I have a real function $f$ over vector $v$ and a multivariate normal probability distribution over these vectors, $p(v) \sim \cal{N}(\mu, \Sigma)$. I want to calculate approximation of this integral:
$$\int_{v} f(v) \ p(v) \ \mathrm{d}v$$
I know there are numerical methods which will sample and choose the values $v$ where to sample it, but my situation is different. I already have a given set of points $v$ (which is distributed according to different distribution, say $v \sim \cal{N}(\mu_v, \Sigma_v)$), and I am able to calculate $f(v)$'s for them. Ideally, I would like to find weights $w(v)$ so that the following sum approximates the integral:
$$\sum_{v} f(v) \cdot w(v)$$
Is that possible? How to set the weights $w(v)$? I guess it would be a function of $\mbox{multivatiate_normal_PDF}(v|\mu, \Sigma)$ .... I would like a very easy computation that I can do myself e.g. in R.
I think I found a solution, I hope it's correct:
$$w'(v) = {\mbox{multivatiate_normal_PDF}(v|\mu, \Sigma) \over \mbox{multivatiate_normal_PDF}(v|\mu_v, \Sigma_v)}$$
which will remove the "actual" distribution of $v$'s and scale it to the "desired" distribution. The weights then need to be normalized, i.e. the final weights would be:
$$w(v) = { w'(v) \over \sum_{v \in V} w'(v)}$$