Rejection Sampling

steps (coda) for some unnormalized target failure density, which is our target (and nominal trajectory p\left(\tau\right)):

\begin{equation} \bar{p} \left(\tau \mid \tau \not \in \psi\right) = \mathbb{1}\left\{\tau \not \in \psi\right\} p\left(\tau\right) \end{equation}

sample \tau \sim q\left(\cdot\right) where q is the proposal distribution where you start generating your samples; you want this to be as close as you can to the target failure distribution.
reject if cq\left(\tau\right) r > \bar{p}\left(\tau\right) first, choose a normalizing constant c which makes

\begin{equation} \bar{p} \left(\tau\right) \leq c q\left(\tau\right) \end{equation}

true. This allows us to rescale our proposal distribution to be at least as big as our target distribution.
In particular keep a sample of \tau if:

\begin{equation} r < \frac{\bar{p}\left(\tau\right)}{cq\left(\tau\right)} \end{equation}

where c is a constant that makes the inequality true; \bar{p} is the not normalized density (a la a Failure Distribution).
drawbacks selecting an appropriate proposal distribution \tau \sim q\left(\cdot\right) is hard selecting an appropriate value for c 109’s definition Sample a ton; perform factor conditioning; then count the observation you’d like.
by rejecting those who don’t match