules.13.9 Existence of Optimal Stopping Rules for Problems of Sampling with Recall and Sampling without Recall.13.10 Martingales.13.11 Stopping Rules for Martingales.13.12 Uniformly Integrable Sequences of Random Variables.13.13 Martingales Formed from Sums and Products of Random Variables.13.14 Regular Supermartingales.13.15 Supermartingales and General Problems of Optimal Stopping.13.16 Markov Processes.13.17 Stationary Stopping Rules for Markov Processes.13.18 Entrance-fee Problems.13.19 The Functional Equation for a Markov Process.Exercises.Chapter 14. Sequential Choice of Experiments.14.1 Introduction.14.2 Markovian Decision Processes with a Finite Number of Stages.14.3 Markovian Decision Processes with an Infinite Number of Stages.14.4 Some Betting Problems.14.5 Two-armed-bandit Problems.14.6 Two-armed-bandit Problems When the Value of One Parameter Is Known.14.7 Two-armed-bandit Problems When the Parameters Are Dependent.14.8 Inventory Problems.14.9 Inventory Problems with an Infinite Number of Stages.14.10 Control Problems.14.11 Optimal Control When the Process Cannot Be Observed without Error.14.12 Multidimensional Control Problems.14.13 Control Problems with Actuation Errors.14.14 Search Problems.14.15 Search Problems with Equal Costs.14.16 Uncertainty Functions and Statistical Decision Problems.14.17 Sufficient Experiments.14.18 Examples of Sufficient Experiments.Exercises.References.Supplementary Bibliography.Name Index.Subject Index.

This book presents a philosophical approach to probability and probabilistic thinking, considering the underpinnings of probabilistic reasoning and modeling, which effectively underlie everything in data science. The ultimate goal is to call into question many standard tenets and lay the philosophical and probabilistic groundwork and infrastructure for statistical modeling. It is the first book devoted to the philosophy of data aimed at working scientists and calls for a new consideration in the practice of probability and statistics to eliminate what has been referred to as the Cult of Statistical Significance. The book explains the philosophy of these ideas and not the mathematics, though there are a handful of mathematical examples. The topics are logically laid out, starting with basic philosophy as related to probability, statistics, and science, and stepping through the key probabilistic ideas and concepts, and ending with statistical models. Its jargon-free approach asserts that standard methods, such as out-of-the-box regression, cannot help in discovering cause. This new way of looking at uncertainty ties together disparate fields probability, physics, biology, the soft sciences, computer science because each aims at discovering cause (of effects). It broadens the understanding beyond frequentist and Bayesian methods to propose a Third Way of modeling. Presents a complete argument showing why probability should be treated as a part of logic Broadens understanding beyond frequentist and Bayesian methods, proposing a Third Way of modeling Proposes that p-values should die, and along with them, hypothesis testing William M. Briggs, PhD, is Adjunct Professor of Statistics at Cornell University. Having earned both his PhD in Statistics and MSc in Atmospheric Physics from Cornell University, he served as the editor of the American Meteorological Society journal and has published over 60 papers. He studies the philosophy of science, the use and misuses of uncertainty - from truth to modeling. Early in life, he began his career as a cryptologist for the Air Force, then slipped into weather and climate forecasting, and later matured into an epistemologist. Currently, he has a popular, long-running blog on the subjects written about here, with about 70,000 - 90,000 monthly readers

