Doubly Robust Deep Kernel Tests for Structured Outcomes

Robin Leman; Bariscan Bozkurt; Sofiane Tanji; Arthur Gretton; Houssam Zenati

stat.MLOct 27, 2026

Doubly Robust Deep Kernel Tests for Structured Outcomes

Authors: Robin Leman, Bariscan Bozkurt, Sofiane Tanji, Arthur Gretton, Houssam Zenati

Organizations: University College London · Gatsby Computational Neuroscience Unit, University College London · ICTEAM/INMA, UCLouvain

Abstract

We propose a test of whether two interventions induce the same outcome distribution when outcomes are structured and high-dimensional, such as images, text and graphs. We represent the interventional distributions as mean embeddings on learned deep features, such that the treatment effect is a doubly robust representation of the difference in mean embeddings. Our test, DR-KTE-Deep, optimises the learned features so as to maximise an estimated power criterion on this treatment effect representation. To motivate our learning objective, we show that the power criterion characterises the asymptotic signal-to-noise ratio of the test under fixed alternatives, and we derive a uniform finite-sample bound for its regularised estimate. We show that, with the learned kernel fixed, a held-out test is asymptotically valid under the null given suitable nuisance-estimation rates. Empirically, DR-KTE-Deep achieves substantially higher power than fixed and bandwidth-optimised kernel baselines across image, text and graph outcomes, while maintaining well-calibrated Type-I error.

Figures & tables

Figure 1: Semi-synthetic medical-image experiments adapted from MedMNIST ( Yang et al., 2021 ; Yang et al., 2023 ) . Each pair shows a treated potential outcome $Y^{1}$ and its ground-truth individual treatment effect (ITE).

Figure 2: Synthetic example where treatment assignment is driven by an indicator of disease severity $X$ . The potential-outcome distributions are Gaussians with the same mean, but the traditional-treatment distribution is isotropic while the experimental-treatment distribution is anisotropic. The deep kernel adapts to the geometry of the distributions, while the Gaussian kernel treats points uniformly throughout the outcome space.

Scenario	DR-KTE	DR-KTE-Opt	DR-KTE-Deep
Breast-Tumor	$0.749{\scriptstyle\pm 0.018}/0.053{\scriptstyle\pm 0.003}$	$0.761{\scriptstyle\pm 0.017}/0.051{\scriptstyle\pm 0.003}$	$\mathbf{0.989{\scriptstyle\pm 0.002}}/0.054{\scriptstyle\pm 0.003}$
Chest-Cardiomegaly	$0.156{\scriptstyle\pm 0.007}/0.050{\scriptstyle\pm 0.003}$	$0.158{\scriptstyle\pm 0.008}/0.050{\scriptstyle\pm 0.003}$	$\mathbf{0.928{\scriptstyle\pm 0.046}}/0.051{\scriptstyle\pm 0.004}$
Chest-Effusion	$0.231{\scriptstyle\pm 0.009}/0.053{\scriptstyle\pm 0.004}$	$0.245{\scriptstyle\pm 0.011}/0.053{\scriptstyle\pm 0.004}$	$\mathbf{0.999{\scriptstyle\pm 0.000}}/0.048{\scriptstyle\pm 0.004}$
Chest-Mass	$0.436{\scriptstyle\pm 0.018}/0.050{\scriptstyle\pm 0.003}$	$0.461{\scriptstyle\pm 0.019}/0.050{\scriptstyle\pm 0.003}$	$\mathbf{1.000{\scriptstyle\pm 0.000}}/0.056{\scriptstyle\pm 0.004}$
Derma-Melanoma	$0.171{\scriptstyle\pm 0.009}/0.051{\scriptstyle\pm 0.002}$	$0.172{\scriptstyle\pm 0.009}/0.050{\scriptstyle\pm 0.002}$	$\mathbf{0.898{\scriptstyle\pm 0.040}}/0.053{\scriptstyle\pm 0.004}$
OCT-CNV	$0.907{\scriptstyle\pm 0.014}/0.052{\scriptstyle\pm 0.003}$	$0.917{\scriptstyle\pm 0.014}/0.051{\scriptstyle\pm 0.003}$	$\mathbf{1.000{\scriptstyle\pm 0.000}}/0.053{\scriptstyle\pm 0.003}$

Table 1: Power (

H_{1}

) and Type-I error (

H_{0}

) for medical-image outcomes at

N=1500

and

\alpha=0.05

. Each cell reports

H_{1}/H_{0}

, with means

\pm

one standard error across 20 independent repetitions.

