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. 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.

• The paper introduces L_p-error rate bounds for RQMC self-normalized importance sampling of unbounded integrands.
• It develops a novel multi-dimensional projection operator and general transport maps to overcome dimensionality issues.
• Numerical experiments in Bayesian inference confirm improved accuracy and performance over traditional methods.

Overview

"The $L_p$-error rate for randomized quasi-Monte Carlo self-normalized importance sampling of unbounded integrands" (2511.10599) presents advanced methodologies for addressing the limitations in traditional importance sampling techniques, specifically focusing on unbounded integrands and domains using randomized quasi-Monte Carlo (RQMC) methods. This work significantly broadens the scope of sample generation strategies by establishing error bounds for self-normalized importance sampling estimators in cases where integrands are unbounded, a scenario often encountered in practical computational challenges.

Importance Sampling Background

Monte Carlo (MC) methods are foundational for approximating expectations in probabilistic systems, especially when sampling directly from the target distribution is infeasible. Importance Sampling (IS) becomes critical when the normalizing constants of posterior distributions in Bayesian inference are unknown. The standard IS and its self-normalized variant (SNIS) are explored, with SNIS being preferable when $\pi(x)$ is known up to a normalizing constant.

For bounded integrands, both $L_1$-error (bias) and $L_2$-error (RMSE) estimates are well-documented. However, when dealing with unbounded test functions, these results become less applicable. The paper extends existing theories by developing $L_p$-error rates for RQMC-SNIS estimators, allowing integration over unbounded domains with improved efficacy compared to traditional MC and QMC methods.

Methodological Advancements

A significant contribution is the extension of $L_p$-error rates to RQMC integration, accommodating a broader class of transport maps beyond the typical inversion transformation. This is achieved by utilizing a novel projection operator defined in multi-dimensions, which circumvents dimensional dependency and facilitates smoother integration bounds.

The investigation encompasses:

• $L_p$-error bounds: Derived for both MC-SNIS and RQMC-SNIS methodologies, enabling error rate $\mathcal{O}(N^{-\beta + \epsilon})$ where $\epsilon>0$ is negligible, outlining how these bounds can be practically implemented.
• General transport maps: The paper successfully demonstrates the versatility in choosing transport maps, including linear and non-linear transformations, accommodating critical quadratic growth conditions.
• Higher-order moment analysis: Detailed the implications of tail-behavior estimations, pivotal for skewness and kurtosis considerations, directly influencing confidence intervals within Bayesian formulations.

Numerical Validation

Through numerical experiments, the paper highlights practical implementations in Bayesian inference scenarios including logistic regression, effectively evaluating the impact of dimension, skewness, kurtosis, and proposal distributions on estimator accuracy. These validations underscore the theoretical predictions and demonstrate the adaptability of RQMC-SNIS in real-world applications.

Conclusions

This research bridges significant gaps in the application of RQMC methods within Bayesian computational frameworks, particularly for unbounded test functions. By establishing $L_p$-error rates and demonstrating effective sample strategies, this study not only enhances theoretical understanding but also paves the way for more efficient computational practices in Bayesian statistics and related fields. Future developments may focus on dimension-independent performance improvements to further refine these techniques.