  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.