Figure 3: Results for OCT-Drusen. (a) Type-I error under

H_{0}

, (b) average power when varying

N

while keeping

\delta=0.4

fixed, (c) average power when varying

\delta

while keeping

N=1500

fixed. Shaded bands show

\pm

one standard error across 20 independent repetitions. The dotted grey line marks

\alpha=0.05

Scenario	DR-KTE	DR-KTE-Opt	DR-KTE-Deep
Oncology Notes	$0.371{\scriptstyle\pm 0.015}/0.048{\scriptstyle\pm 0.003}$	$0.371{\scriptstyle\pm 0.015}/0.048{\scriptstyle\pm 0.003}$	$\mathbf{0.809{\scriptstyle\pm 0.023}}/0.055{\scriptstyle\pm 0.003}$
Movie Reviews	$0.255{\scriptstyle\pm 0.012}/0.050{\scriptstyle\pm 0.003}$	$0.254{\scriptstyle\pm 0.012}/0.050{\scriptstyle\pm 0.003}$	$\mathbf{0.449{\scriptstyle\pm 0.026}}/0.053{\scriptstyle\pm 0.004}$
Clinical Notes	$0.058{\scriptstyle\pm 0.004}/0.052{\scriptstyle\pm 0.003}$	$0.058{\scriptstyle\pm 0.004}/0.051{\scriptstyle\pm 0.003}$	$\mathbf{0.939{\scriptstyle\pm 0.013}}/0.058{\scriptstyle\pm 0.004}$

Table 2: Power (

H_{1}

) and Type-I error (

H_{0}

) for text outcomes, represented as a sequence of tokens, at

N=1500

and

\alpha=0.05

. Each cell reports

H_{1}/H_{0}

, with means

\pm

one standard error across 20 independent repetitions.

Figure 4: Results for Movie Reviews. (a) Type-I error under

H_{0}

, (b) average test power when varying

N

while keeping

\delta=1.0

fixed, (c) average test power when varying

\delta

while keeping

N=1500

fixed. Shaded bands show

\pm

one standard error across 20 independent repetitions. The dotted grey line marks

\alpha=0.05

Scenario	DR-KTE	DR-KTE-Opt	DR-KTE-Deep
LLM Code Repairs	$0.968{\scriptstyle\pm 0.008}/0.048{\scriptstyle\pm 0.004}$	$0.963{\scriptstyle\pm 0.009}/0.048{\scriptstyle\pm 0.004}$	$\mathbf{1.000{\scriptstyle\pm 0.000}}/0.051{\scriptstyle\pm 0.003}$
Contact Networks	$0.959{\scriptstyle\pm 0.006}/0.052{\scriptstyle\pm 0.003}$	$0.958{\scriptstyle\pm 0.006}/0.052{\scriptstyle\pm 0.003}$	$\mathbf{0.962{\scriptstyle\pm 0.006}}/0.051{\scriptstyle\pm 0.002}$

Table 3: Power (

H_{1}

) and Type-I error (

H_{0}

) for graph outcomes at

N=1500

. Each cell reports

H_{1}/H_{0}

, with means

\pm

one standard error across 20 outer repetitions.

Appendix figures & tables29 assets

Supplementary material from the paper’s appendix.

Appendix

Hyperparameter	Value
Training variance regularisation $\lambda$	$10^{-4}$
Validation and test variance regularisation	$0$
CME regularisation $\lambda_{\mu}$	$10^{-3}$
EMA decay $\gamma$	$0.95$
Minimum mixture weight $w_{\min}$	$10^{-9}$
Learning-rate schedule	Cosine decay to $1\%$ of initial rate

Appendix

Table 4: Shared hyperparameters for our experiments.

