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

2025/11/13 by Jiarui Du, Du, Jiarui, Zhijian He +1
Computer Science · Mathematics · #FOS: Mathematics #Gaussian Processes and Bayesian Inference #Markov Chains and Monte Carlo Methods #Mathematical Approximation and Integration #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2511.10599

openalex publication_date 2025/11/13 · openalex created_date 2025/11/15 · openalex updated_date 2026/07/28

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 L₁-error (mean absolute error) and L₂-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≥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 \(O(N^{-β+ ε})\) for RQMC-SNIS estimators, where ε>0 is arbitrarily small, N is the sample size, and \(β∈ (0,1]\) depends on the boundary growth rate of the resulting integrand. Numerical experiments validate the theoretical results.

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