In numerical integration, cubature methods are effective, especially when the integrands can be well-approximated by known test functions, such as polynomials. However, the construction of cubature formulas has not generally been known, and existing examples only represent the particular domains of integrands, such as hypercubes and spheres. In this study, we show that cubature formulas can be constructed for probability measures provided that we have an i.i.d. sampler from the measure and the mean values of given test functions. Moreover, the proposed method also works as a means of data compression, even if sufficient prior information of the measure is not available.

中文翻译：

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蒙特卡洛文化建设

在数值积分中，体积法是有效的，尤其是当被积函数可以通过已知的测试函数（例如多项式）很好地逼近时。然而，体积公式的构造尚未为人所知，现有的例子仅代表被积函数的特定域，例如超立方体和球体。在这项研究中，我们表明，只要我们有来自度量和给定测试函数的平均值的 iid 采样器，就可以为概率度量构建体积公式。此外，即使没有足够的测量先验信息，所提出的方法也可以作为数据压缩的一种手段。