Introduction: Introduction. - Types of Uncertainty. - Taylor Series Expansion. - Applications. - Problems. - Data Description and Treatment: Introduction.- Classification of Data. - Graphical Description of Data. - Histograms and Frequency Diagrams. - Descriptive Measures. - Applications. - Problems. - Fundamentals Of Probability: Introduction. - Sample Spaces, Sets, and Events. - Mathematics of Probability. - Random Variables and Their Probability Distributions. - Moment.- Common Discrete Probability Distributions. - Common Continuous Probability Distributions. - Applications. - Problems. - Multiple Random Variables: Introduction. - Joint Random Variables and Their Probability Distributions. - Functions of Random Variables. - Applications. - Problems. - Fundamentals of Statistical Analysis: Introduction. - Estimation of Parameters. - Sampling Distributions. - Hypothesis Testing: Procedure. - Hypothesis Tests of Means. - Hypothesis Tests of Variances. - Confidence Intervals. - Sample-Size Determination. - Selection of Model Probability Distributions. - Applications. Problems. - Curve Fitting and Regression Analysis: Introduction. - Correlation Analysis. - Introduction to Regression. - Principle of Least Squares. - Reliability of the Regression Equation. - Reliability of Point Estimates of the Regression Coefficients. - Confidence Intervals of the Regression Equation. - Correlation Versus Regression. - Applications of Bivariate Regression Analysis. - Multiple Regression Analysis. - Regression Analysis of Nonlinear Models. - Applications. Problems. - Simulation: Introduction. - Monte Carlo Simulation. - Random Numbers. - Generation of Random Variables. - Generation of Selected Discrete Random Variables. - Generation of Selected Continuous Random Variables. - Applications. - Problems. - Reliability and Risk Analysis: Introduction. - Time to Failure. - Reliability of Components. - Reliability of Systems. - Risk-Based Decision Analysis. - Applications. - Problems. - Bayesian Methods: Introduction. - Bayesian Probabilities. - Bayesian Estimation of Parameters. - Bayesian Statistics. - Applications. - Problems. - Appendix A: Probability and Statistics Tables. - Appendix B: Values of the Gamma Function. - Subject Index.

