The $L_p$-error rate for randomized quasi-Monte Carlo self-normalized importance sampling of unbounded integrands
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915622160433152 |
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| author | Du, Jiarui He, Zhijian |
| author_facet | Du, Jiarui He, Zhijian |
| contents | 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 \(\mathcal{O}(N^{-β+ ε})\) for RQMC-SNIS estimators, where $ε>0$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_10599 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $L_p$-error rate for randomized quasi-Monte Carlo self-normalized importance sampling of unbounded integrands Du, Jiarui He, Zhijian Numerical Analysis 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 \(\mathcal{O}(N^{-β+ ε})\) for RQMC-SNIS estimators, where $ε>0$ 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. |
| title | The $L_p$-error rate for randomized quasi-Monte Carlo self-normalized importance sampling of unbounded integrands |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2511.10599 |