How to get a better importance sampling algorithm

Question

I am trying to code some importance sampling algorithms, but I have a question about how importance sampling works.

Say we want to estimate the expected value of some function $h(x)$, with samples drawn from the distribution defined by the pdf $\pi(x)$, which is hard to sample, so we have the proposal distribution defined by the pdf $g(x)$. When does this algorithm work better, when $g(x)$ is similar to $\pi(x)$ or similar to $|h(x)|\pi(x)$?

Answer 1

The most relevant result in that area is that the choice of $g$ that leads to the minimal variance is$$g^\star(x) = \dfrac{|h(x)\pi(x)|}{\int_\mathfrak X }$$