Probabilistic Risk Analysis and Bayesian Decision Theory (SpringerBriefs in Statistics)

✍ Scribed by Marcel van Oijen, Mark Brewer

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 118
• Category: Library

✦ Synopsis

The book shows how risk, defined as the statistical expectation of loss, can be formally decomposed as the product of two terms: hazard probability and system vulnerability. This requires a specific definition of vulnerability that replaces the many fuzzy definitions abounding in the literature. The approach is expanded to more complex risk analysis with three components rather than two, and with various definitions of hazard. Equations are derived to quantify the uncertainty of each risk component and show how the approach relates to Bayesian decision theory. Intended for statisticians, environmental scientists and risk analysts interested in the theory and application of risk analysis, this book provides precise definitions, new theory, and many examples with full computer code. The approach is based on straightforward use of probability theory which brings rigour and clarity. Only a moderate knowledge and understanding of probability theory is expected from the reader.

✦ Table of Contents

Preface
Why This Book?
Who Is this Book for?
Notation
Outline of Chapters
Acknowledgements
Contents
1 Introduction to Probabilistic Risk Analysis (PRA)
1.1 From Risk Matrices to PRA
1.2 Basic Equations for PRA
1.3 Decomposition of Risk: 2 or 3 Components
1.4 Resolution of PRA: Single-Threshold, Multi-Threshold, Categorical, Continuous
1.4.1 Single-Threshold PRA
1.4.2 Multi-Threshold PRA
1.4.3 Categorical PRA
1.4.4 Continuous PRA
1.5 Implementation of PRA: Distribution-Based, Sampling-Based, Model-Based
2 Distribution-Based Single-Threshold PRA
2.1 Conditional Distributions for z
2.1.1 Conditions for V Being Constant
2.2 Example of Distribution-Based PRA: Gaussian p[x,z]
2.2.1 Hazard Probability and Conditional Distributions
2.2.2 Conditional Expectations and PRA
2.3 Approximation Formulas for the Conditional Bivariate Gaussian Expectations
3 Sampling-Based Single-Threshold PRA
3.1 Example of Sampling-Based PRA: Linear Relationship
3.1.1 Varying the Threshold
3.2 Example of Sampling-Based PRA: Nonlinear Relationship
4 Sampling-Based Single-Threshold PRA: Uncertainty Quantification (UQ)
4.1 Uncertainty in p[H]
4.2 Uncertainty in V
4.3 Uncertainty in R
4.4 Extension of R-Code for PRA: Adding the UQ
4.5 PRA with UQ on the Nonlinear Data Set
4.6 Verification of the UQ by Simulating Multiple Data Sets
4.6.1 UQ-Verification: Nonlinear Relationship
4.6.2 UQ-Verification: Linear Relationship
4.7 Approximation Formulas for the Conditional Bivariate Gaussian Variances
5 Density Estimation to Move from Sampling- to Distribution-Based PRA
6 Copulas for Distribution-Based PRA
6.1 Sampling from Copulas and Carrying out PRA
6.2 Copula Selection
6.3 Using Copulas in PRA
7 Bayesian Model-Based PRA
7.1 Linear Example: Full Bayesian PRA with Uncertainty
7.1.1 Checking the MCMC
7.1.2 PRA
7.2 Nonlinear Example: Full Bayesian PRA with Uncertainty
7.3 Advantages of the Bayesian Modelling Approach
8 Sampling-Based Multi-Threshold PRA:Gaussian Linear Example
9 Distribution-Based Continuous PRA: Gaussian Linear Example
10 Categorical PRA with Other Splits than for Threshold-Levels: Spatio-Temporal Example
10.1 Spatio-Temporal Environmental Data: x(s,t)
10.2 Spatio-Temporal System Data: z(s,t)
10.3 Single-Category Single-Threshold PRA for the Spatio-Temporal Data
10.4 Two-Category Single-Threshold PRA for Spatio-Temporal Data
11 Three-Component PRA
11.1 Three-Component PRA for Spatio-Temporal Data
11.2 Country-Wide Application of Three-Component PRA
11.3 UQ for Three-Component PRA
12 Introduction to Bayesian Decision Theory (BDT)
12.1 Example of BDT in Action
13 Implementation of BDT Using Bayesian Networks
13.1 Three Ways to Specify a Multivariate Gaussian
13.1.1 Switching Between the Three Different Specifications of the Multivariate Gaussian
13.2 Sampling from a GBN and Bayesian Updating
13.2.1 Updating a GBN When Information About Nodes Becomes Available
13.3 A Linear BDT Example Implemented as a GBN
13.4 A Linear BDT Example Implemented Using \texttt{Nimble}
13.4.1 Varying IRRIG to Identify the Value for Which E[U] Is Maximized
13.5 A Nonlinear BDT Example Implemented Using \texttt{Nimble}
14 A Spatial Example: Forestry in Scotland
14.1 A Decision Problem: Forest Irrigation in Scotland
14.2 Computational Demand of BDT and Emulation
14.3 Data
14.4 A Simple Model for Forest Yield Class (YC)
14.5 Emulation
14.6 Application of the Emulator
15 Spatial BDT Using Model and Emulator
15.1 Multiple Action Levels
16 Linkages Between PRA and BDT
16.1 Risk Management
16.2 The Relationship Between Utility Maximisation in BDT and Risk Assessment in PRA: R_c
16.3 Simplified Accounting for Both Benefits and Costs of the Action: R_b
16.4 Only Correcting for Costs: R_a
17 PRA vs. BDT in the Spatial Example
18 Three-Component PRA in the Spatial Example
19 Discussion
19.1 PRA and Its Application
19.2 Data and Computational Demand of PRA
19.3 BDT
19.4 Computational Demand of BDT
19.5 PRA as a Tool for Simplifying and Elucidating BDT
19.6 Parameter and Model Uncertainties
19.7 Modelling and Decision-Support for Forest Response to Hazards
19.8 Spatial Statistics
References
Index