# When and Why Do Linear Bias Probes Fail? A Geometric and Statistical Theory of Bias Detectability in Large Language Model Representations

arXiv: https://arxiv.org/abs/2609.22337 · HTML: https://mlsanity.com/paper/2609.22337/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 45 of its 72 references that are in the map.
- First posted on arXiv 21 Jul 2026.

## Builds on (most cited first, 30 of 45)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [Efficient Estimation of Word Representations in Vector Space](https://mlsanity.com/paper/1301.3781/) (2013)
- [Sparse Autoencoders Find Highly Interpretable Features in Language Models](https://mlsanity.com/paper/2309.08600/) (2023)
- [Understanding intermediate layers using linear classifier probes](https://mlsanity.com/paper/1610.01644/) (2016)
- [Representation Engineering: A Top-Down Approach to AI Transparency](https://mlsanity.com/paper/2310.01405/) (2023)
- [Man is to Computer Programmer as Woman is to Homemaker? Debiasing Word Embeddings](https://mlsanity.com/paper/1607.06520/) (2016)
- [Inference-Time Intervention: Eliciting Truthful Answers from a Language Model](https://mlsanity.com/paper/2306.03341/) (2023)
- [BERT Rediscovers the Classical NLP Pipeline](https://mlsanity.com/paper/1905.05950/) (2019)
- [Refusal in Language Models Is Mediated by a Single Direction](https://mlsanity.com/paper/2406.11717/) (2024)
- [Language (Technology) is Power: A Critical Survey of "Bias" in NLP](https://mlsanity.com/paper/2005.14050/) (2020)
- [StereoSet: Measuring stereotypical bias in pretrained language models](https://mlsanity.com/paper/2004.09456/) (2020)
- [Semantics derived automatically from language corpora contain human-like biases](https://mlsanity.com/paper/1608.07187/) (2016)
- [Probing Classifiers: Promises, Shortcomings, and Advances](https://mlsanity.com/paper/2102.12452/) (2021)
- [How Contextual are Contextualized Word Representations? Comparing the Geometry of BERT, ELMo, and GPT-2 Embeddings](https://mlsanity.com/paper/1909.00512/) (2019)
- [The Linear Representation Hypothesis and the Geometry of Large Language Models](https://mlsanity.com/paper/2311.03658/) (2023)
- [The Geometry of Truth: Emergent Linear Structure in Large Language Model Representations of True/False Datasets](https://mlsanity.com/paper/2310.06824/) (2023)
- [CrowS-Pairs: A Challenge Dataset for Measuring Social Biases in Masked Language Models](https://mlsanity.com/paper/2010.00133/) (2020)
- [BBQ: A Hand-Built Bias Benchmark for Question Answering](https://mlsanity.com/paper/2110.08193/) (2021)
- [Intrinsic Dimensionality Explains the Effectiveness of Language Model Fine-Tuning](https://mlsanity.com/paper/2012.13255/) (2020)
- [Gender Bias in Coreference Resolution: Evaluation and Debiasing Methods](https://mlsanity.com/paper/1804.06876/) (2018)
- [Designing and Interpreting Probes with Control Tasks](https://mlsanity.com/paper/1909.03368/) (2019)
- [Bias and Fairness in Large Language Models: A Survey](https://mlsanity.com/paper/2309.00770/) (2023)
- [What you can cram into a single vector: Probing sentence embeddings for linguistic properties](https://mlsanity.com/paper/1805.01070/) (2018)
- [Null It Out: Guarding Protected Attributes by Iterative Nullspace Projection](https://mlsanity.com/paper/2004.07667/) (2020)
- [Gender Bias in Coreference Resolution](https://mlsanity.com/paper/1804.09301/) (2018)
- [On Measuring Social Biases in Sentence Encoders](https://mlsanity.com/paper/1903.10561/) (2019)
- [Linguistic Knowledge and Transferability of Contextual Representations](https://mlsanity.com/paper/1903.08855/) (2019)
- [Emergent Linear Representations in World Models of Self-Supervised Sequence Models](https://mlsanity.com/paper/2309.00941/) (2023)
- [Language Models Represent Space and Time](https://mlsanity.com/paper/2310.02207/) (2023)

## Abstract (arXiv)

Linear probing is the standard instrument for detecting social biases in the hidden representations of large language models. Yet reported probe accuracies come almost exclusively from counterfactual evaluations in which every input carries an explicit demographic marker. Once only a fraction α of inputs carries demographic information, performance degrades sharply, and a weak probe may reflect either an unbiased model or an underpowered detector. We develop a theory that resolves this ambiguity. Modeling representations as two class-conditional clusters with Mahalanobis separation s on a manifold of curvature kap, we prove: (i) a finite-sample generalization bound governed by the manifold's extrinsic radius with a matching smash√dB/n minimax lower bound; (ii) an exact purity law for the maximum linear-probe AUC, strictly increasing in α; (iii) a curvature ceiling: ambient chordal separation on a space form cannot exceed 2/√kap; and (iv) a detectability threshold below which no audit can distinguish probe output from chance. Every theorem is validated on synthetic manifolds with known ground truth and on six open-weight models × four bias dimensions, where the purity law predicts entire AUC--α curves from a single cross-fitted hat s measured at α=1, with no parameters fitted to those curves. The framework turns bias auditing into a power analysis: given a target purity and effect size, it prescribes the sample budget n(α) for a conclusive audit.