For a trajectory p\left(\tau\right), the failure distribution is p \left(τ | τ ¬ ∈ψ\right)—the probability of a particular trajectory given that its a failure:
\begin{equation} p \left( \tau \mid \tau \not \in \psi\right) = \frac{\mathbb{1}\left\{\tau \not \in \psi\right\} p\left(\tau\right)}{ \int \mathbb{1}\left\{\tau \not \in \psi\right\} p\left(\tau\right) \dd{\tau}} \end{equation}
This bottom integral could be very difficult to compute; but the numerator may take a bit more work to compute! So ultimately we can also give up and don’t normalize (and then use systems that allows us to draw samples from unnormalized probability densities:
\begin{equation} \hat{p} \left( \tau \mid \tau \not \in \psi\right) = {\mathbb{1}\left\{\tau \not \in \psi\right\} p\left(\tau\right)} \end{equation}
so we can implicitly represents the failure distirbution using the drawn samples. some ways of sampling from failure distribution Rejection Sampling Markov Chain Monte-Carlo