CHAPTER 1 Describing Data: Graphical 1.1 Decision Making in an Uncertain Environment 1.2 Classification of Variables 1.3 Graphs to Describe Categorical Variables 1.4 Graphs to Describe Time-Series Data 1.5 Graphs to Describe Numerical Variables 1.6 Tables and Graphs to Describe Relationships Between Variables 1.7 Data Presentation Errors CHAPTER 2 Describing Data: Numerical 2.1 Measures of Central Tendency 2.2 Measures of Variability 2.3 Weighted Mean and Measures of Grouped Data 2.4 Measures of Relationships Between Variables CHAPTER 3 Probability 3.1 Random Experiment, Outcomes, Events 3.2 Probability and Its Postulates 3.3 Probability Rules 3.4 Bivariate Probabilities 3.5 Bayes' Theorem CHAPTER 4 Discrete Random Variables and Probability Distributions 4.1 Random Variables 4.2 Probability Distributions for Discrete Random Variables 4.3 Properties of Discrete Random Variables 4.4 Binomial Distribution 4.5 Hypergeometric Distribution 4.6 The Poisson Probability Distribution 4.7 Jointly Distributed Discrete Random Variables CHAPTER 5 Continuous Random Variables and Probability Distributions 5.1 Continuous Random Variables 5.2 Expectations for Continuous Random Variables 5.3 The Normal Distribution 5.4 Normal Distribution Approximation for Binomial Distribution 5.5 The Exponential Distribution 5.6 Jointly Distributed Continuous Random Variables CHAPTER 6 Sampling and Sampling Distributions 6.1 Sampling from a Population 6.2 Sampling Distributions of Sample Means 6.3 Sampling Distributions of Sample Proportions 6.4 Sampling Distributions of Sample Variances CHAPTER 7 Estimation: Single Population 7.1 Properties of Point Estimators 7.2 Confidence Interval Estimation of the Mean of a Normal Distribution: Population Variance Known 7.3 Confidence Interval Estimation of the Mean of a Normal Distribution: Population Variance Unknown 7.4 Confidence Interval Estimation of Population Proportion 7.5 Confidence Interval Estimation of the Variance of a Normal Distribution 7.6 Confidence Interval Estimation: Finite Populations CHAPTER 8 Estimation: Additional Topics 8.1 Confidence Interval Estimation of the Difference Between Two Normal Population Means: Dependent Samples 8.2 Confidence Interval Estimation of the Difference Between Two Normal Population Means: Independent Samples 8.3 Confidence Interval Estimation of the Difference Between Two Population Proportions 8.4 Sample Size Determination: Large Populations 8.5 Sample Size Determination: Finite Populations CHAPTER 9 Hypothesis Testing: Single Population 9.1 Concepts of Hypothesis Testing 9.2 Tests of the Mean of a Normal Distribution: Population Variance Known 9.3 Tests of the Mean of a Normal Distribution: Population Variance Unknown 9.4 Tests of the Population Proportion 9.5 Assessing the Power of a Test 9.6 Tests of the Variance of a Normal Distribution CHAPTER 10 Hypothesis Testing: Additional Topics 10.1 Tests of the Difference Between Two Population Means: Dependent Samples 10.2 Tests of the Difference Between Two Normal Population Means: Independent Samples 10.3 Tests of the Difference Between Two Population Proportions 10.4 Tests of the Equality of the Variances Between Two Normally Distributed Populations 10.5 Some Comments on Hypothesis Testing CHAPTER 11 Simple Regression 11.1 Overview of Linear Models 11.2 Linear Regression Model 11.3 Least Squares Coefficient Estimators 11.4 The Explanatory Power of a Linear Regression Equation 11.5 Statistical Inference: Hypothesis Tests and Confidence Intervals 11.6 Prediction 11.7 Correlation Analysis 11.8 Beta Measure of Financial Risk 11.9 Graphical Analysis CHAPTER 12 Multiple Regression 12.1 The Multiple Regression Model 12.2 Estimation of Coefficients 12.3 Explanatory Power of a Multiple Regression Equation 12.4 Confidence Intervals and Hypothesis Tests for Individual Regression Coefficients 12.5 Tests on Regression Coefficients 12.6 Prediction 12.7 Transformations for Nonlinear Regression Models 12.8 Dummy Variables for Regression Models 12.9 Multiple Regression Analysis Application Procedure CHAPTER 13 Additional Topics in Regression Analysis 13.1 Model-Building Methodology 13.2 Dummy Variables and Experimental Design 13.3 Lagged Values of the Dependent Variables as Regressors 13.4 Specification Bias 13.5 Multicollinearity 13.6 Heteroscedasticity 13.7 Autocorrelated Errors CHAPTER 14 ANALYSIS OF CATEGORICAL DATA 14.1 Goodness-of-Fit Tests: Specified Probabilities 14.2 Goodness-of-Fit Tests: Population Parameters Unknown 14.3 Contingency Tables 14.4 Sign Test and Confidence Interval 14.5 Wilcoxon Signed Rank Test 14.6 Mann--Whitney U Test 14.7 Wilcoxon Rank Sum Test 14.7 Spearman Rank Correlation CHAPTER 15 Analysis of Variance 15.1 Comparison of Several Population Means 15.2 One-Way Analysis of Variance 15.3 The Kruskal--Wallis Test 15.4 Two-Way Analysis of Variance: One Observation per Cell, Randomized Blocks 15.5 Two-Way Analysis of Variance: More Than One Observation per Cell CHAPTER 16 Time-Series Analysis and Forecasting 16.1 Index Numbers 16.2 A Nonparametric Test for Randomness 16.3 Components of a Time Series 16.4 Moving Averages 16.5 Exponential Smoothing 16.6 Autoregressive Models 16.7 Autoregressive Integrated Moving Average Models CHAPTER 17 Sampling: Additional Topics 17.1 Stratified Sampling 17.2 Other Sampling Methods CHAPTER 18 Statistical Decision Theory 18.1 Decision Making Under Uncertainty 18.2 Solutions Not Involving Specification of Probabilities 18.3 Expected Monetary Value TreePlan 18.4 Sample Information: Bayesian Analysis and Value 18.5 Allowing for Risk: Utility Analysis APPENDIX TABLES 1. Cumulative Distribution Function of the Standard Normal Distribution 2. Probability Function of the Binomial Distribution 3. Cumulative Binomial Probabilities 4. Values of e --lambda 5. Individual Poisson Probabilities 6. Cumulative Poisson Probabilities 7. Cutoff Points of the Chi-Square Distribution Function 8. Cutoff Points for the Student's t Distribution 9. Cutoff Points for the F Distribution 10. Cutoff Points for the Distribution of the Wilcoxon Test Statistic 11. Cutoff Points for the Distribution of Spearman Rank Correlation Coefficient 12. Cutoff Points for the Distribution of the Durbin--Watson Test Statistic 13 Critical Values of the Studentized Range Q (page 964 965 Applied Statistical Methods Carlson, Thorne Prentice Hall 1997) 14. Cumulative Distribution Function of the Runs Test Statistic ANSWERS TO SELECTED EVEN-NUMBERED EXERCISES INDEX I-1