Experiment	Epochs	Ensemble Size $M$	Feature LR	Other LR
Image outcomes	500	3	$2\times 10^{-2}$	$10^{-3}$
Text outcomes (token sequences)	1000	3	$10^{-3}$	$10^{-4}$
Graph outcomes	1000	3	$2\times 10^{-2}$	$10^{-3}$
Text outcomes (embedded)	1000	3	$10^{-5}$	$10^{-4}$
Scalar outcomes	1000	5	$2\times 10^{-2}$	$10^{-3}$
Causal-Blob	1000	5	$2\times 10^{-2}$	$10^{-2}$

Appendix

Table 5: Experiment-specific hyperparameters.

Figure 5: Four-layer fully connected feature extractor for scalar outcomes, Causal-Blob, and text outcomes under an embedded representation.

Figure 6: Four-layer convolutional feature extractor for image outcomes.

Figure 7: Two-layer transformer feature extractor for text outcomes represented as a sequence of tokens.

Figure 8: Three-layer message-passing feature extractor for graph outcomes.

Figure 9: Results for OCT-DME. From left to right: (a) Type-I error under

H_{0}

(

\delta=0

), (b) average test power when varying

N

while keeping

\delta=0.4

fixed, (c) average test power when varying

\delta

while keeping

N=1500

fixed. Shaded bands show

\pm

one standard error across 20 independent outer repetitions. The dotted grey line marks

\alpha=0.05

Figure 10: Results for Derma-Melanoma. From left to right: (a) Type-I error under

H_{0}

(

\delta=0

), (b) average test power when varying

N

with

\delta=1.4

fixed, (c) average test power when varying

\delta

with

N=1500

fixed. Shaded bands show

\pm

one standard error across 20 independent outer repetitions. The dotted grey line marks

\alpha=0.05

Panel of Figure 11 — Figure 11: Data-generating process for MNIST, RetinaMNIST and BUSI. The covariate $X$ and treatment $A$ generate a scalar pseudo-outcome $\tilde{Y}$ , which is discretised to select the label of the sampled image outcome $Y$ .

Figure 12: MNIST (Scenario II, left) and RetinaMNIST (Scenario III, right). Panels (a) and (c) show label distributions under each treatment. Panels (b) and (d) show sampled image outcomes.

Dataset	$N$	Scenario	DR-KTE	DR-KTE-Opt	DR-KTE-Deep
MNIST	1500	I	$0.046{\scriptstyle\pm 0.003}$	$0.046{\scriptstyle\pm 0.003}$	$0.051{\scriptstyle\pm 0.003}$
		II	$0.731{\scriptstyle\pm 0.026}$	$0.731{\scriptstyle\pm 0.026}$	$\mathbf{0.870{\scriptstyle\pm 0.024}}$
		III	$0.721{\scriptstyle\pm 0.024}$	$0.721{\scriptstyle\pm 0.024}$	$\mathbf{0.803{\scriptstyle\pm 0.023}}$
	3000	I	$0.049{\scriptstyle\pm 0.003}$	$0.049{\scriptstyle\pm 0.003}$	$0.049{\scriptstyle\pm 0.004}$
		II	$0.948{\scriptstyle\pm 0.008}$	$0.948{\scriptstyle\pm 0.008}$	$\mathbf{0.986{\scriptstyle\pm 0.002}}$
		III	$0.938{\scriptstyle\pm 0.010}$	$0.937{\scriptstyle\pm 0.010}$	$\mathbf{0.973{\scriptstyle\pm 0.006}}$

Appendix

Table 6: Average rejection rates for MNIST, RetinaMNIST, and BUSI image outcomes at

\alpha=0.05

. Scenario I reports Type-I error; Scenarios II and III report power. Effect strengths are

\delta=1.0

and

1.5

for MNIST II and III,

\delta=2.0

and

4.0

for RetinaMNIST II and III, and

\delta=4.0

for BUSI III. Values are means

\pm

one standard error across 20 independent outer repetitions.

