For a trajectory \(p\qty(\tau)\), the failure distribution is $p \qty(τ | τ ¬ ∈ψ)$—the probability of a particular trajectory given that its a failure:
\begin{equation} p \qty( \tau \mid \tau \not \in \psi) = \frac{\mathbb{1}\qty {\tau \not \in \psi} p\qty(\tau)}{ \int \mathbb{1}\qty {\tau \not \in \psi} p\qty(\tau) \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} \qty( \tau \mid \tau \not \in \psi) = {\mathbb{1}\qty {\tau \not \in \psi} p\qty(\tau)} \end{equation}
so we can implicitly represents the failure distirbution using the drawn samples.