The field of uncertainty quantification is evolving rapidly because of increasing emphasis on models that require quantified uncertainties for large-scale applications, novel algorithm development, and new computational architectures that facilitate implementation of these algorithms. Uncertainty Quantification: Theory, Implementation, and Applications provides readers with the basic concepts, theory, and algorithms necessary to quantify input and response uncertainties for simulation models arising in a broad range of disciplines. The book begins with a detailed discussion of applications where uncertainty quantification is critical for both scientific understanding and policy. It then covers concepts from probability and statistics, parameter selection techniques, frequentist and Bayesian model calibration, propagation of uncertainties, quantification of model discrepancy, surrogate model construction, and local and global sensitivity analysis. The author maintains a complementary web page where readers can find data used in the exercises and other supplementary material. Uncertainty Quantification: Theory, Implementation, and Applications includes a large number of definitions and examples that use a suite of relatively simple models to illustrate concepts; numerous references to current and open research issues; and exercises that illustrate basic concepts and guide readers through the numerical implementation of algorithms for prototypical problems. It also features a wide range of applications, including weather and climate models, subsurface hydrology and geology models, nuclear power plant design, and models for biological phenomena, along with recent advances and topics that have appeared in the research literature within the last 15 years, including aspects of Bayesian model calibration, surrogate model development, parameter selection techniques, and global sensitivity analysis.

