Embedding-Bias in Conditional Independence Testing

The gist The paper studies embedding-bias in conditional independence testing by formalizing and quantifying how conditioning on an embedding, instead of the original variable, can invalidate tests for conditional independence when sufficiency is not guaranteed.

How it works

The core problem addressed is that testing conditional independence of X and Y given Z via an embedding ψ(Z) may not preserve validity because the test relies on the embedding being sufficient, which is generally not true for pre-trained encoders The paper formalizes this by considering the surrogate hypothesis Hψ0: ψX(X) ⊥⊥ ψY (Y) ψZ(Z) and examining how the embedding discards information

The authors define a measure of conditional dependence, ρψ(f, g), which is the absolute correlation of the residuals f − E[f ψ(Z)] and g − E[g ψ(Z)] Under the null hypothesis Hψ0, this measure vanishes if X ⊥⊥ Y ψ(Z), but when sufficiency fails, ρψ can be non-zero even under Hψ0

The bias under the null hypothesis is quantified by an identity that states 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 is not identified from data

Measures of conditional dependence

The paper introduces several measures of conditional dependence, including ρψ(f, g), which captures different forms of conditional dependence depending on the classes of functions F and G used The measure ρψ(F, G) is defined as the supremum over f∈Fψ and g∈Gψ of ρψ(f, g)

The paper defines the partial R2 value of a function f with respect to ψ(Z) as R2ψ(f), which is defined as E[∆ψ(f) 2] / σ2ψ(f), where ∆ψ(f):= E[f Z]− E[f ψ(Z)] Theorem 2.3 provides a bias bound relating ρψ to ρid and the partial R2 values

The corollary derived from Theorem 2.3 states that under Hψ0, ρψ(F, G) = 0 if and only if λψ(f, g) = 0 for all f and g This means the embedding creates no dependence exactly when the discarded parts are uncorrelated for all pairs

Testing with an insufficient embedding

The paper proposes a residual correlation test inspired by the Generalised Covariance Measure, which uses cross-fitting to estimate conditional means This test is valid under Hψ0 whenever λ = 0, meaning the discarded parts are uncorrelated

The residual correlation test statistic is defined as T:= n−1/2 Pn i=1 Ri sˆR, where Ri:= ˆrX,irˆY,i This statistic is asymptotically equivalent to the GCM test when Hψ0 holds

To create a more robust test, the authors introduce a robust test that controls the level if an analyst assumes q R2ψ(X)R2ψ(Y) ≤ B for some B ≥ 0 This tolerance is called an insufficiency tolerance and is valid if this bound holds

Benchmarking with richer embeddings

The paper establishes a benchmark lower bound for any valid tolerance B by considering a second embedding ψ', where ρψ→ψ′ is the insufficiency demonstrated by ψ' Proposition 3.6 states that every valid tolerance is at least ρψ→ψ'

The authors suggest using prognostic scores, which are one-dimensional embeddings fitted on a split of the sample that is not used for testing, as they demonstrate insufficiency more reliably than a richer embedding

Application

In experiments with text data from the ChangeMyView corpus, the embedded ρˆψ stays close to the unconditional value because embeddings do not fully recover the effect of the comment The results show that while a plain residual correlation test can be biased by an insufficient embedding, a robust test with correct tolerance approximately holds its level and has power

The experiments demonstrate that the robust test with the true bias as tolerance detects almost the same alternatives as the reference test, while the conservative test using the true insufficiency as tolerance only starts rejecting at the strongest alternatives

The paper concludes that a plain residual correlation test can be biased by an insufficient embedding, while the robust test with correct tolerance approximately holds its level and has power, but underestimating the tolerance leads to a biased test and overestimating it to a loss of power

Appendix

The appendix provides detailed proofs for Theorem 2.3, including Lemma A.1 which states that R2ψ(L2(PX,Z)) = 0 if and only if X ⊥⊥ Z ψ(Z) It also details the construction of the robust test p-value pB and its asymptotic validity under a declared tolerance B

The appendix further shows that identification with replicates can be used to identify the sensitivity parameters E[X | Z] and E[Y | Z] These parameters are identified when conditionally independent replicates are available The appendix also details implementation details for the algorithm, including cross-fitting regressions and standard errors

The paper acknowledges that without a tolerance justified by domain knowledge, a rejection is therefore hard to distinguish from an artifact of the embedding The AI use statement details how generative AI tools were used for drafting and editing text and proofs

The acknowledgments state that Nikolaj Thams is supported by the Novo Nordisk Foundation (grant NNF26OC0114284) The regenerated texts of Experiment 3 were computed on the Gefion AI supercomputer, operated by the Danish Centre for AI Innovation (DCAI)

The references list various works related to conditional independence testing, embeddings, and sensitivity analysis The paper includes a section on experimental design and computation details The final discussion highlights the need for better benchmarks for the insufficiency of embeddings

The paper is published as arXiv:2610.11584v1 [stat.

1. Improve conditional independence testing by using a robust residual correlation test that controls bias under an insufficiency tolerance. This test rejects H0 when a calculated p-value, derived from the bias-aware pB formula, is less than a threshold α, ensuring the test's validity is not solely dependent on unidentifiable data.

2. Develop AI systems capable of detecting dependence even when using embeddings that are insufficient for the task. The paper shows that a plain residual correlation test can be biased by an insufficient embedding, and a robust test with a correctly specified tolerance approximately holds its level and has power.

3. Implement sensitivity analysis to quantify bias caused by discarded information, treating it as an omitted variable. This allows researchers to understand the effect of the embedding on testing validity using sensitivity parameters like partial R2 values, which are identified through conditional independence replicates.

4. Construct prognostic scores derived from embeddings that demonstrably capture all relevant information about the conditioning variable Z. The paper suggests taking embeddings to be such predictions, known as prognostic scores (Hansen, 2008), because accurate predictions demonstrate all of the insufficiency.

5. Design generative AI systems that produce conditional independence tests where the null hypothesis is exact under specific conditions. For text generated by a language model under an exact null hypothesis, every embedding, even the generator’s own states, biases the plain embedded test.

To test conditional independence of X and Y given a text or an image Z, one conditions on an embedding ψ(Z) in place of Z. The embedded test is valid if Z is independent of X or of Y given ψ(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 E[X Z] and E[Y Z] missed by E[X ψ(Z)] and E[Y ψ(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 R squared 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.