Abstract
In this paper, we propose a novel and generic family of multiple importance sampling estimators. We first revisit the celebrated balance heuristic estimator, a widely used Monte Carlo technique for the approximation of intractable integrals. Then, we establish a generalized framework for the combination of samples simulated from multiple proposals. Our approach is based on considering as free parameters both the sampling rates and the combination coefficients, which are the same in the balance heuristics estimator. Thus our novel framework contains the balance heuristic as a particular case. We study the optimal choice of the free parameters in such a way that the variance of the resulting estimator is minimized. A theoretical variance study shows the optimal solution is always better than the balance heuristic estimator (except in degenerate cases where both are the same). We also give sufficient conditions on the parameter values for the new generalized estimator to be better than the balance heuristic estimator, and one necessary and sufficient condition related to divergence. Using five numerical examples, we first show the gap in the efficiency of both new and classical balance heuristic estimators, for equal sampling and for several state of the art sampling rates. Then, for these five examples, we find the variances for some notable selection of parameters showing that, for the important case of equal count of samples, our new estimator with an optimal selection of parameters outperforms the classical balance heuristic. Finally, new heuristics are introduced that exploit the theoretical findings.
Highlights
Multiple importance sampling (MIS) is a Monte Carlo technique widely used in the literature of signal processing, computational statistics, and computer graphics for approximating complicated integrals
Since the publication of [1], the celebrated balance heuristic estimator has been extensively used in the Monte Carlo literature, with an unprecedented success in the computer graphics industry (Eric Veach has been awarded with several prizes because of his contributions in the MIS literature, where the balance heuristic is arguably the most relevant one)
We have proposed a multiple importance sampling estimator that combines samples simulated from different techniques
Summary
Multiple importance sampling (MIS) is a Monte Carlo technique widely used in the literature of signal processing, computational statistics, and computer graphics for approximating complicated integrals. In the balance heuristic method, different samples are simulated from each proposal and the traditional IS weight is assigned to each of them. Its superiority in terms of variance with regard to other traditional combination schemes has been recently shown in [2], where a framework is established for sampling and weighting in MIS under equal number of samples per technique. We relax this constraint providing a generalized weighting/combining family of estimators that has the balance heuristic as a particular case. We study four different cases fixing some of these coefficients (sampling and/or weighting), and we give the optimal solution for the rest of coefficients in such a way the variance of the MIS estimator is minimized.
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