Preface. Part I: Theory and methods. 1 What is a risk analysis? 1.1 Why risk analysis?. 1.2 Risk management. 1.2.1 Decision-making under uncertainty. 1.3 Examples: decision situations. 1.3.1 Risk analysis for a tunnel. 1.3.2 Risk analysis for an offshore installation. 1.3.3 Risk analysis related to a cash depot. 2 What is risk? 2.1 Vulnerability. 2.2 How to describe risk quantitatively. 2.2.1 Description of risk in a financial context. 2.2.2 Description of risk in a safety context. 3 The risk analysis process: planning. 3.1 Problem definition. 3.2 Selection of analysis method. 3.2.1 Checklist-based approach. 3.2.2 Risk-based approach. 4 The risk analysis process: risk assessment. 4.1 Identification of initiating events. 4.2 Cause analysis. 4.3 Consequence analysis. 4.4 Probabilities and uncertainties. 4.5 Risk picture: Risk presentation. 4.5.1 Sensitivity and robustness analyses. 4.5.2 Risk evaluation. 5 The risk analysis process: risk treatment. 5.1 Comparisons of alternatives. 5.1.1 How to assess measures? 5.2 Management review and judgement. 6 Risk analysis methods. 6.1 Coarse risk analysis. 6.2 Job safety analysis. 6.3 Failure modes and effects analysis. 6.3.1 Strengths and weaknesses of an FMEA. 6.4 Hazard and operability studies. 6.5 SWIFT. 6.6 Fault tree analysis. 6.6.1 Qualitative analysis. 6.6.2 Quantitative analysis. 6.7 Event tree analysis. 6.7.1 Barrier block diagrams. 6.8 Bayesian networks. 6.9 Monte Carlo simulation. Part II Examples of applications. 7 Safety measures for a road tunnel. 7.1 Planning. 7.1.1 Problem definition. 7.1.2 Selection of analysis method. 7.2 Risk assessment. 7.2.1 Identification of initiating events. 7.2.2 Cause analysis. 7.2.3 Consequence analysis. 7.2.4 Risk picture. 7.3 Risk treatment. 7.3.1 Comparison of alternatives. 7.3.2 Management review and decision. 8 Risk analysis process for an offshore installation. 8.1 Planning. 8.1.1 Problem definition. 8.1.2 Selection of analysis method. 8.2 Risk analysis. 8.2.1 Hazard identification. 8.2.2 Cause analysis. 8.2.3 Consequence analysis. 8.3 Risk picture and comparison of alternatives. 8.4 Management review and judgement. 9 Production assurance. 9.1 Planning. 9.2 Risk analysis. 9.2.1 Identification of failures. 9.2.2 Cause analysis. 9.2.3 Consequence analysis. 9.3 Risk picture and comparison of alternatives. 9.4 Management review and judgement. Decision. 10 Risk analysis process for a cash depot. 10.1 Planning. 10.1.1 Problem definition. 10.1.2 Selection of analysis method. 10.2 Risk analysis. 10.2.1 Identification of hazards and threats. 10.2.2 Cause analysis. 10.2.3 Consequence analysis. 10.3 Risk picture. 10.4 Risk-reducing measures. 10.4.1 Relocation of the NOKAS facility. 10.4.2 Erection of a wall. 10.5 Management review and judgment. Decision. 10.6 Discussion. 11 Risk analysis process for municipalities. 11.1 Planning . 11.1.1 Problem definition. 11.1.2 Selection of analysis method. 11.2 Risk assessment. 11.2.1 Hazard and threat identification. 11.2.2 Cause and consequence analysis. Risk picture. 11.3 Risk treatment. 12 Risk analysis process for the entire enterprise. 12.1 Planning. 12.1.1 Problem definition. 12.1.2 Selection of analysis method. 12.2 Risk analysis. 12.2.1 Price risk. 12.2.2 Operational risk. 12.2.3 Health, Environment and Safety (HES). 12.2.4 Reputation risk. 12.3 Overall risk picture. 12.4 Risk treatment. 13 Discussion. 13.1 Risk analysis as a decision support tool. 13.2 Risk is more than the calculated probabilities and expected values. 13.3 Risk analysis has both strengths and weaknesses. 13.3.1 Precision of a risk analysis: uncertainty and sensitivity analysis. 13.3.2 Terminology. 13.3.3 Risk acceptance criteria (tolerability limits). 13.4 Reflection on approaches, methods and results. 13.5 Limitations of the causal chain approach. 13.6 Risk perspectives. 13.7 Scientific basis. 13.8 The implications of the limitations of risk assessment. 13.9 Critical systems and activities. 13.10 Conclusions. A Probability calculus and statistics. A.1 The meaning of a probability. A.2 Probability calculus. A.3 Probability distributions: expected value. A.3.1 Binomial distribution. A.4 Statistics (Bayesian statistics). B Introduction to reliability analysis. B.1 Reliability of systems composed of components. B.2 Production system. B.3 Safety system. C Approach for selecting risk analysis methods. C.1 Expected consequences. C.2 Uncertainty factors. C.3 Frame conditions. C.4 Selection of a specific method. D Terminology. D.1 Risk management: relationships between key terms. Bibliography. Index.

Formal ways of representing uncertainty and various logics for reasoning about it; updated with new material on weighted probability measures, complexity-theoretic considerations, and other topics. In order to deal with uncertainty intelligently, we need to be able to represent it and reason about it. In this book, Joseph Halpern examines formal ways of representing uncertainty and considers various logics for reasoning about it. While the ideas presented are formalized in terms of definitions and theorems, the emphasis is on the philosophy of representing and reasoning about uncertainty. Halpern surveys possible formal systems for representing uncertainty, including probability measures, possibility measures, and plausibility measures; considers the updating of beliefs based on changing information and the relation to Bayes' theorem; and discusses qualitative, quantitative, and plausibilistic Bayesian networks. This second edition has been updated to reflect Halpern's recent research. New material includes a consideration of weighted probability measures and how they can be used in decision making; analyses of the Doomsday argument and the Sleeping Beauty problem; modeling games with imperfect recall using the runs-and-systems approach; a discussion of complexity-theoretic considerations; the application of first-order conditional logic to security. Reasoning about Uncertainty is accessible and relevant to researchers and students in many fields, including computer science, artificial intelligence, economics (particularly game theory), mathematics, philosophy, and statistics.

