# Model-X Sequential Testing for Conditional Independence via Testing by Betting

arXiv: https://arxiv.org/abs/2210.00354 · HTML: https://mlsanity.com/paper/2210.00354/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 63 references that are in the map.
- First posted on arXiv 1 Oct 2022.

## Builds on (most cited first, 28 of 28)

- [Coordinate Descent Algorithms](https://mlsanity.com/paper/1502.04759/) (2015)
- [Estimating means of bounded random variables by betting](https://mlsanity.com/paper/2010.09686/) (2020)
- [Panning for Gold: Model-X Knockoffs for High-dimensional Controlled Variable Selection](https://mlsanity.com/paper/1610.02351/) (2016)
- [Safe Testing](https://mlsanity.com/paper/1906.07801/) (2019)
- [Controlling the false discovery rate via knockoffs](https://mlsanity.com/paper/1404.5609/) (2014)
- [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)
- [DeepPINK: reproducible feature selection in deep neural networks](https://mlsanity.com/paper/1809.01185/) (2018)
- [Deep Knockoffs](https://mlsanity.com/paper/1811.06687/) (2018)
- [Conditional Independence Testing using Generative Adversarial Networks](https://mlsanity.com/paper/1907.04068/) (2019)
- [Testing exchangeability: fork-convexity, supermartingales, and e-processes](https://mlsanity.com/paper/2102.00630/) (2021)
- [Parameter-Free Online Convex Optimization with Sub-Exponential Noise](https://mlsanity.com/paper/1902.01500/) (2019)
- [Double Generative Adversarial Networks for Conditional Independence Testing](https://mlsanity.com/paper/2006.02615/) (2020)
- [Sequential Nonparametric Testing with the Law of the Iterated Logarithm](https://mlsanity.com/paper/1506.03486/) (2015)
- [Fast and Powerful Conditional Randomization Testing via Distillation](https://mlsanity.com/paper/2006.03980/) (2020)
- [AdaPT: An interactive procedure for multiple testing with side information](https://mlsanity.com/paper/1609.06035/) (2016)
- [Robust inference with knockoffs](https://mlsanity.com/paper/1801.03896/) (2018)
- [Gene Hunting with Knockoffs for Hidden Markov Models](https://mlsanity.com/paper/1706.04677/) (2017)
- [Fast Nonlinear Vector Quantile Regression](https://mlsanity.com/paper/2205.14977/) (2022)
- [Derandomized knockoffs: leveraging e-values for false discovery rate control](https://mlsanity.com/paper/2205.15461/) (2022)
- [Knockoffs for the mass: new feature importance statistics with false discovery guarantees](https://mlsanity.com/paper/1807.06214/) (2018)
- [Anytime Valid Tests of Conditional Independence Under Model-X](https://mlsanity.com/paper/2209.12637/) (2022)
- [Online control of the familywise error rate](https://mlsanity.com/paper/1910.04900/) (2019)
- [Causal Inference in Genetic Trio Studies](https://mlsanity.com/paper/2002.09644/) (2020)
- [Interactive rank testing by betting](https://mlsanity.com/paper/2009.05892/) (2020)
- [STAR: A general interactive framework for FDR control under structural constraints](https://mlsanity.com/paper/1710.02776/) (2017)
- [Pearson Chi-squared Conditional Randomization Test](https://mlsanity.com/paper/2111.00027/) (2021)
- [Learning to Increase the Power of Conditional Randomization Tests](https://mlsanity.com/paper/2207.01022/) (2022)

## Built on by (most cited first, 14 of 14)

- [Protected Test-Time Adaptation via Online Entropy Matching: A Betting Approach](https://mlsanity.com/paper/2408.07511/) (2024)
- [Sequential Predictive Two-Sample and Independence Testing](https://mlsanity.com/paper/2305.00143/) (2023)
- [Sequential Kernelized Independence Testing](https://mlsanity.com/paper/2212.07383/) (2022)
- [Prediction-Powered E-Values](https://mlsanity.com/paper/2502.04294/) (2025)
- [WATCH: Adaptive Monitoring for AI Deployments via Weighted-Conformal Martingales](https://mlsanity.com/paper/2505.04608/) (2025)
- [Online Detection of LLM-Generated Texts via Sequential Hypothesis Testing by Betting](https://mlsanity.com/paper/2410.22318/) (2024)
- [E-Valuating Classifier Two-Sample Tests](https://mlsanity.com/paper/2210.13027/) (2022)
- [Deep anytime-valid hypothesis testing](https://mlsanity.com/paper/2310.19384/) (2023)
- [Optimistic Interior Point Methods for Sequential Hypothesis Testing by Betting](https://mlsanity.com/paper/2502.07774/) (2025)
- [Testing For Distribution Shifts with Conditional Conformal Test Martingales](https://mlsanity.com/paper/2602.13848/) (2026)
- [I Bet You Did Not Mean That: Testing Semantic Importance via Betting](https://mlsanity.com/paper/2405.19146/) (2024)
- [A complete characterization of sequential testability and change detectability in i.i.d. models](https://mlsanity.com/paper/2609.05752/) (2026)
- [Sequential Kernel-based Conditional Independence Testing via Adaptive Betting](https://mlsanity.com/paper/2606.18993/) (2026)
- [NxN E-valuation: Hypothesis Certification via a Conformal CRT Null](https://mlsanity.com/paper/2608.06621/) (2026)

## Abstract (arXiv)

This paper develops a model-free sequential test for conditional independence. The proposed test allows researchers to analyze an incoming i.i.d. data stream with any arbitrary dependency structure, and safely conclude whether a feature is conditionally associated with the response under study. We allow the processing of data points online, as soon as they arrive, and stop data acquisition once significant results are detected, rigorously controlling the type-I error rate. Our test can work with any sophisticated machine learning algorithm to enhance data efficiency to the extent possible. The developed method is inspired by two statistical frameworks. The first is the model-X conditional randomization test, a test for conditional independence that is valid in offline settings where the sample size is fixed in advance. The second is testing by betting, a ``game-theoretic'' approach for sequential hypothesis testing. We conduct synthetic experiments to demonstrate the advantage of our test over out-of-the-box sequential tests that account for the multiplicity of tests in the time horizon, and demonstrate the practicality of our proposal by applying it to real-world tasks.