论文标题
关于随机准蒙特卡洛的重要性采样错误率
On the error rate of importance sampling with randomized quasi-Monte Carlo
论文作者
论文摘要
重要性抽样（IS）对于减少许多领域的蒙特卡洛采样方差（包括财务，稀有事件模拟和贝叶斯推论）是有价值的。将准蒙特卡洛（QMC）方法与IS达到更快的收敛速度相结合是自然而明显的。但是，用QMC幼稚地更换蒙特卡洛可能无法正常工作。本文研究了基于随机QMC的随机收敛速率，用于估算高斯度量的积分，其中IS度量是高斯或$ t $分布。我们证明，如果目标函数满足所谓的边界生长条件，并且IS密度的协方差矩阵的特征值不小于1，则使用高斯提案的随机QMC具有$ o（n^{ - 1+ε}）的根平方误差，用于任意小$> $>> 0 $。也建立了$ t $分销的类似结果。这些足够的条件有助于评估QMC中IS的有效性。对于某些特定的应用，我们发现Laplace是一种非常通用的方法，可以通过其模式周围的二次泰勒近似值近似目标函数，其特征值小于1，从而使所得的积分对QMC有利。从这个角度来看，当将高斯分布用作IS提议时，尽管蒙特卡洛采样的差异降低了，但通过拉普拉斯的度量变化可能会转化为QMC不利的集成。我们还举了一些例子来验证我们的命题，并警告不要以QMC为基础的MC幼稚地替代。数值结果表明，使用laplace具有$ t $发行量比高斯分布更强大。
Importance sampling (IS) is valuable in reducing the variance of Monte Carlo sampling for many areas, including finance, rare event simulation, and Bayesian inference. It is natural and obvious to combine quasi-Monte Carlo (QMC) methods with IS to achieve a faster rate of convergence. However, a naive replacement of Monte Carlo with QMC may not work well. This paper investigates the convergence rates of randomized QMC-based IS for estimating integrals with respect to a Gaussian measure, in which the IS measure is a Gaussian or $t$ distribution. We prove that if the target function satisfies the so-called boundary growth condition and the covariance matrix of the IS density has eigenvalues no smaller than 1, then randomized QMC with the Gaussian proposal has a root mean squared error of $O(N^{-1+ε})$ for arbitrarily small $ε>0$. Similar results of $t$ distribution as the proposal are also established. These sufficient conditions help to assess the effectiveness of IS in QMC. For some particular applications, we find that the Laplace IS, a very general approach to approximate the target function by a quadratic Taylor approximation around its mode, has eigenvalues smaller than 1, making the resulting integrand less favorable for QMC. From this point of view, when using Gaussian distributions as the IS proposal, a change of measure via Laplace IS may transform a favorable integrand into unfavorable one for QMC although the variance of Monte Carlo sampling is reduced. We also give some examples to verify our propositions and warn against naive replacement of MC with QMC under IS proposals. Numerical results suggest that using Laplace IS with $t$ distributions is more robust than that with Gaussian distributions.
