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Multiple importance sampling (MIS) methods approximate moments of complicated distributions by drawing samples from a set of proposal distributions. Several ways to compute the importance weights assigned to each sample have been recently proposed, with the so-called deterministic mixture (DM) weights providing the best performance in terms of variance, at the expense of an increase in the computational cost. A recent work has shown that it is possible to achieve a tradeoff between variance reduction and computational effort by performing an a priori random clustering of the proposals (partial DM algorithm). In this paper, we propose a novel "heretical" MIS framework, where the clustering is performed a posteriori with the goal of reducing the variance of the importance sampling weights. This approach yields biased estimators with a potentially large reduction in variance. Numerical examples show that heretical MIS estimators can outperform, in terms of mean squared error, both the standard and the partial MIS estimators, achieving a performance close to that of DM with less computational cost.
biased estimation; deterministic mixture (dm); monte carlo methods; multiple importance sampling (mis); population monte-carlo