# BR-SNIS: Bias Reduced Self-Normalized Importance Sampling

arXiv: https://arxiv.org/abs/2207.06364 · HTML: https://mlsanity.com/paper/2207.06364/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 61 references that are in the map.
- First posted on arXiv 13 Jul 2022.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Importance Weighted Autoencoders](https://mlsanity.com/paper/1509.00519/) (2015)
- [Elements of Sequential Monte Carlo](https://mlsanity.com/paper/1903.04797/) (2019)
- [Policy Optimization via Importance Sampling](https://mlsanity.com/paper/1809.06098/) (2018)
- [Markovian Score Climbing: Variational Inference with KL(p||q)](https://mlsanity.com/paper/2003.10374/) (2020)
- [Non-asymptotic Analysis of Biased Stochastic Approximation Scheme](https://mlsanity.com/paper/1902.00629/) (2019)
- [Confident Off-Policy Evaluation and Selection through Self-Normalized Importance Weighting](https://mlsanity.com/paper/2006.10460/) (2020)
- [Mixing time estimation in reversible Markov chains from a single sample path](https://mlsanity.com/paper/1708.07367/) (2017)
- [Particle Gibbs with Ancestor Sampling](https://mlsanity.com/paper/1401.0604/) (2014)
- [Asymptotic Bias of Stochastic Gradient Search](https://mlsanity.com/paper/1709.00291/) (2017)
- [Optimal Bayesian Transfer Learning](https://mlsanity.com/paper/1801.00857/) (2018)
- [Energy-Inspired Models: Learning with Sampler-Induced Distributions](https://mlsanity.com/paper/1910.14265/) (2019)
- [Fast Variational AutoEncoder with Inverted Multi-Index for Collaborative Filtering](https://mlsanity.com/paper/2109.05773/) (2021)
- [Unbiased Smoothing using Particle Independent Metropolis-Hastings](https://mlsanity.com/paper/1902.01781/) (2019)
- [External Patch-Based Image Restoration Using Importance Sampling](https://mlsanity.com/paper/1807.03018/) (2018)
- [Robust Importance Sampling for Error Estimation in the Context of Optimal Bayesian Transfer Learning](https://mlsanity.com/paper/2109.02150/) (2021)

## Built on by (most cited first, 12 of 12)

- [On the Closed-Form of Flow Matching: Generalization Does Not Arise from Target Stochasticity](https://mlsanity.com/paper/2506.03719/) (2025)
- [Explicit Flow Matching: On The Theory of Flow Matching Algorithms with Applications](https://mlsanity.com/paper/2402.03232/) (2024)
- [Non-asymptotic Analysis of Biased Adaptive Stochastic Approximation](https://mlsanity.com/paper/2402.02857/) (2024)
- [TRADE: Transfer of Distributions between External Conditions with Normalizing Flows](https://mlsanity.com/paper/2410.19492/) (2024)
- [Importance Weighted Score Matching for Diffusion Samplers with Enhanced Mode Coverage](https://mlsanity.com/paper/2505.19431/) (2025)
- [Analytical Correction for Subsampling Bias in Drifting Models](https://mlsanity.com/paper/2604.27239/) (2026)
- [Reducing the Probability of Undesirable Outputs in Language Models Using Probabilistic Inference](https://mlsanity.com/paper/2510.21184/) (2025)
- [Convergence of a class of gradient-free optimisation schemes when the objective function is noisy, irregular, or both](https://mlsanity.com/paper/2512.03225/) (2025)
- [Principled Synthetic Data Enables the First Scaling Laws for LLMs in Recommendation](https://mlsanity.com/paper/2602.07298/) (2026)
- [Wasserstein Distributionally Robust Regret Optimization for Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2605.00155/) (2026)
- [Particle Monte Carlo Tree Search](https://mlsanity.com/paper/2605.08982/) (2026)
- [A Probabilistic Interpretation of KV Cache Eviction](https://mlsanity.com/paper/2608.28293/) (2026)

## Abstract (arXiv)

Importance Sampling (IS) is a method for approximating expectations under a target distribution using independent samples from a proposal distribution and the associated importance weights. In many applications, the target distribution is known only up to a normalization constant, in which case self-normalized IS (SNIS) can be used. While the use of self-normalization can have a positive effect on the dispersion of the estimator, it introduces bias. In this work, we propose a new method, BR-SNIS, whose complexity is essentially the same as that of SNIS and which significantly reduces bias without increasing the variance. This method is a wrapper in the sense that it uses the same proposal samples and importance weights as SNIS, but makes clever use of iterated sampling--importance resampling (ISIR) to form a bias-reduced version of the estimator. We furnish the proposed algorithm with rigorous theoretical results, including new bias, variance and high-probability bounds, and these are illustrated by numerical examples.