direct estimation of the probability of failure: perform a rollout of the system label the outcome as 1 if the trajectory is a failure, and 0 otherwise this is just Direct Sampling. From there, we can just go about estimating this using standard parameter estimation (i.e. using MLE estimation or Baysian estimation.) maximum-likelihood estimation of failure distribution \begin{equation} \hat{p}{\text{fail}} = \frac{1}{m} \sum{i=1}^{m} 1\left{\tau_{i} \not \in \psi\right} = \frac{n}{m} \end{equation} for n failures and m rollouts, where \tau \sim p\left(\cdot\right). Bayesian estimation of failure distribution \begin{equation} p \left(\theta | D\right) = \frac{p\left(D| \theta\right) p\left(\theta\right)}{\int_{\theta} p\left(D | \theta\right) p\left(\theta\right) \dd{\theta}} \end{equation} whereby \theta := p_{\text{fail}}, n is the number of failures, m is the number of total trials