This book gives an overview of rigorous risk analysis, linking it to Bayesian decision theory and demonstrating practical implementations. Risk analysis is the final step before decision-making. But what is risk, and how do we quantify the key components of risk? This book aims to provide clear definitions, formulas and algorithms.

Risk is the expectation of loss due to a hazard. It is high when both hazard probability and system vulnerability are high. We show how these terms should be defined to allow a formal decomposition of risk as the mathematical product of hazard probability and system vulnerability. From these definitions we derive a comprehensive theory of risk analysis and its links with Bayesian decision theory (BDT). We present formulas for risk and its components, and for quantifying the uncertainties associated with these estimates. We also show how the formulas for PRA and BDT can be implemented in R. All the computer code used in this book, including that for producing tables and figures, can be downloaded from the book's public GitHub repository.

This book is intended for researchers and decision-makers with an interest in rigorous risk analysis. Some familiarity with probability theory will make the book easier to digest, but it includes simple introductions to both probabilistic risk analysis and Bayesian decision theory. Jargon is avoided as much as possible, and all terms are defined within the book itself. Many examples of applications are included, mostly using simple data sets. The examples are from the environmental sciences, but the analytical methods are completely generic. The theory is applicable to both discrete event hazards (e.g. earthquakes) and continuous hazards (e.g. pollution or water availability).

This second edition adds new sections on uncertainty quantification, continuous risk analysis, interacting hazards, dataset quality, and the Value of Information concept in decision theory. Literature references have been updated throughout, and exercises have been added to each chapter, all with solutions.

​Marcel van Oijen studied mathematical biology at the University of Utrecht, graduating cum laude in 1985. He completed his PhD in plant disease epidemiology at Wageningen University, where he then worked on modelling the impacts of environmental change on crops. In 1999, he moved to Edinburgh where he was a senior scientist for the UK’s Natural Environment Research Council, focusing on the use of Bayesian methods in the modelling of ecosystem services provided by grasslands, forests and agroforestry systems. He is now an independent researcher and this is his second book, following the publication in 2020 of ‘Bayesian Compendium’, an introductory guide to the universality of Bayesian methods. 

Mark Brewer is director of BioSS (Biomathematics and Statistics Scotland). His first degree was in Probability and Statistics from the University of Sheffield, and Mark subsequently studied for a PhD in statistics - specialising in MCMC and graphical models - at the University of Edinburgh. After three years working in statistical consultancy at the University of Aberdeen and five years as a lecturer in statistics at the University of Exeter, in 2001 Mark moved to BioSS as a senior statistician. He has worked mainly in ecological and environmental applications, conducting research in spatio-temporal and Bayesian modelling. He became head of BioSS in 2018, and has seen the organisation increase both its funding and staffing complement since that time. Mark acted as co-Editor for Biometrics (2019-2021) and was previously on the Executive Board of the International Biometric Society (2017-2020).