Risk analysis is often the first step before decision-making. But what is risk, and how can we rigorously analyse it? How do we quantify the key components of risk? There is considerable confusion about these questions in the academic literature, and this book is our attempt to provide answers. We aim to provide clear definitions, formulas and algorithms.

Risk involves a stress factor or hazard and a system that is vulnerable to that hazard. The risk is high when both hazard probability and system vulnerability are high. These ideas are expressed in an influential United Nations report on disaster management (UN, 1992) which defined risk as "expected losses" to be calculated as "the product of hazard and vulnerability." Likewise, the Intergovernmental Panel on Climate Change (IPCC, 2014) represented risk as "the probability of occurrence of hazardous events or trends multiplied by the impacts if these events or trends occur." These definitions make intuitive sense, but they need to be formalised to make risk analysis unambiguous and uniquely quantifiable. Unfortunately, the literature has been deficient in that respect, with especially the definition of vulnerability lacking any consensus (Ionescu et al., 2009).

In earlier work, we showed how risk, hazard probability and system vulnerability should be defined to allow formal decomposition of risk as the mathematical product of the other two terms (Van Oijen et al., 2013). We applied this risk-decomposition to the problem of present and future drought risk of vegetation across Europe (Van Oijen et al., 2014). We also applied the method to forest data sets and introduced formulas for quantifying uncertainties associated with sampling-based estimates of risk and its two components (Van Oijen and Zavala, 2019). These studies form a useful basis for rigorous probabilistic risk analysis that is slowly becoming adopted more widely (e.g. Kuhnert et al. (2017); Zhou et al. (2018); He et al. (2021); Nandintsetseg et al. (2021)). However, the applications have so far been restricted to 2-factor sampling-based risk analysis. This book expands the approach to a comprehensive theory of risk analysis and elucidates the links with Bayesian decision theory. We derive formulas for estimating risk and its components, and for quantifying the uncertainties associated with all terms.