2017/07/26 by Jacopo Pantaleoni, Pantaleoni, Jacopo, Eric Heitz +1
Decision Sciences · Mathematics · #FOS: Computer and information sciences #FOS: Mathematics #Graphics (cs.GR) #Mathematical Approximation and Integration #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design #Statistical Methods and Inference
paper · pdf · doi:10.48550/arxiv.1707.08358
openalex publication_date 2017/07/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Eg[f/g], as well as the related problem of finding optimal mixture weights to approximate and importance sample a target mixture distribution f = ∑i αi fi with components fi in a family F, through a corresponding mixture of importance sampling densities gi that are only approximately proportional to fi. We further show that in both cases the optimal projection is different from the commonly used ℓ1 projection, and provide an intuitive explanation for the difference.