$N$	Scenario	DR-KTE	DR-KTE-Opt	DR-KTE-Deep
1500	$H_{0}$	$0.051{\scriptstyle\pm 0.003}$	$0.052{\scriptstyle\pm 0.003}$	$0.050{\scriptstyle\pm 0.004}$
1500	$H_{1}$	$0.225{\scriptstyle\pm 0.008}$	$0.229{\scriptstyle\pm 0.008}$	$\mathbf{0.380{\scriptstyle\pm 0.050}}$
3000	$H_{0}$	$0.054{\scriptstyle\pm 0.004}$	$0.054{\scriptstyle\pm 0.004}$	$0.050{\scriptstyle\pm 0.003}$
3000	$H_{1}$	$0.502{\scriptstyle\pm 0.012}$	$0.514{\scriptstyle\pm 0.012}$	$\mathbf{0.961{\scriptstyle\pm 0.019}}$

Appendix

Table 7: Type-I error (

H_{0}

) and power (

H_{1}

) for Fake-MNIST at

\alpha=0.05

. Values are means

\pm

one standard error across 20 independent outer repetitions.

	Power/Type-I error ( $H_{1}/H_{0}$ )
$N$	DR-KTE	DR-KTE-Opt	DR-KTE-Deep
1500	$0.322{\scriptstyle\pm 0.014}/0.049{\scriptstyle\pm 0.004}$	$0.351{\scriptstyle\pm 0.016}/0.051{\scriptstyle\pm 0.003}$	$\mathbf{0.382}{\scriptstyle\pm 0.024}/0.050{\scriptstyle\pm 0.004}$
3000	$0.710{\scriptstyle\pm 0.020}/0.046{\scriptstyle\pm 0.003}$	$0.780{\scriptstyle\pm 0.022}/0.048{\scriptstyle\pm 0.003}$	$\mathbf{0.891}{\scriptstyle\pm 0.019}/0.045{\scriptstyle\pm 0.003}$

Appendix

Table 8: Power and Type-I error for Causal-Blob at

\alpha=0.05

. Each cell reports

H_{1}/H_{0}

, with means

\pm

one standard error across 20 independent outer repetitions.

		Power/Type-I error ( $H_{1}/H_{0}$ )
$N$	Scenario	DR-KTE	DR-KTE-Opt	DR-KTE-Deep
1500	II	$0.999{\scriptstyle\pm 0.000}/0.055{\scriptstyle\pm 0.004}$	$\mathbf{1.000}{\scriptstyle\pm 0.000}/0.059{\scriptstyle\pm 0.004}$	$0.995{\scriptstyle\pm 0.002}/0.055{\scriptstyle\pm 0.004}$
	III	$0.611{\scriptstyle\pm 0.028}/0.055{\scriptstyle\pm 0.004}$	$\mathbf{0.653}{\scriptstyle\pm 0.027}/0.059{\scriptstyle\pm 0.004}$	$0.614{\scriptstyle\pm 0.028}/0.055{\scriptstyle\pm 0.004}$
	IV	$0.980{\scriptstyle\pm 0.007}/0.055{\scriptstyle\pm 0.004}$	$\mathbf{0.985}{\scriptstyle\pm 0.005}/0.059{\scriptstyle\pm 0.004}$	$0.981{\scriptstyle\pm 0.004}/0.055{\scriptstyle\pm 0.004}$
3000	II	$\mathbf{1.000}{\scriptstyle\pm 0.000}/0.048{\scriptstyle\pm 0.003}$	$\mathbf{1.000}{\scriptstyle\pm 0.000}/0.046{\scriptstyle\pm 0.003}$	$\mathbf{1.000}{\scriptstyle\pm 0.000}/0.048{\scriptstyle\pm 0.003}$
	III	$0.903{\scriptstyle\pm 0.013}/0.048{\scriptstyle\pm 0.003}$	$\mathbf{0.927}{\scriptstyle\pm 0.012}/0.046{\scriptstyle\pm 0.003}$	$0.914{\scriptstyle\pm 0.013}/0.048{\scriptstyle\pm 0.003}$

Appendix

Table 9: Power and Type-I error for scalar outcomes at

\alpha=0.05

, with

\delta=0.8

and

M=5

. Each cell reports

H_{1}/H_{0}

, with means

\pm

one standard error across 20 independent outer repetitions.

