Notes on optimal approximations for importance sampling
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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 = ∑_i α_i f_i with components f_i in a family 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 ℓ_1 projection, and provide an intuitive explanation for the difference.
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