Bias point selection in the importance sampling Monte Carlo simulation of systems

We consider the issue of whether it is better to bias the random variables at the input, at the output, or at some intermediate point of a system. We show that in a very general setting, the closer to the output that we can bias our system simulation variables, the better off we will be. We show that surprisingly, in some important special cases, the performance can be equal no matter where the bias point is selected. In the second part of the paper, we present a very general large deviation-type theorem on the variance rates of importance sampling estimators. We then use this theorem to consider, in a quantitative fashion, what the difference in the variance rates can be for input versus output formulations. We present several examples illustrating the developed theory.