Embedding-Bias in Conditional Independence Testing

Nikolaj Thams; Anton Rask Lundborg

stat.MLOct 8, 2026

Embedding-Bias in Conditional Independence Testing

Authors: Nikolaj Thams, Anton Rask Lundborg

Organizations: Department of Mathematical Sciences University of Copenhagen Copenhagen, Denmark

Abstract

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 $\mathbb{E}[X \mid Z]$ and $\mathbb{E}[Y \mid Z]$ missed by $\mathbb{E}[X \mid ψ(Z)]$ and $\mathbb{E}[Y \mid ψ(Z)]$ 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 $R^2$ 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.

Figures & tables

Panel of Figure 1 — Figure 1: The conceptual example: the topic and the opinion of a post $Z$ both influence its reply $X$ and its reception $Y$ , and the embedded test uses $f(X)$ , $g(Y)$ , and $\psi(Z)$ . Since $\psi(Z)$ represents the topic but not the opinion, the dashed path through the opinion stays open and $X$ and $Y$ are dependent given $\psi(Z)$ although $X\mathrel{\perp\!\!\!\perp}Y\mid Z$ .

Panel of Figure 2 — Figure 2: Rejection rate under $H_{0}$ against $n$ in Experiment 1 (left, $\psi=\psi_{5}$ ) and Experiment 2 (right, $\psi=E_{1}$ ) for the residual correlation test (standard) and the robust test, run at $\lambda=1$ , with the embeddings and tolerances named in the rows. Dashed cells are fitted by ridge regression, since $n$ is below the dimension of $\psi(Z)$ .

Figure 3: ( Left ): Experiment 3, rejection rate under

H_{0}

over 200 null datasets drawn from the replicates, for the residual correlation test and for the robust test with the tolerance equal to half the true insufficiency, to the true bias

|\lambda|\sqrt{R^{2}_{\psi}(X)R^{2}_{\psi}(Y)}

and to the true insufficiency (

B^{*}

in the legend);

\psi(Z)

is 64 to 384 principal components of the generator’s mean hidden state over the prompt tokens at its last layer,

E_{1}

E_{2}

. ( Middle ): The same tests with

\psi=E_{1}

against a range of alternatives. We plot a reference test (dashed) that conditions on the post through the unused draws. ( Right ): Reddit data application. For each pair of

X

(politeness rating, log length, politeness lexicon) and

Y

(signed log score, delta indicator) we plot the unconditional correlation

\hat{\rho}_{0}

(diamonds),

\hat{\rho}_{\id}

from the baseline, which conditions on the comment being replied to (squares), and

\hat{\rho}_{\psi}

from the embedded test with

\psi=E_{1}

(circles), all with 95% intervals.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Figure D.1: Experiment 1,

\hat{\rho}_{\psi}

under

H_{0}

against

\sqrt{R^{2}_{\psi}(X)R^{2}_{\psi}(Y)}

for five values of

\lambda

. Solid lines are

\lambda\sqrt{R^{2}_{\psi}(X)R^{2}_{\psi}(Y)}

, the value under

H_{0}

by ( 1 ); the dashed line is the bound.

Figure D.2: Experiment 1. Left: power at

n=1000

against the effect size

\gamma

of the alternative

Y=\theta_{Y}^{\top}Z+\gamma X+\varepsilon_{Y}

, with the sufficient embedding, for the residual correlation test and for the robust test with tolerances

B\in\{0.05,0.1,0.2\}

. Right:

\hat{\rho}_{\psi}

(labelled

\mathrm{RV}=|\hat{\rho}|

in the legend, the robustness value of the point estimate) and

\RV_{\alpha}

against

n

under

H_{0}

with the insufficient embedding at

\lambda=1

; the dashed line is the population value

\rho_{\psi_{q}}=0.1

Figure D.3: Experiment 2,

\hat{\rho}_{\psi}

(median and interquartile range; labelled

\mathrm{RV}=|\hat{\rho}|

in the legend), median

\RV_{\alpha}

, and the estimated benchmark

\hat{\rho}_{E_{1}\to E_{2}}

against

n

with

\psi=E_{1}

and

\lambda=1

. The dashed line is the oracle bound, which equals the population benchmark.

