Variance Analysis of Multi‐sample and One‐sample Multiple Importance Sampling

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Abstract
We reexamine in this paper the variance for the Multiple Importance Sampling (MIS) estimator for multi‐sample and one‐sample model. As a result of our analysis we can obtain the optimal estimator for the multi‐sample model for the case where the weights do not depend on the count of samples. We extend the analysis to include the cost of sampling. With these results in hand we find a better estimator than balance heuristic with equal count of samples. Further, we show that the variance for the one‐sample model is larger or equal than for the multi‐sample model, and that there are only two cases where the variance is the same. Finally, we study on four examples the difference of variances for equal count as used by Veach, our new estimator, and a recently introduced heuristic.