# Uncertainty in Risk Assessment: The Representation and Treatment of Uncertainties by Probabilistic and Non-Probabilistic Methods

## Books

Explores methods for the representation and treatment of uncertainty in risk assessment

In providing guidance for practical decision-making situations concerning high-consequence technologies (e.g., nuclear, oil and gas, transport, etc.), the theories and methods studied in Uncertainty in Risk Assessment have wide-ranging applications from engineering and medicine to environmental impacts and natural disasters, security, and financial risk management. The main focus, however, is on engineering applications.

While requiring some fundamental background in risk assessment, as well as a basic knowledge of probability theory and statistics, Uncertainty in Risk Assessment can be read profitably by a broad audience of professionals in the field, including researchers and graduate students on courses within risk analysis, statistics, engineering, and the physical sciences.

Uncertainty in Risk Assessment:

• Illustrates the need for seeing beyond probability to represent uncertainties in risk assessment contexts.
• Provides simple explanations (supported by straightforward numerical examples) of the meaning of different types of probabilities, including interval probabilities, and the fundamentals of possibility theory and evidence theory.
• Offers guidance on when to use probability and when to use an alternative representation of uncertainty.
• Presents and discusses methods for the representation and characterization of uncertainty in risk assessment.
• Uses examples to clearly illustrate ideas and concepts.

Preface ix

PART I INTRODUCTION 1

1 Introduction 3

1.1 Risk 4

1.1.1 The concept of risk 4

1.1.2 Describing/measuring risk 6

1.1.3 Examples 6

1.2 Probabilistic risk assessment 8

1.3 Use of risk assessment: The risk management and decision-making context 11

1.4 Treatment of uncertainties in risk assessments 13

1.5 Challenges: Discussion 15

1.5.1 Examples 16

1.5.2 Alternatives to the probability-based approaches to risk and uncertainty assessment 17

References – Part I 21

PART II METHODS 27

2 Probabilistic approaches for treating uncertainty 29

2.1 Classical probabilities 30

2.2 Frequentist probabilities 31

2.3 Subjective probabilities 35

2.3.1 Betting interpretation 36

2.3.2 Reference to a standard for uncertainty 36

2.4 The Bayesian subjective probability framework 37

2.5 Logical probabilities 39

3 Imprecise probabilities for treating uncertainty 41

4 Possibility theory for treating uncertainty 45

4.1 Basics of possibility theory 45

4.2 Approaches for constructing possibility distributions 49

4.2.1 Building possibility distributions from nested probability intervals 49

4.2.2 Justification for using the triangular possibility distribution 51

4.2.3 Building possibility distributions using Chebyshev’s inequality 52

5 Evidence theory for treating uncertainty 53

6 Methods of uncertainty propagation 59

6.1 Level 1 uncertainty propagation setting 61

6.1.1 Level 1 purely probabilistic framework 62

6.1.2 Level 1 purely possibilistic framework 64

6.1.3 Level 1 hybrid probabilistic–possibilistic framework 67

6.2 Level 2 uncertainty propagation setting 71

6.2.1 Level 2 purely probabilistic framework 73

6.2.2 Level 2 hybrid probabilistic–evidence theory framework 75

7 Discussion 79

7.1 Probabilistic analysis 80

7.2 Lower and upper probabilities 82

7.3 Non-probabilistic representations with interpretations other than lower and upper probabilities 84

7.4 Hybrid representations of uncertainty 85

7.5 Semi-quantitative approaches 87

References – Part II 93

PART III PRACTICAL APPLICATIONS 99

8 Uncertainty representation and propagation in structural reliability analysis 101

8.1 Structural reliability analysis 101

8.1.1 A model of crack propagation under cyclic fatigue 101

8.2 Case study 102

8.3 Uncertainty representation 104

8.4 Uncertainty propagation 105

8.5 Results 107

8.6 Comparison to a purely probabilistic method 107

9 Uncertainty representation and propagation in maintenance performance assessment 111

9.1 Maintenance performance assessment 111

9.2 Case study 113

9.3 Uncertainty representation 116

9.4 Uncertainty propagation 118

9.4.1 Maintenance performance assessment in the case of no epistemic uncertainty on the parameters 118

9.4.2 Application of the hybrid probabilistic–theory of evidence uncertainty propagation method 122

9.5 Results 123

10 Uncertainty representation and propagation in event tree analysis 127

10.1 Event tree analysis 127

10.2 Case study 128

10.3 Uncertainty representation 134

10.4 Uncertainty propagation 135

10.5 Results 137

10.6 Comparison of the results to those obtained by using other uncertainty representation and propagation methods 138

10.6.1 Purely probabilistic representation and propagation of the uncertainty 138

10.6.2 Purely possibilistic representation and propagation of the uncertainty 138

10.7 Result comparison 141

10.7.1 Comparison of results 141

10.7.2 Comparison of the results for the probability of occurrence of a severe consequence accident 145

11 Uncertainty representation and propagation in the evaluation of the consequences of industrial activity 147

11.1 Evaluation of the consequences of undesirable events 147

11.2 Case study 148

11.3 Uncertainty representation 150

11.4 Uncertainty propagation 152

11.5 Results 152

11.6 Comparison of the results to those obtained using a purely probabilistic approach 153

12 Uncertainty representation and propagation in the risk assessment of a process plant 155

12.1 Introduction 155

12.2 Case description 155

12.3 The “textbook” Bayesian approach (level 2 analysis) 156

12.4 An alternative approach based on subjective probabilities (level 1 analysis) 159

References – Part III 163

PART IV CONCLUSIONS 167

13 Conclusions 169

References – Part IV 173

Appendix A Operative procedures for the methods of uncertainty propagation 175

A.1 Level 1 hybrid probabilistic–possibilistic framework 175

A.2 Level 2 purely probabilistic framework 176

Appendix B Possibility–probability transformation 179

Reference 181

Index 183

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