Dynamic splitting simulation

Splitting is a method for zooming in to the important region of an intractable probability distribution

I have just spent so much time writing about this that I had better pause for a while and leave this as a placeholder.

Splitting for static problems

We have a density \(f\) on \(\bb R^d\) and a quasi-monotone importance function \(S: \bb R^d\rightarrow\bb R.\) We wish to simulate from a conditional

\[ \begin{aligned} f^*(\vv x) &:=f(\vv x|S(\vv x)\geq\gamma)\\ &=\frac{1}{\ell(\gamma)}f(\vv x)\bb I\{S(\vv x)\geq\gamma)\} \end{aligned} \]

where \(\ell(\gamma)=\bb P[S(\vv X)\geq\gamma)],\quad \vv X \sim f.\)

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