The $L_p$-error rate for randomized quasi-Monte Carlo self-normalized importance sampling of unbounded integrands

Abstract

Self-normalized importance sampling (SNIS) is a fundamental tool in Bayesian inference when the posterior distribution involves an unknown normalizing constant. In many applications, both the test function of interest and the underlying state space are unbounded, making direct $L_1$-error (mean absolute error) and $L_2$-error (root mean square error) estimates challenging for SNIS under randomized quasi-Monte Carlo (RQMC) sampling. In this work, we derive the $L_p$-error rate $(p\ge 1)$ for RQMC-based SNIS (RQMC-SNIS) estimators with unbounded integrands on unbounded domains. A key step in our analysis is to first establish the $L_p$-error rate for plain RQMC integration. Our results allow for a broader class of transport maps used to generate samples from RQMC points. Under mild function boundary growth conditions, we further establish the \(L_p\)-error rate of order \(\mathcal{O}(N^{-β+ ε})\) for RQMC-SNIS estimators, where $\hat{I}\mu >0$ is arbitrarily small, $N$ is the sample size, and \(β\in (0,1]\) depends on the boundary growth rate of the resulting integrand. Numerical experiments validate the theoretical results.