Defining what is a probability of failure for systems modelled by stochastic simulators

Reliability analysis is a field of uncertainty quantification primarily concerned with estimating the probability that a system response exceeds a critical threshold, resulting in failure. By design, such a probability of failure is small. Consequently, accurately computing it requires many evaluations of the so-called limit state function, a computational model that classifies whether the system fails or not, and that is often expensive to evaluate. Historically, limit-state functions have been considered deterministic simulators, a particular class of models that have a fixed response for a given input. Some systems, however, are inherently stochastic and cannot be deterministically modelled. Instead, so-called stochastic simulators must be used in their modelling. Examples of these simulators include disease propagation models or wind turbine simulators. In contrast to their deterministic counterparts, different outputs may be obtained when repeating their evaluation for the same input. Because the current definition of reliability analysis relies on deterministic limit-state functions, defining a reliability index for stochastic simulators is not straightforward. In this contribution, we aim to broaden the definition of reliability analysis, to comply with stochastic simulators. With that aim, we discuss suitable reliability measures and estimate them using a Monte-Carlo approach. Additionally, we showcase our results using suitably designed benchmark problems that serve as academic validation, while emulating credible scenarios. We show that when considering stochastic limit-state functions, the failure probability is not always a scalar value. Instead, it becomes a random variable that may be fully characterized by its probability density function. Moreover, we show that the stochasticity of the model severely affects the estimated probability of failure, causing it to vary up to several orders of magnitude