	Power/Type-I error ( $H_{1}/H_{0}$ )
Scenario	DR-KTE-Deep	IPW-Cross	KPT
Scenario II	$\mathbf{0.961}{\scriptstyle\pm 0.007}/0.048{\scriptstyle\pm 0.003}$	$0.920{\scriptstyle\pm 0.012}/0.085{\scriptstyle\pm 0.009}$	$0.638{\scriptstyle\pm 0.022}/0.070{\scriptstyle\pm 0.013}$
Scenario III	$\mathbf{0.739}{\scriptstyle\pm 0.011}/0.048{\scriptstyle\pm 0.003}$	$0.574{\scriptstyle\pm 0.008}/0.085{\scriptstyle\pm 0.009}$	$0.291{\scriptstyle\pm 0.017}/0.070{\scriptstyle\pm 0.013}$
Scenario IV	$\mathbf{0.945}{\scriptstyle\pm 0.009}/0.048{\scriptstyle\pm 0.003}$	$0.903{\scriptstyle\pm 0.013}/0.085{\scriptstyle\pm 0.009}$	$0.607{\scriptstyle\pm 0.027}/0.070{\scriptstyle\pm 0.013}$

Appendix

Table 10: Power and Type-I error at

\alpha=0.05

N=1600

and

M=1

for different choices of estimator. Each cell reports

H_{1}/H_{0}

, with means

\pm

one standard error across 20 independent outer repetitions.

	Power/Type-I error ( $H_{1}/H_{0}$ )
Environment	Fitted $\hat{\mu}_{\theta}$	$\hat{\mu}_{\theta}\equiv 0$
Causal-Blob	$0.318{\scriptstyle\pm 0.024}/0.047{\scriptstyle\pm 0.006}$	$0.247{\scriptstyle\pm 0.017}/0.047{\scriptstyle\pm 0.007}$
Scenario II	$0.999{\scriptstyle\pm 0.001}/0.056{\scriptstyle\pm 0.004}$	$0.927{\scriptstyle\pm 0.012}/0.057{\scriptstyle\pm 0.003}$
Scenario III	$0.616{\scriptstyle\pm 0.015}/0.056{\scriptstyle\pm 0.004}$	$0.349{\scriptstyle\pm 0.013}/0.057{\scriptstyle\pm 0.003}$
Scenario IV	$1.000{\scriptstyle\pm 0.000}/0.056{\scriptstyle\pm 0.004}$	$0.999{\scriptstyle\pm 0.001}/0.057{\scriptstyle\pm 0.003}$

Appendix

Table 11: Power and Type-I error at

\alpha=0.05

N=1500

and

M=1

. Each cell reports

H_{1}/H_{0}

, with means

\pm

one standard error across 20 independent outer repetitions.

Scenario		Random Weights	Pixel Autoencoder	DR-KTE-Deep
OCT-Drusen	$H_{1}$	$0.057\pm 0.004$	$0.052\pm 0.004$	$\mathbf{0.371\pm 0.080}$
OCT-Drusen	$H_{0}$	$0.054\pm 0.004$	$0.050\pm 0.003$	$0.058\pm 0.005$
Fake-MNIST	$H_{1}$	$0.204\pm 0.008$	$0.200\pm 0.012$	$\mathbf{0.380\pm 0.050}$
Fake-MNIST	$H_{0}$	$0.053\pm 0.004$	$0.055\pm 0.004$	$0.050\pm 0.004$

Appendix

Table 12: Power under

H_{1}

and Type-I error under

H_{0}

N=1500

. Values are means and one standard error across 20 independent outer repetitions.

