On the error rate of importance sampling with randomized quasi-Monte Carlo
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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 also a Gaussian. 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 has a mean error of O(N^-1+ϵ) for arbitrarily small ϵ>0. 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, 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 study the effect of positivization trick on the error rate when the integrand has mixed sign. If the smooth positivization proposed by Owen and Zhou (2000) is used, the rate O(N^-1+ϵ) is retained. This is not the case if taking the positive and negative parts of the integrand. We also give some examples to verify our propositions and warn against naive replacement of MC with QMC under IS proposals.
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