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^{*}\qty(\tau) = \frac{p\qty(\tau) 1\qty {\tau \not \in \psi}}{p_{\text{fail}}} = \frac{p\qty(\tau) 1\qty {\tau \not \in \psi}}{\int 1 \qty {\tau \not\in \psi} p\qty(\tau) \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