# Bayesian Approaches

The most common approach to analyze the risk of accidents in the energy sector relies on quantitative measures such as aggregated risk indicators (e.g., fatality rates), allowing to compare energy chains in comprehensive manner. In this context, the calculation of objective risk indicators comprises a valuable element to support decision makers in the assessment of current and future technology portfolios. Hence, there is a clear need for more and better data, and accordingly, improved uncertainty quantification for accident risk indicators, because they are the basis to support decision-makers and risk managers in their efforts to design and implement better risk management strategies, including both preventive measures (pre-event) and mitigation processes (post-event).

For risk assessment of accidents in the energy sector, only recently activities related to the treatment of uncertainties in a consistent way have been developed. Within the PSI activities on the assessment of risk indicators and their uncertainty, Bayesian inference has been demonstrated to be useful, since uncertainty is intrinsically assessed through it. In fact, a Bayesian analysis is a fully probabilistic approach that accounts for both epistemic and aleatory uncertainties. The Bayesian Theorem can be written as follow:

p(θ│y)∝L(y; θ)p(θ)

The posterior distribution p(θ│y) in a Bayesian analysis is given by the product of:

• a priori distribution p(θ), which describes what is known about the parameter of interest before observing any data, i.e., it contributes to the epistemic uncertainty, since it intrinsically describes the lack of knowledge on the parameter of interest;
• a likelihood function L(y; θ), which describes the process-giving rise to data in terms of the unknown parameter of interest, i.e., it contributes to the random variability of the parameter of interest, and thus describes the aleatory uncertainty.

Furthermore, Bayesian analysis can be extended to, among others:

• model distributions in a robust way for subsets with scarce data (Bayesian Hierarchical Model);
• for quantile regression to assess the indicators representing extreme risk;
• for trend analysis;
• model distributions for tiny datasets, i.e., very small datasets (Approximate Bayesian Computation, ABC).

A selection of references on the topic are shown below, while other publications can be found here.

Kalinina, A., Spada, M., Burgherr, P., 2020. Probabilistic Analysis of Dam Accidents Worldwide: Risk Assessment for Dams of Different Purposes in OECD and Non‐OECD Countries with Focus on Time Trend Analysis. Risk Analysis. http://dx.doi.org/10.1111/risa.13536

Spada, M., Burgherr, P., 2020. Comparative Risk Assessment for Fossil Energy Chains Using Bayesian Model Averaging. Energies 13(2). http://dx.doi.org/10.3390/en13020295

Spada, M., Burgherr, P., 2019. A Hierarchical Approximate Bayesian Computation (HABC) for Accident Risk in the Energy Sector triggered by Natural Events, in: Beer, M., Zio, E. (Eds.), 29th European Safety and Reliability Conference. Research Publishing, Hannover, Germany, pp. 1423-1430. http://dx.doi.org/10.3850/978-981-11-2724-3_0758-cd

Kalinina, A., Spada, M., Burgherr, P., 2018. Application of a Bayesian hierarchical modeling for risk assessment of accidents at hydropower dams. Safety Science 110, 164-177. http://dx.doi.org/10.1016/j.ssci.2018.08.006

Spada, M., Burgherr, P., 2016. An aftermath analysis of the 2014 coal mine accident in Soma, Turkey: Use of risk performance indicators based on historical experience. Accident Analysis & Prevention 87, 134-140. http://dx.doi.org/10.1016/j.aap.2015.11.020