带减法的蒙特卡罗积分
Monte Carlo Integration with Subtraction
摘要：
本文研究了紧致域上d维函数的一类Monte Carlo数值积分算法。我们构造一个直方图近似的功能使用分区的集成域到一组箱指定的一些参数。然后，我们考虑两个适应;第一个是减去直方图近似，其积分，我们可以很容易地明确评估，从功能和集成的差异使用Monte Carlo;第二个是修改bin参数，以使方差的Monte Carlo估计的积分相同的所有箱。这允许我们使用Student t检验作为重新分组的触发器，我们声称这比通常用于此目的的χ平方检验更稳定。我们提供了一个程序，我们已经用来研究的情况下，直方图表示为一维直方图的产品的算法。我们讨论的假设和近似值，以及给无数的方式，任何这样的蒙特卡罗积分程序的结果可能会误导教学讨论。
英文摘要：
This paper investigates a class of algorithms for numerical integration of a function in d dimensions over a compact domain by Monte Carlo methods. We construct a histogram approximation to the function using a partition of the integration domain into a set of bins specified by some parameters. We then consider two adaptations; the first is to subtract the histogram approximation, whose integral we may easily evaluate explicitly, from the function and integrate the difference using Monte Carlo; the second is to modify the bin parameters in order to make the variance of the Monte Carlo estimate of the integral the same for all bins. This allows us to use Student's t-test as a trigger for rebinning, which we claim is more stable than the χ-squared test that is commonly used for this purpose. We provide a program that we have used to study the algorithm for the case where the histogram is represented as a product of one-dimensional histograms. We discuss the assumptions and approximations made, as well as giving a pedagogical discussion of the myriad ways in which the results of any such Monte Carlo integration program can be misleading.
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