Importance Sampling (IS), an effective variance reduction strategy in Monte Carlo (MC) simulation, is frequently utilized for Bayesian inference and other statistical challenges. Quasi-Monte Carlo (QMC) replaces the random samples in MC with low discrepancy points and has the potential to substantially enhance error rates. In this paper, we integrate IS with a randomly shifted rank-1 lattice rule, a widely used QMC method, to approximate posterior expectations arising from Bayesian Inverse Problems (BIPs) where the posterior density tends to concentrate as the intensity of noise diminishes. Within the framework of weighted Hilbert spaces, we first establish the convergence rate of the lattice rule for a large class of unbounded integrands. This method extends to the analysis of QMC combined with IS in BIPs. Furthermore, we explore the robustness of the IS-based randomly shifted rank-1 lattice rule by determining the quadrature error rate with respect to the noise level. The effects of using Gaussian distributions and $t$-distributions as the proposal distributions on the error rate of QMC are comprehensively investigated. We find that the error rate may deteriorate at low intensity of noise when using improper proposals, such as the prior distribution. To reclaim the effectiveness of QMC, we propose a new IS method such that the lattice rule with $N$ quadrature points achieves an optimal error rate close to $O(N^{-1})$, which is insensitive to the noise level. Numerical experiments are conducted to support the theoretical results.

翻译：重要性采样（IS）作为蒙特卡洛（MC）模拟中一种有效的方差缩减策略，常被用于贝叶斯推断及其他统计难题。准蒙特卡洛（QMC）方法采用低差异点替代MC中的随机样本，有望显著提升误差收敛速率。本文通过将IS与广泛应用的QMC方法——随机移位秩1格点法则相结合，以逼近贝叶斯反问题（BIPs）中的后验期望。在该类问题中，后验密度会随噪声强度降低而趋于集中。我们首先在加权希尔伯特空间框架下，针对一大类无界被积函数建立了格点法则的收敛速率，并将该方法推广至BIPs中QMC与IS联合分析的场景。进一步，我们通过确定格点法则随噪声水平的误差阶，探讨了基于IS的随机移位秩1格点法则的鲁棒性。系统研究了以高斯分布和t分布作为提议分布对QMC误差速率的影响，发现若采用不当提议分布（如先验分布），误差速率在低噪声强度下可能恶化。为恢复QMC的有效性，我们提出一种新型IS方法，使得含N个求积点的格点法则达到接近O(N^{-1})的最优误差速率，且该速率对噪声水平不敏感。数值实验结果验证了理论结论的可靠性。