I'm looking through my notes on importance sampling.
$\mu = \int h(x)\pi(x)dx = \int [h(x) \frac{\pi(x)}{g(x)}]g(x)dx$
draw $x^{(1)},...,x^{(m)}$ i.i.d. from proposal distribution $g(x)$
Calculate the importance weights:
$w^{(i)}=\frac{\pi(x^{(i)})}{g(x^{(i)})},$ for $i = 1,...,m$
This is the part where I'm confused. What is the distribution $\pi(x)$? Where did it come from? Do we choose it, similarly to how we chose $g(x)$?
As an example, I'm asked the following: Suppose $X \sim Unif[0,1]$ and we want to estimate $E[sin(\sqrt{X})]$....
So... $h(x) = \sin(\sqrt{X})$. Is $\pi(x)$ = 1 in this case? Since $X \sim Unif[0,1]$
Edit: I think I'm correct about the above stuff. Would $Unif[0,40]$ be a good choice for $g(x)$? $Unif[0,40]$ covers a lot more of $sin(\sqrt{X})$ than $Unif[0,1]$. Or are we only concerned with the portion of $h(x)$ bounded by [0,1]?