I first met Leo Breiman in 1979 at the beginning of his third career, Professor of Statistics at Berkeley. He obtained his PhD with Loéve at Berkeley in 1957. His first career was as a probabilist in the Mathematics Department at UCLA. After distinguished research, including the Shannon--Breiman--MacMillan Theorem and getting tenure, he decided that his real interest was in applied statistics, so he resigned his position at UCLA and set up as a consultant. Before doing so he produced two classic texts, Probability, now reprinted as a SIAM Classic in Applied Mathematics, and Statistics. Both books reflected his strong opinion that intuition and rigor must be combined. He expressed this in his probability book which he viewed as a combination of his learning the right hand of probability, rigor, from Loéve, and the left-hand, intuition, from David Blackwell.

Many doctors, patients, journalists, and politicians alike do not understand what health statistics mean or draw wrong conclusions without noticing. Collective statistical illiteracy refers to the widespread inability to understand the meaning of numbers. For instance, many citizens are unaware that higher survival rates with cancer screening do not imply longer life, or that the statement that mammography screening reduces the risk of dying from breast cancer by 25% in fact means that 1 less woman out of 1,000 will die of the disease. We provide evidence that statistical illiteracy (a) is common to patients, journalists, and physicians; (b) is created by nontransparent framing of information that is sometimes an unintentional result of lack of understanding but can also be a result of intentional efforts to manipulate or persuade people; and (c) can have serious consequences for health. The causes of statistical illiteracy should not be attributed to cognitive biases alone, but to the emotional nature of the doctor-patient relationship and conflicts of interest in the healthcare system. The classic doctor-patient relation is based on (the physician's) paternalism and (the patient's) trust in authority, which make statistical literacy seem unnecessary; so does the traditional combination of determinism (physicians who seek causes, not chances) and the illusion of certainty (patients who seek certainty when there is none). We show that information pamphlets, Web sites, leaflets distributed to doctors by the pharmaceutical industry, and even medical journals often report evidence in nontransparent forms that suggest big benefits of featured interventions and small harms. Without understanding the numbers involved, the public is susceptible to political and commercial manipulation of their anxieties and hopes, which undermines the goals of informed consent and shared decision making. What can be done? We discuss the importance of teaching statistical thinking and transparent representations in primary and secondary education as well as in medical school. Yet this requires familiarizing children early on with the concept of probability and teaching statistical literacy as the art of solving real-world problems rather than applying formulas to toy problems about coins and dice. A major precondition for statistical literacy is transparent risk communication. We recommend using frequency statements instead of single-event probabilities, absolute risks instead of relative risks, mortality rates instead of survival rates, and natural frequencies instead of conditional probabilities. Psychological research on transparent visual and numerical forms of risk communication, as well as training of physicians in their use, is called for. Statistical literacy is a necessary precondition for an educated citizenship in a technological democracy. Understanding risks and asking critical questions can also shape the emotional climate in a society so that hopes and anxieties are no longer as easily manipulated from outside and citizens can develop a better-informed and more relaxed attitude toward their health.

We study percolation in the following random environment: let $Z$ be a Poisson process of constant intensity in the plane, and form the Voronoi tessellation of the plane with respect to $Z$. Colour each Voronoi cell black with probability $p$, independently of the other cells. We show that the critical probability is 1/2. More precisely, if $p>1/2$ then the union of the black cells contains an infinite component with probability 1, while if $p<1/2$ then the distribution of the size of the component of black cells containing a given point decays exponentially. These results are analogous to Kesten's results for bond percolation in the square lattice. The result corresponding to Harris' Theorem for bond percolation in the square lattice is known: Zvavitch noted that one of the many proofs of this result can easily be adapted to the random Voronoi setting. For Kesten's results, none of the existing proofs seems to adapt. The methods used here also give a new and very simple proof of Kesten's Theorem for the square lattice; we hope they will be applicable in other contexts as well.

Organisms must act in the face of sensory, motor, and reward uncertainty stemming from a pandemonium of stochasticity and missing information. In many tasks, organisms can make better decisions if they have at their disposal a representation of the uncertainty associated with task-relevant variables. We formalize this problem using Bayesian decision theory and review recent behavioral and neural evidence that the brain may use knowledge of uncertainty, confidence, and probability.