	Power/Type-I error ( $H_{1}/H_{0}$ )
Scenario	DR-KTE	DR-KTE-Opt	DR-KTE-Deep
Oncology Notes	$\mathbf{0.828}{\scriptstyle\pm 0.035}/0.057{\scriptstyle\pm 0.010}$	$0.827{\scriptstyle\pm 0.035}/0.058{\scriptstyle\pm 0.010}$	$\mathbf{0.828}{\scriptstyle\pm 0.036}/0.058{\scriptstyle\pm 0.010}$
Movie Reviews	$0.892{\scriptstyle\pm 0.009}/0.029{\scriptstyle\pm 0.004}$	$\mathbf{0.895}{\scriptstyle\pm 0.008}/0.029{\scriptstyle\pm 0.004}$	$\mathbf{0.895}{\scriptstyle\pm 0.008}/0.029{\scriptstyle\pm 0.004}$
Clinical Notes	$\mathbf{0.894}{\scriptstyle\pm 0.027}/0.049{\scriptstyle\pm 0.007}$	$\mathbf{0.894}{\scriptstyle\pm 0.027}/0.049{\scriptstyle\pm 0.007}$	$\mathbf{0.894}{\scriptstyle\pm 0.027}/0.050{\scriptstyle\pm 0.007}$

Appendix

Table 13: Power (

H_{1}

) and Type-I error (

H_{0}

) at

N=1500

and

\alpha=0.05

for text outcomes represented as text embeddings.

Dataset	Statistic	$M=1$	$M=2$	$M=3$	$M=5$
Causal-Blob	Power	$0.350{\scriptstyle\pm 0.003}$	$0.373{\scriptstyle\pm 0.007}$	$0.383{\scriptstyle\pm 0.007}$	$0.394{\scriptstyle\pm 0.008}$
	SD	$0.150{\scriptstyle\pm 0.011}$	$0.157{\scriptstyle\pm 0.015}$	$0.150{\scriptstyle\pm 0.014}$	$0.151{\scriptstyle\pm 0.014}$
OCT-Drusen	Power	$0.993{\scriptstyle\pm 0.000}$	$0.995{\scriptstyle\pm 0.000}$	$0.995{\scriptstyle\pm 0.000}$	$0.996{\scriptstyle\pm 0.001}$
	SD	$0.011{\scriptstyle\pm 0.001}$	$0.009{\scriptstyle\pm 0.002}$	$0.009{\scriptstyle\pm 0.002}$	$0.007{\scriptstyle\pm 0.001}$

Appendix

Table 14: Mean and standard deviation (SD) of empirical power at

N=1500

and

\delta=1.0

. Each statistic is computed across 20 independent outer repetitions within each batch; values are means

\pm

one standard error across five independent batches.

Figure 13: Average standard deviation of empirical power against ensemble size at

N=1500

and

\delta=1.0

. Shaded bands show

\pm

one standard error across five independent batches.

	Power/Type-I error ( $H_{1}/H_{0}$ )
Scenario	DR-KTE-WL	DR-KTE-WL-Deep
LLM Code Repairs	$0.743{\scriptstyle\pm 0.023}/0.059{\scriptstyle\pm 0.006}$	$\mathbf{0.964}{\scriptstyle\pm 0.008}/0.050{\scriptstyle\pm 0.004}$
Contact Networks	$0.971{\scriptstyle\pm 0.005}/0.057{\scriptstyle\pm 0.003}$	$\mathbf{0.972}{\scriptstyle\pm 0.005}/0.056{\scriptstyle\pm 0.004}$

Appendix

Table 15: Power (

H_{1}

) and Type-I error (

H_{0}

) for graph outcomes using Weisfeiler-Lehman base kernels at

N=1500

. Each cell reports

H_{1}/H_{0}

, with means

\pm

one standard error across 20 outer repetitions. Gaussian-kernel results are given in Table 3 .

