Understand
In this manuscript, we derive optimal conditions for building function approximations that minimize variance when used as importance sampling estimators for Monte Carlo integration problems.
- Particularly, we study the problem of finding the optimal projection $g$ of an integrand $f$ onto certain classes of piecewise constant functions, in order to minimize the variance of the unbiased importance sampling estimator $E_g[f/g]$, as well as the related problem of finding optimal mixture weights to approximate and importance sample a target mixture distribution $f = \sum_i \alpha_i f_i$ with components $f_i$ in a family $\mathcal{F}$, through a corresponding mixture of importance sampling densities $g_i$ that are only approximately proportional to $f_i$.
- We further show that in both cases the optimal projection is different from the commonly used $\ell_1$ projection, and provide an intuitive explanation for the difference.