Figure D.4: Experiment 2, power at

n=2000

against the effect size

\gamma

with

\psi=E_{1}

, for the residual correlation test and for the robust test with tolerance equal to the oracle bound. Left:

\lambda=0

, where

E_{1}

is insufficient but the residual correlation test is unbiased. Right:

\lambda=1

, where the residual correlation test rejects at rate 0.96 under

H_{0}

Figure D.5: Experiment 3, the four embeddings of the generator’s state. Left:

\hat{\rho}_{\psi}

under

H_{0}

with 95% intervals against the dimension of

\psi(Z)

, for the principal components of the generator’s states (mean over the prompt tokens and last prompt token, at the last and the middle layer), for

E_{1}

and

E_{2}

, and the unadjusted correlation (dotted); hollow markers are the true insufficiency

\sqrt{R^{2}_{\psi}(X)R^{2}_{\psi}(Y)}

of each embedding, identified from the replicates. Right: rejection rate under

H_{0}

over 200 null datasets drawn from the replicates, for the residual correlation test and for the robust test with the tolerance equal to half the true insufficiency, to the true bias

|\lambda|\sqrt{R^{2}_{\psi}(X)R^{2}_{\psi}(Y)}

and to the true insufficiency (

B^{*}

in the legend), for the mean and the last-token state at the last layer. All regressions are least squares.

Figure D.6: Experiment 3, identification check. For every embedding

\psi

\hat{\rho}_{\psi}

under

H_{0}

(mean over the 200 null datasets, with twice its standard deviation) against the prediction

|\lambda|\sqrt{R^{2}_{\psi}(X)R^{2}_{\psi}(Y)}

of Theorem 2.3 (filled) and the bound

\sqrt{R^{2}_{\psi}(X)R^{2}_{\psi}(Y)}

(hollow), both identified from the replicates; the line is the diagonal. Every filled marker lies on the diagonal, to within 0.006, and every hollow marker to its right.

Figure D.7: Experiment 3, rejection rates against the coupled alternatives for

\psi=E_{1}

E_{2}

and 384 principal components of the generator’s mean state at the last layer, by the conditional correlation

\rho_{\id}

of comment and summary given the post, over 200 datasets per value: the residual correlation test, the robust test with the tolerance equal to half the true insufficiency, to the true bias and to the true insufficiency (

B^{*}

in the legend), and the reference test that conditions on the post through the unused draws ( Section C.7 ).

Figure D.8: Application, politeness against score (left) and log length against score (right) under every embedding of the comment being replied to:

E_{1}

E_{2}

[E_{1},E_{2}]

, the surface statistics

S

, the metadata

M

, the TF-IDF directions

L

, the post embedding

P

, combinations with

E_{1}

, and the prognostic score. Circles:

\hat{\rho}_{\psi}

with 95% intervals; squares: the insufficiency of

E_{1}

demonstrated by the added embedding,

\hat{\rho}_{E_{1}\to\psi^{\prime}}

; line and band: the baseline with its 95% interval; dotted: the unconditional correlation.

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Conditional independence tests (CITs) test for conditional dependence between two random objects

X

and

Y

given a third random object

Z

. Existing CITs have limited applicability to high-dimensional data, especially multimodal data like text. However, we show that such tests are of interest for large language model (LLM) outputs, where we test whether an output

X

generated from a source text

Z

carries information about an attribute

Y

beyond

Z

itself. For this purpose, we propose embedded CITs (eCITs), which embed

X

and

Z

and apply an existing CIT to the resulting representations and to

Y

. We show that, provided the embedding of

Z

is sufficient, i.e. retains the information

Z

carries about either

Y

or the representation of

X

, the null hypothesis transfers from

X

and

Z

to their representations, so that a CIT valid for the embedded hypothesis is valid for the original one. We further give conditions for equivalence of the two hypotheses, and show that sufficiency weakens to mean sufficiency when the embedded test targets conditional mean independence. We propose a semi-synthetic simulation design to assess type I error (T1E) control and power of the eCITs for given embedding maps on a specific dataset and task, and use it to evaluate them on our application. Applying the eCITs to German Parliament speeches, we find for all combinations of embedding maps considered that the summaries of two LLMs contain information about the speaker's faction and gender beyond the speech they were generated from.

Abstract

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