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 its basic configuration, it works by drawing random samples from several proposal distributions (also called techniques) and weighting them appropriately in such a way that an estimator built with the pairs of weighted samples is consistent. 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). In the balance heuristic method, different samples are simulated from each proposal and the traditional IS weight is assigned to each of them. Unlike the standard IS estimator, all the weighted samples are combined with an extra weighting, in such a way the resulting estimator typically shows a reduced variance. 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. The balance heuristic, also called the deterministic mixture [3], has been widely used in the literature of MIS. Further efficient variance reduction techniques are proposed in [4,5,6] also in the context of MIS, still with equal counts from each technique. Provably better estimators [7] and heuristically better ones [8,9] have been presented that use a different count of samples than equal count for all techniques. In [10], it has been shown the relationship of a better count of samples with generalized weighted means. The balance heuristic is also present in most of successful adaptive IS (AIS) methods, see [11,12,13,14,15], in particular in the case where all techniques are used to simulate the same number of samples. Recently, MIS has returned to the main focus in computer graphics in [16], where it is shown that allowing weights to be negative, the reduction over balance heuristics of the resulting MIS estimator can be higher than the one predicted by Veach bounds [17]; in [18], where one of the sampling techniques is optimized to decrease the overall variance of the resulting MIS estimator; in [19], where the weights are made proportional to the quotients of second moments divided by the variances of the independent techniques; in [20], where MIS is generalized to uncountably infinite sets of techniques.

Interestingly, the balance heuristic has two properties in the assigned weights. First, all techniques appear at the denominator of the weight of a specific technique. Second, they appear in a form of a mixture, with coefficients proportional to the number of samples simulated from each technique. In this paper, we relax this constraint providing a generalized weighting/combining family of estimators that has the balance heuristic as a particular case. First, we show that it is possible to use a specific set of coefficients to decide the amount of samples per technique, and a different set of coefficients to be applied as the importance weight. Second, 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. Note that, the novel estimator always outperforms the balance heuristic under the optimal choice of those coefficients. In five numerical examples we show that, under an adequate choice of parameters, the novel estimator outperforms the celebrated balance heuristic.

The rest of the paper is structured as follows. Section 2 revisits the balance heuristic estimator. In Section 3, we propose the new family of estimators that generalizes the balance heuristic. We address five cases of special interest, depending on which parameters are free and the number of samples simulated from each technique. We then, in Section 4, discuss the different singular points in the variances of the estimators considered. In Section 5, we give a necessary and sufficient condition, in terms of ${\chi }^2$ divergence (and its approximation as a Kullback–Leibler divergence), for the variance of the new estimator to be better than the one of balance heuristics. Finally, we conclude the paper with five numerical examples in Section 6, propose some heuristics, and give some conclusions in Section 7.

The goal in IS is usually the estimation of the value of integral $\mu =\int f\left(\mathbf{x}\right)\mathrm{d}\mathbf{x}$. In MIS, $n_i$ samples, ${\left\{{\mathbf{X}}_{i,j}\right\}}_{j=1}^{n_i}$, are simulated from a set of available probability density functions (pdfs), ${\left\{p_i\right\}}_{i=1}^n$. The MIS estimator introduced by Veach and Guibas [1] is given by where $w_i\left(\mathbf{x}\right)$ is a weight function associated to the i-th proposal that fulfills both following conditions. First, the weights must sum up to one in all points of the domain where the value of the function is different from zero, i.e., ${\sum }_{i=1}^n{w}_i\left(\mathbf{x}\right)=1$, $\forall x$ where $f\left(\mathbf{x}\right)\ne 0$, Second, for all x where $p_i\left(\mathbf{x}\right)=0$, then $w_i\left(\mathbf{x}\right)=0$. In this paper, we consider only weighting functions that yield Z unbiased.

The balance heuristic estimator is a particular case of Equation (1) where the weight function is given by which can be written too as where $n_i={\alpha }_{i}N$. In this case, the estimator in Equation (1) becomes the balance heuristic or deterministic mixture estimator given by

