Error Bounds for Importance Sampling with Estimated Proposal Distributions
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| Format: | Preprint |
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2026
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| author | Aeckerle-Willems, Cathrine Klebanov, Ilja Weissmann, Simon |
| author_facet | Aeckerle-Willems, Cathrine Klebanov, Ilja Weissmann, Simon |
| contents | Importance sampling with data-driven proposal distributions is widely used in practice. A common workflow first generates an auxiliary sample of size $N$ from an approximation of the target distribution, constructs a density estimate $\hat q$ such as a kernel density estimator (KDE), and then draws $n$ importance samples from this learned proposal. Despite its practical relevance, the theoretical properties of this hierarchical procedure remain poorly understood, since classical importance sampling theory assumes a fixed proposal.
We address this gap by deriving non-asymptotic error bounds for standard, defensive, and self-normalized importance sampling estimators with random proposals. Our results separate the Monte Carlo error, scaling as $n^{-1/2}$, from the proposal approximation error measured through the mean integrated absolute and squared errors (MIAE and MISE) of $\hat q$. To obtain explicit convergence rates in $(N,n)$, we establish MIAE and MISE bounds for KDEs constructed from geometrically ergodic Markov chains in stationary and non-stationary regimes. Combining these results yields quantitative guarantees for importance sampling with KDE-based proposals. Our theory provides practical guidance for selecting defensive mixture weights in a nonparametric importance sampling framework. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_19989 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Error Bounds for Importance Sampling with Estimated Proposal Distributions Aeckerle-Willems, Cathrine Klebanov, Ilja Weissmann, Simon Statistics Theory Numerical Analysis 62G07, 65C05, 65C40, 62G05 Importance sampling with data-driven proposal distributions is widely used in practice. A common workflow first generates an auxiliary sample of size $N$ from an approximation of the target distribution, constructs a density estimate $\hat q$ such as a kernel density estimator (KDE), and then draws $n$ importance samples from this learned proposal. Despite its practical relevance, the theoretical properties of this hierarchical procedure remain poorly understood, since classical importance sampling theory assumes a fixed proposal. We address this gap by deriving non-asymptotic error bounds for standard, defensive, and self-normalized importance sampling estimators with random proposals. Our results separate the Monte Carlo error, scaling as $n^{-1/2}$, from the proposal approximation error measured through the mean integrated absolute and squared errors (MIAE and MISE) of $\hat q$. To obtain explicit convergence rates in $(N,n)$, we establish MIAE and MISE bounds for KDEs constructed from geometrically ergodic Markov chains in stationary and non-stationary regimes. Combining these results yields quantitative guarantees for importance sampling with KDE-based proposals. Our theory provides practical guidance for selecting defensive mixture weights in a nonparametric importance sampling framework. |
| title | Error Bounds for Importance Sampling with Estimated Proposal Distributions |
| topic | Statistics Theory Numerical Analysis 62G07, 65C05, 65C40, 62G05 |
| url | https://arxiv.org/abs/2605.19989 |