We revisit the multiple importance sampling (MIS) estimator and investigate the bound on the efficiency\nimprovement over balance heuristic estimator with equal count of samples established in Veach�s thesis. We revise\nthe proof for this and come to the conclusion that there is no such bound and henceforth it makes sense to look for\nnew estimators that improve on balance heuristic estimator with equal count of samples. Next, we examine a recently\nintroduced non-balance heuristic MIS estimator that is provably better than balance heuristic with equal count of\nsamples, and we improve it both in variance and efficiency. We then obtain an equally provably better one-sample\nbalance heuristic estimator, and finally, we introduce a heuristic for the count of samples that can be used when the\nindividual techniques are biased. All in all, we present three new sampling strategies to improve on both variance and\nefficiency on the balance heuristic using non-equal count of samples.\nOur scheme requires the previous knowledge of several quantities, but those can be obtained in an adaptive way. The\nresults also show that by a careful examination of the variance and properties of the estimators, even better\nestimators could be discovered in the future. We present examples that support our theoretical findings.
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