# Risk-limiting Financial Audits via Weighted Sampling without Replacement

arXiv: https://arxiv.org/abs/2305.06884 · HTML: https://mlsanity.com/paper/2305.06884/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 37 references that are in the map.
- First posted on arXiv 8 May 2023.

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

- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [Estimating means of bounded random variables by betting](https://mlsanity.com/paper/2010.09686/) (2020)
- [Black-Box Reductions for Parameter-free Online Learning in Banach Spaces](https://mlsanity.com/paper/1802.06293/) (2018)
- [Safe Testing](https://mlsanity.com/paper/1906.07801/) (2019)
- [There are natural scores: Full comment on Shafer, "Testing by betting: A strategy for statistical and scientific communication"](https://mlsanity.com/paper/2102.05569/) (2021)
- [Admissible anytime-valid sequential inference must rely on nonnegative martingales](https://mlsanity.com/paper/2009.03167/) (2020)
- [Anytime-valid off-policy inference for contextual bandits](https://mlsanity.com/paper/2210.10768/) (2022)
- [Sequential estimation of quantiles with applications to A/B-testing and best-arm identification](https://mlsanity.com/paper/1906.09712/) (2019)
- [Confidence sequences for sampling without replacement](https://mlsanity.com/paper/2006.04347/) (2020)
- [Always Valid Inference: Bringing Sequential Analysis to A/B Testing](https://mlsanity.com/paper/1512.04922/) (2015)
- [Catoni-style confidence sequences for heavy-tailed mean estimation](https://mlsanity.com/paper/2202.01250/) (2022)
- [On Confidence Sequences for Bounded Random Processes via Universal Gambling Strategies](https://mlsanity.com/paper/2207.12382/) (2022)
- [Huber-Robust Confidence Sequences](https://mlsanity.com/paper/2301.09573/) (2023)
- [RiLACS: Risk-Limiting Audits via Confidence Sequences](https://mlsanity.com/paper/2107.11323/) (2021)
- [Catoni-style Confidence Sequences under Infinite Variance](https://mlsanity.com/paper/2208.03185/) (2022)
- [A lower confidence sequence for the changing mean of non-negative right heavy-tailed observations with bounded mean](https://mlsanity.com/paper/2210.11133/) (2022)
- [Sets of Half-Average Nulls Generate Risk-Limiting Audits: SHANGRLA](https://mlsanity.com/paper/1911.10035/) (2019)

## Built on by (most cited first, 6 of 6)

- [Auditing Fairness by Betting](https://mlsanity.com/paper/2305.17570/) (2023)
- [Active, anytime-valid risk controlling prediction sets](https://mlsanity.com/paper/2406.10490/) (2024)
- [Probably Approximately Correct Labels](https://mlsanity.com/paper/2506.10908/) (2025)
- [Sequentially Auditing Differential Privacy](https://mlsanity.com/paper/2509.07055/) (2025)
- [Efficient Sequential Evaluation of Large Language Models](https://mlsanity.com/paper/2607.17409/) (2026)
- [Sequential Audit Sampling for Finite Populations with Exact and Simulation-based Guarantee](https://mlsanity.com/paper/2604.06116/) (2026)

## Abstract (arXiv)

We introduce the notion of a risk-limiting financial auditing (RLFA): given N transactions, the goal is to estimate the total misstated monetary fraction~(m^*) to a given accuracy ε, with confidence 1-δ. We do this by constructing new confidence sequences (CSs) for the weighted average of N unknown values, based on samples drawn without replacement according to a (randomized) weighted sampling scheme. Using the idea of importance weighting to construct test martingales, we first develop a framework to construct CSs for arbitrary sampling strategies. Next, we develop methods to improve the quality of CSs by incorporating side information about the unknown values associated with each item. We show that when the side information is sufficiently predictive, it can directly drive the sampling. Addressing the case where the accuracy is unknown a priori, we introduce a method that incorporates side information via control variates. Crucially, our construction is adaptive: if the side information is highly predictive of the unknown misstated amounts, then the benefits of incorporating it are significant; but if the side information is uncorrelated, our methods learn to ignore it. Our methods recover state-of-the-art bounds for the special case when the weights are equal, which has already found applications in election auditing. The harder weighted case solves our more challenging problem of AI-assisted financial auditing.