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

TL;DR

This paper derives $L_p$-error rates for randomized quasi-Monte Carlo self-normalized importance sampling with unbounded integrands, expanding theoretical understanding and validating results through numerical experiments.

Contribution

It establishes $L_p$-error rates for RQMC-SNIS estimators with unbounded integrands, broadening the theoretical framework for importance sampling.

Findings

$L_p$-error rate of order $ ext{O}(N^{-eta + ext{epsilon}})$ under mild conditions

Extension of $L_p$-error analysis to unbounded integrands on unbounded domains

Numerical experiments confirm the theoretical error rates

Abstract

Self-normalized importance sampling (SNIS) is a fundamental tool in Bayesian inference when the posterior distribution involves an unknown normalizing constant. Although $L_1$-error (bias) and $L_2$-error (root mean square error) estimates of SNIS are well established for bounded integrands, results for unbounded integrands remain limited, especially under randomized quasi-Monte Carlo (RQMC) sampling. In this work, we derive $L_p$-error rate $(p\ge1)$ 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 \(L_p\)-error rate of order \(\mathcal{O}(N^{-\beta + \epsilon})\) for RQMC-SNIS estimators, where $\epsilon>0$ is arbitrarily small, $N$ is the sample size, and \(\beta \in (0,1]\) depends on the boundary growth rate of the resulting integrand. Numerical experiments validate the theoretical results.