Monte Carlo Sampling on Integrand with Poles How does one do MC for integrand which has poles?
$I=\int \frac{e^{-S(x)}}{f(x)}dx$
where $x$ would be multi-dimensional, and hence this is a multi-dimensional integral over coordinates such as $x_1,x_2,x_3$ and so on. 
I want to do importance sampling. Suppose this function has poles at $x_i$, such that $f(x_i)=0$. How does one perform importance sampling on functions with poles? Generally, how does one perform MC sampling on integrand with pole(s)?
 A: Just do normal Monte Carlo as usual
First let's assume that the integral is well defined! If the integral is ill defined or does not exist then you have other bigger problems with the underlying mathematics.
So taking the integral to exist, lets assume the integral is over a finite domain, which let's hope is scaled to the unit hypercube (if the integral limits are over all of $\mathbb{R}^n$ then you should perform a substitution to a finite domain).
You'll never hit a pole
Why even worry about the poles? If a sample $x_\omega $ is drawn, then it will almost surely not be at a pole (assume you're poles don't form a dense set!), where $\mathbb{P}(x_\omega \in \{x_i\}) = 0$. Of course this is a mathematical nicety. However, if you're using actual numbers drawn from a pseudo random number generator in finite precision then the number produced may be at a pole. This is a  technical pitfall of real-world implementations, and it will be up to you to decide what to do in this case (discard, resample, perturb, etc). Hopefully for any reasonable sample size this will almost never happen. 