	Power/Type-I error ( $H_{1}/H_{0}$ )
Scenario	DR-ATE	DR-KTE-Deep
Breast-Tumor	$0.052{\scriptstyle\pm 0.003}/0.047{\scriptstyle\pm 0.004}$	$\mathbf{0.989}{\scriptstyle\pm 0.002}/0.054{\scriptstyle\pm 0.003}$
Chest-Cardiomegaly	$0.044{\scriptstyle\pm 0.003}/0.043{\scriptstyle\pm 0.003}$	$\mathbf{0.928}{\scriptstyle\pm 0.046}/0.051{\scriptstyle\pm 0.004}$
Chest-Effusion	$0.046{\scriptstyle\pm 0.005}/0.049{\scriptstyle\pm 0.003}$	$\mathbf{0.999}{\scriptstyle\pm 0.000}/0.048{\scriptstyle\pm 0.004}$
Chest-Mass	$0.052{\scriptstyle\pm 0.003}/0.051{\scriptstyle\pm 0.004}$	$\mathbf{1.000}{\scriptstyle\pm 0.000}/0.056{\scriptstyle\pm 0.004}$
Derma-Melanoma	$0.045{\scriptstyle\pm 0.004}/0.048{\scriptstyle\pm 0.002}$	$\mathbf{0.898}{\scriptstyle\pm 0.040}/0.053{\scriptstyle\pm 0.004}$

Appendix

Table 16: Comparison with DR-ATE for medical-image outcomes at

N=1500

. Each cell reports

H_{1}/H_{0}

, with means

\pm

one standard error across 20 outer repetitions.

Dataset	Scenario	DR-ATE	DR-KTE-Deep
MNIST	I	$0.045{\scriptstyle\pm 0.003}$	$0.051{\scriptstyle\pm 0.003}$
	II	$0.677{\scriptstyle\pm 0.027}$	$\mathbf{0.870}{\scriptstyle\pm 0.024}$
	III	$0.695{\scriptstyle\pm 0.023}$	$\mathbf{0.803}{\scriptstyle\pm 0.023}$
RetinaMNIST	I	$0.052{\scriptstyle\pm 0.003}$	$0.051{\scriptstyle\pm 0.004}$
	II	$0.434{\scriptstyle\pm 0.016}$	$\mathbf{0.647}{\scriptstyle\pm 0.022}$
	III	$0.235{\scriptstyle\pm 0.010}$	$\mathbf{0.732}{\scriptstyle\pm 0.035}$

Appendix

Table 17: Comparison with DR-ATE for MNIST, RetinaMNIST, BUSI and Fake-MNIST at

N=1500

. Scenario I and Fake-MNIST

H_{0}

report Type-I error; Scenarios II and III and Fake-MNIST

H_{1}

report power. Values are means

\pm

one standard error across 20 outer repetitions.

	Power/Type-I error ( $H_{1}/H_{0}$ )
Scenario	DR-ATE	DR-KTE-Deep
Oncology Notes	$0.375{\scriptstyle\pm 0.015}/0.051{\scriptstyle\pm 0.003}$	$\mathbf{0.809}{\scriptstyle\pm 0.023}/0.055{\scriptstyle\pm 0.003}$
Movie Reviews	$0.141{\scriptstyle\pm 0.009}/0.048{\scriptstyle\pm 0.003}$	$\mathbf{0.449}{\scriptstyle\pm 0.026}/0.053{\scriptstyle\pm 0.004}$
Clinical Notes	$0.053{\scriptstyle\pm 0.004}/0.053{\scriptstyle\pm 0.003}$	$\mathbf{0.939}{\scriptstyle\pm 0.013}/0.058{\scriptstyle\pm 0.004}$

Appendix

Table 18: Comparison with DR-ATE for text outcomes at

N=1500

. Each cell reports

H_{1}/H_{0}

, with means

\pm

one standard error across 20 outer repetitions.

