On the convergence conditions of Laplace importance sampling with randomized quasi-Monte Carlo

TL;DR

This paper investigates the convergence conditions of Laplace importance sampling combined with randomized quasi-Monte Carlo methods, providing theoretical error bounds and analyzing factors affecting convergence in high-dimensional integrals.

Contribution

It introduces a theoretical framework for the convergence of Laplace importance sampling with RQMC, including error bounds and the impact of problem dimensions and smoothness.

Findings

IS with randomly shifted lattice rule can achieve near O(N^{-1}) error bound

Theoretical convergence rate depends on problem smoothness and dimension

Error bounds are derived using reproducing kernel Hilbert space (RKHS) techniques

Abstract

The study further explores randomized QMC (RQMC), which maintains the QMC convergence rate and facilitates computational efficiency analysis. Emphasis is laid on integrating randomly shifted lattice rules, a distinct RQMC quadrature, with IS,a classic variance reduction technique. The study underscores the intricacies of establishing a theoretical convergence rate for IS in QMC compared to MC, given the influence of problem dimensions and smoothness on QMC. The research also touches on the significance of IS density selection and its potential implications. The study culminates in examining the error bound of IS with a randomly shifted lattice rule, drawing inspiration from the reproducing kernel Hilbert space (RKHS). In the realm of finance and statistics, many problems boil down to computing expectations, predominantly integrals concerning a Gaussian measure. This study considers optimal drift importance sampling (ODIS) and Laplace importance sampling (LapIS) as common importance densities. Conclusively, the paper establishes that under certain conditions, the IS-randomly shifted lattice rule can achieve a near $O(N^{-1})$ error bound.