In classic balance heuristic, both the number of samples ${\{n_i={\alpha }_{i}N\}}_{i=1}^n$ and the weights of the mixture of pdfs, ${\left\{{\alpha }_i\right\}}_{i=1}^n$, are given by the same set of parameters. Let us consider now the estimator of Equation (4), where we relax the dependence between the number of samples $n_i$, and the associated coefficient ${\alpha }_i$, i.e., now $n_i={\beta }_{i}N$, ${\beta }_i>0$, ${\sum }_{i=1}^n{\beta }_i=1$, where in general ${\alpha }_i\ne {\beta }_i$ (otherwise, we recover F). We now define the estimator Note that G is a particular case of Z, with weights $w_i=\frac{{\alpha }_i{p}_i\left(\mathbf{x}\right)}{{\sum }_{k=1}^n{\alpha }_k{p}_k\left({\mathbf{X}}_{i,j}\right)}$ in Equation (1). Note also that the balance heuristic F is a particular case of G, i.e., in general we do not impose the restriction of ${\alpha }_i=\frac{n_i}{N}$. Although the allocation of samples will be different for F and G, we remark that both estimators require the same number of simulations and evaluations. Finally, we note that a close examination of (16) allows us to re-interpret the importance weight associated to each sample. The mixture parametrized with ${\{n_i={\alpha }_{i}N\}}_{i=1}^n$ appears in its denominator, but since this mixture does not reflect the sampling procedure (the true mixture proposal in sampling is the one parametrized by ${\{n_i={\beta }_{i}N\}}_{i=1}^n$), the importance weight of a sample simulated from the technique ${\beta }_i$, is corrected by the factor ${\alpha }_i/{\beta }_i$ in order to account for this mismatch.

Estimator G can be rewritten as $G={\sum }_{i=1}^n{\alpha }_i{G}_i$, where Note also that G depends of two sets of parameters, ${\left\{{\alpha }_i\right\}}_{i=1}^n and {\left\{{\beta }_i\right\}}_{i=1}^n$. In the particular case where ${\beta }_i={\alpha }_i,\forall i$, the estimator G becomes F. Let us consider the case with $n_i=1$. Then, with expectation and variance Observe that $E\left[G_i\right]=E\left[G_i^{\prime }\right]={\mu }_i^{\prime }$, and $V\left[G_i\right]=\frac{1}{n_i}V\left[G_i^{\prime }\right]=\frac{1}{n_i}{{\sigma }^{\prime }}_i^2$.

The variance of G is given by

For the sake of the theoretical analysis, we define $G^1$, a normalized version of G with $N=1$ (see Section 2.1), with variance Next we study four special cases of estimator $G^1$.

The variance of $\mathcal{F}$ can be written as [7] The optimal parameters ${\left\{{\alpha }_i\right\}}_{i=1}^n$ happen when for all i, ${{\sigma }^{\prime }}_i^2+{{\mu }^{\prime }}_i^2$ equal, which gives a local minimum for $V\left[\mathcal{F}\right]$ (in this case a global minimum too, as $V\left[\mathcal{F}\right]$ is convex). Observe that this makes, in the third inequality, all terms factored by $\left\{{\alpha }_i\right\}$ equal, and also corresponds to equality in the Cauchy–Schwartz inequality with equality only when, for all i, ${{\sigma }^{\prime }}_i^2+{{\mu }^{\prime }}_i^2$ are equal.

If there exist ${\left\{{\alpha }_i^{\star }\right\}}_{i=1}^n$ values such that for all i, ${\mu }_i^{\prime }=\mu$ and all ${{\sigma }^{\prime }}_i$ are equal, these values correspond to the global optimum $V\left[{\mathcal{F}}^{\star }\right]=V\left[F^{\star }\right]=V\left[G^{\star }\right]$, and $F^{\star }\equiv G^{\star }$, but $F^{\star }\not\equiv {\mathcal{F}}^{\star }$. Suppose there exist ${\left\{{\alpha }_i^{\star }\right\}}_{i=1}^n$ such that for all i, ${\mu }_i^{\prime }=\mu$, then we can not improve the F estimator with a G estimator with a suitable selection of ${\left\{{\beta }_i\right\}}_{i=1}^n$ parameters, because the optimal $\left\{{\alpha }_i\right\}$ parameters for the G estimator when ${\left\{{\beta }_i\right\}}_{i=1}^n={\left\{{\alpha }_i^{\star }\right\}}_{i=1}^n$ are precisely $\left\{{\alpha }_i^{\star }\right\}$. This is, the optimal of $V\left[G({\left\{{\alpha }_i\right\}}_{i=1}^n,{\left\{{\beta }_i\right\}}_{i=1}^n={\left\{{\alpha }_i^{\star }\right\}}_{i=1}^n)\right]$ is for ${\left\{{\alpha }_i\right\}}_{i=1}^n={\left\{{\alpha }_i^{\star }\right\}}_{i=1}^n$.