The standard Monte Carlo estimator $\widehat{I}_N^{\mathrm{MC}}$ of $\int fdω$ relies on independent samples from $ω$ and has variance of order $1/N$. Replacing the samples with a determinantal point process (DPP), a repulsive distribution, makes the estimator consistent, with variance rates that depend on how the DPP is adapted to $f$ and $ω$. We examine two existing DPP-based estimators: one by Bardenet & Hardy (2020) with a rate of $\mathcal{O}(N^{-(1+1/d)})$ for smooth $f$, but relying on a fixed DPP. The other, by Ermakov & Zolotukhin (1960), is unbiased with rate of order $1/N$, like Monte Carlo, but its DPP is tailored to $f$. We revisit these estimators, generalize them to continuous settings, and provide sampling algorithms.

翻译：标准蒙特卡罗估计量 $\widehat{I}_N^{\mathrm{MC}}$ 用于计算 $\int fdω$，其依赖于从 $ω$ 中独立采样的样本，方差阶数为 $1/N$。将样本替换为行列式点过程（DPP）（一种排斥性分布）后，该估计量具有一致性，其方差速率取决于DPP如何适配 $f$ 和 $ω$。我们考察了两种基于DPP的现有估计量：一种是Bardenet & Hardy（2020）提出的方法，对于光滑函数 $f$，其收敛速率为 $\mathcal{O}(N^{-(1+1/d)})$，但依赖于固定的DPP；另一种是Ermakov & Zolotukhin（1960）提出的方法，该估计量无偏且速率阶数为 $1/N$（与蒙特卡罗方法相同），但其DPP需针对 $f$ 定制。我们重新审视了这些估计量，将其推广至连续情形，并提供了相应的采样算法。