proposal distribution — Jemoka Knowledge Base

We define the optimal proposal distribution as the one that minimizes the variance of the estimator of the Probability of Failure. Sadly, the best proposal distributions is…

\begin{equation} q^{*}\left(\tau\right) = \frac{p\left(\tau\right) 1\left\{\tau \not \in \psi\right\}}{p_{\text{fail}}} = \frac{p\left(\tau\right) 1\left\{\tau \not \in \psi\right\}}{\int 1 \left\{\tau \not\in \psi\right\} p\left(\tau\right) \dd{\tau }} \end{equation}

but wait this is just the Failure Distribution! But our entire point is trying to estimate p_{\text{fail}}. notice that this is exactly the DEFINITION OF THE FAILURE DISTRIBUTION. et, we were trying to estimate p_{\text{fail}} in the first place? Recall; we are able to sample from the Failure Distribution, fit a model and nice. Yet, this brings two challenges sampling from Failure Distribution is quite hard it maybe difficult to produce a good fit with higher dimensional systems see adaptive cross entropy method with adaptive importance sampling population monte-carlo what if you are doing multiple importance sampling and so you need a whole bunch of proposals? let’s just keep around a bunch of proposals select an initial populating of proposals draw a sample from each proposal compute the importance weight for each sample resample based on importance weights create new proposal distribution centered at the samples—perhaps with constant variance