# Embedding-Bias in Conditional Independence Testing

arXiv: https://arxiv.org/abs/2610.11584 · HTML: https://mlsanity.com/paper/2610.11584/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 41 references that are in the map.
- First posted on arXiv 8 Oct 2026.

## Builds on (most cited first, 5 of 5)

- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Long Story Short: Omitted Variable Bias in Causal Machine Learning](https://mlsanity.com/paper/2112.13398/) (2021)
- [Doubly Robust Conditional Independence Testing with Generative Neural Networks](https://mlsanity.com/paper/2407.17694/) (2024)
- [The Generalised Kernel Covariance Measure](https://mlsanity.com/paper/2604.03721/) (2026)
- [Embedded Conditional Independence Tests for Large Language Model Generated Text with an Application to German Parliament Speeches](https://mlsanity.com/paper/2609.00946/) (2026)

## Abstract (arXiv)

To test conditional independence of X and Y given a text or an image Z, one conditions on an embedding psi(Z) in place of Z. The embedded test is valid if Z is independent of X or of Y given psi(Z), which cannot be confirmed from data, and when this fails, the rejection probability under the null hypothesis can tend to one. We study this failure, and show that focusing on a specific form of dependence relaxes what the embedding must retain. For a residual correlation test inspired by the Generalised Covariance Measure, validity only requires that the parts of mathbbE[X mid Z] and mathbbE[Y mid Z] missed by mathbbE[X mid psi(Z)] and mathbbE[Y mid psi(Z)] are uncorrelated. Otherwise, we treat the discarded information as an omitted variable. Under the null hypothesis, the bias equals the absolute correlation of the missed parts times the geometric mean of two partial R2 values. This identity yields a robust test valid under a declared tolerance for the geometric mean, which, like a sensitivity parameter, is not identified from the data. On synthetic data and text embeddings, the robust test holds its level approximately. On text generated by a language model, under an exact null hypothesis, every embedding, even the generator's own states, biases the embedded test.