三元恒等式与准蒙特卡罗误差
The Trio Identity for Quasi-Monte Carlo Error
AI总结：
本文探讨了准蒙特卡罗方法中误差的三元恒等式，分析了变分、采样测度偏差和混淆因素的关系，并通过理论和实例说明如何通过低偏差采样和积分重写来减少立方误差。
摘要：
蒙特卡罗方法通过样本均值近似积分。蒙特卡罗方法的误差可表示为三元恒等式：被积函数的变分、采样测度的偏差和混淆因素的乘积。三元恒等式有不同版本，取决于被积函数是确定性还是贝叶斯，以及采样测度是确定性还是随机。尽管变分和偏差在文献中较为常见，但混淆因素相对不为人知。通过理论和实例说明如何通过低偏差采样（定义准蒙特卡罗方法）来减少立方误差。误差也可通过重新表述积分以不同的被积函数来减少。最后，混淆解释了立方误差可能以不同于偏差的速率衰减的原因。
英文摘要：
Monte Carlo methods approximate integrals by sample averages of integrand values. The error of Monte Carlo methods may be expressed as a trio identity: the product of the variation of the integrand, the discrepancy of the sampling measure, and the confounding. The trio identity has different versions, depending on whether the integrand is deterministic or Bayesian and whether the sampling measure is deterministic or random. Although the variation and the discrepancy are common in the literature, the confounding is relatively unknown and under-appreciated. Theory and examples are used to show how the cubature error may be reduced by employing the low discrepancy sampling that defines quasi-Monte Carlo methods. The error may also be reduced by rewriting the integral in terms of a different integrand. Finally, the confounding explains why the cubature error might decay at a rate different from that of the discrepancy.
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