# Anytime Valid Tests of Conditional Independence Under Model-X

arXiv: https://arxiv.org/abs/2209.12637 · HTML: https://mlsanity.com/paper/2209.12637/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Sep 2022.

## Built on by (most cited first, 11 of 11)

- [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)
- [Model-X Sequential Testing for Conditional Independence via Testing by Betting](https://mlsanity.com/paper/2210.00354/) (2022)
- [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)
- [On the Hardness of Conditional Independence Testing In Practice](https://mlsanity.com/paper/2512.14000/) (2025)
- [Optimistic Interior Point Methods for Sequential Hypothesis Testing by Betting](https://mlsanity.com/paper/2502.07774/) (2025)
- [Semi-Supervised Hypothesis Testing by Betting on Predictions](https://mlsanity.com/paper/2605.28533/) (2026)
- [Sequential Kernel-based Conditional Independence Testing via Adaptive Betting](https://mlsanity.com/paper/2606.18993/) (2026)

## Abstract (arXiv)

We propose a sequential, anytime-valid method to test the conditional independence of a response Y and a predictor X given a random vector Z. The proposed test is based on e-statistics and test martingales, which generalize likelihood ratios and allow valid inference at arbitrary stopping times. In accordance with the recently introduced model-X setting, our test depends on the availability of the conditional distribution of X given Z, or at least a sufficiently sharp approximation thereof. Within this setting, we derive a general method for constructing e-statistics for testing conditional independence, show that it leads to growth-rate optimal e-statistics for simple alternatives, and prove that our method yields tests with asymptotic power one in the special case of a logistic regression model. A simulation study is done to demonstrate that the approach is competitive in terms of power when compared to established sequential and nonsequential testing methods, and robust with respect to violations of the model-X assumption.