---
title: "Failure Distribution"
type: concept
related: [Markov Chain Monte Carlo, Rejection Sampling]
source: https://www.jemoka.com/posts/kbhfailure_distribution/
confidence: high
status: active
---

For a trajectory \(p\qty(\tau)\), the failure distribution is $p \qty(τ | τ ¬ ∈ψ)$&mdash;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&rsquo;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.
some ways of sampling from failure distribution Rejection Sampling Markov Chain Monte-Carlo