Abstract The notion of uncertainty is of major importance in machine learning and constitutes a key element of machine learning methodology. In line with the statistical tradition, uncertainty has long been perceived as almost synonymous with standard probability and probabilistic predictions. Yet, due to the steadily increasing relevance of machine learning for practical applications and related issues such as safety requirements, new problems and challenges have recently been identified by machine learning scholars, and these problems may call for new methodological developments. In particular, this includes the importance of distinguishing between (at least) two different types of uncertainty, often referred to as aleatoric and epistemic . In this paper, we provide an introduction to the topic of uncertainty in machine learning as well as an overview of attempts so far at handling uncertainty in general and formalizing this distinction in particular.

Our data are random fields of multivariate Gaussian observations, and we fit a multivariate linear model with common design matrix at each point. We are interested in detecting those points where some of the coefficients are nonzero using classical multivariate statistics evaluated at each point. The problem is to find the $P$-value of the maximum of such a random field of test statistics. We approximate this by the expected Euler characteristic of the excursion set. Our main result is a very simple method for calculating this, which not only gives us the previous result of Cao and Worsley [Ann. Statist. 27 (1999) 925--942] for Hotelling's $T^2$, but also random fields of Roy's maximum root, maximum canonical correlations [Ann. Appl. Probab. 9 (1999) 1021--1057], multilinear forms [Ann. Statist. 29 (2001) 328--371], $\barχ^2$ [Statist. Probab. Lett 32 (1997) 367--376, Ann. Statist. 25 (1997) 2368--2387] and $χ^2$ scale space [Adv. in Appl. Probab. 33 (2001) 773--793]. The trick involves approaching the problem from the point of view of Roy's union-intersection principle. The results are applied to a problem in shape analysis where we look for brain damage due to nonmissile trauma.

We prove that if (X,\mathfrakA,P) is an arbitrary probability space with countably generated σ-algebra \mathfrakA, (Y,\mathfrakB,Q) is an arbitrary complete probability space with a lifting ρand \hat R is a complete probability measure on \mathfrakA \hat \otimes_R \mathfrakB determined by a regular conditional probability {S_y:y\in Y} on \mathfrakA with respect to \mathfrakB, then there exist a lifting πon (X\times Y,\mathfrakA \hat \otimes_R \mathfrakB,\hat R) and liftings σ_y on (X,\hat \mathfrakA_y,\hat S_y), y\in Y, such that, for every E\in\mathfrakA \hat \otimes_R \mathfrakB and every y\in Y, [π(E)]^y=σ_y\bigl([π(E)]^y\bigr). Assuming the absolute continuity of R with respect to P\otimes Q, we prove the existence of a regular conditional probability {T_y:y\in Y} and liftings \varpi on (X\times Y,\mathfrakA \hat \otimes_R \mathfrakB,\hat R), ρ' on (Y,\mathfrakB,\hat Q) and σ_y on (X,\hat \mathfrakA_y,\hat S_y), y\in Y, such that, for every E\in\mathfrakA \hat \otimes_R \mathfrakB and every y\in Y, [\varpi(E)]^y=σ_y\bigl([\varpi(E)]^y\bigr) and \varpi(A\times B)=\bigcup_{y\inρ'(B)}σ_y(A)\times{y}\qquadif A\times B\in\mathfrakA\times\mathfrakB. Both results are generalizations o

Analyses of ecological data should account for the uncertainty in the process(es) that generated the data. However, accounting for these uncertainties is a difficult task, since ecology is known for its complexity. Measurement and/or process errors are often the only sources of uncertainty modeled when addressing complex ecological problems, yet analyses should also account for uncertainty in sampling design, in model specification, in parameters governing the specified model, and in initial and boundary conditions. Only then can we be confident in the scientific inferences and forecasts made from an analysis. Probability and statistics provide a framework that accounts for multiple sources of uncertainty. Given the complexities of ecological studies, the hierarchical statistical model is an invaluable tool. This approach is not new in ecology, and there are many examples (both Bayesian and non-Bayesian) in the literature illustrating the benefits of this approach. In this article, we provide a baseline for concepts, notation, and methods, from which discussion on hierarchical statistical modeling in ecology can proceed. We have also planted some seeds for discussion and tried to show where the practical difficulties lie. Our thesis is that hierarchical statistical modeling is a powerful way of approaching ecological analysis in the presence of inevitable but quantifiable uncertainties, even if practical issues sometimes require pragmatic compromises.