	Power/Type-I error ( $H_{1}/H_{0}$ )
Scenario	DR-ATE	DR-KTE-Deep
LLM Code Repairs	$0.874{\scriptstyle\pm 0.019}/0.056{\scriptstyle\pm 0.004}$	$\mathbf{1.000}{\scriptstyle\pm 0.000}/0.051{\scriptstyle\pm 0.003}$
Contact Networks	$0.923{\scriptstyle\pm 0.009}/0.059{\scriptstyle\pm 0.004}$	$\mathbf{0.962}{\scriptstyle\pm 0.006}/0.051{\scriptstyle\pm 0.002}$

Appendix

Table 19: Comparison with DR-ATE for graph outcomes at

N=1500

. Each cell reports

H_{1}/H_{0}

, with means

\pm

one standard error across 20 outer repetitions.

	Power/Type-I error ( $H_{1}/H_{0}$ )
Scenario	DR-ATE	DR-KTE-Deep
Causal-Blob	$0.049{\scriptstyle\pm 0.003}/0.043{\scriptstyle\pm 0.002}$	$\mathbf{0.382}{\scriptstyle\pm 0.024}/0.050{\scriptstyle\pm 0.004}$

Appendix

Table 20: Comparison with DR-ATE on Causal-Blob at

N=1500

. Each cell reports

H_{1}/H_{0}

, with means

\pm

one standard error across 20 outer repetitions.

	Power/Type-I error ( $H_{1}/H_{0}$ )
Scenario	DR-ATE	DR-KTE-Deep
Scenario II	$0.939{\scriptstyle\pm 0.009}/0.097{\scriptstyle\pm 0.007}$	$\mathbf{0.995}{\scriptstyle\pm 0.002}/0.055{\scriptstyle\pm 0.004}$
Scenario III	$0.050{\scriptstyle\pm 0.004}/0.097{\scriptstyle\pm 0.007}$	$\mathbf{0.614}{\scriptstyle\pm 0.028}/0.055{\scriptstyle\pm 0.004}$
Scenario IV	$0.972{\scriptstyle\pm 0.006}/0.097{\scriptstyle\pm 0.007}$	$\mathbf{0.981}{\scriptstyle\pm 0.004}/0.055{\scriptstyle\pm 0.004}$

Appendix

Table 21: Comparison with DR-ATE for scalar outcomes at

N=1500

. Each alternative is paired with the shared Scenario I null. Each cell reports

H_{1}/H_{0}

, with means

\pm

one standard error across 20 outer repetitions.

Figure 14: Causal-Blob at

N=4000

when varying the outcome dimension, with

\delta=0

under

H_{0}

and

\delta=1.0

under

H_{1}

. Shaded bands show

\pm

one standard error across 20 independent outer repetitions. The dotted grey line marks

\alpha=0.05

	Power/Type-I error ( $H_{1}/H_{0}$ )
Scenario	DR-KTE	DR-KTE-Deep	Joint maximisation
Breast-Tumor	$0.749{\scriptstyle\pm 0.018}/0.053{\scriptstyle\pm 0.003}$	$\mathbf{0.989}{\scriptstyle\pm 0.002}/0.054{\scriptstyle\pm 0.003}$	$\mathbf{0.989}{\scriptstyle\pm 0.003}/0.053{\scriptstyle\pm 0.003}$
Chest-Cardiomegaly	$0.156{\scriptstyle\pm 0.007}/0.050{\scriptstyle\pm 0.003}$	$\mathbf{0.928}{\scriptstyle\pm 0.046}/0.051{\scriptstyle\pm 0.004}$	$0.763{\scriptstyle\pm 0.077}/0.051{\scriptstyle\pm 0.004}$
Chest-Effusion	$0.231{\scriptstyle\pm 0.009}/0.053{\scriptstyle\pm 0.004}$	$\mathbf{0.999}{\scriptstyle\pm 0.000}/0.048{\scriptstyle\pm 0.004}$	$0.998{\scriptstyle\pm 0.001}/0.056{\scriptstyle\pm 0.004}$
Chest-Mass	$0.436{\scriptstyle\pm 0.018}/0.050{\scriptstyle\pm 0.003}$	$\mathbf{1.000}{\scriptstyle\pm 0.000}/0.056{\scriptstyle\pm 0.004}$	$0.999{\scriptstyle\pm 0.000}/0.050{\scriptstyle\pm 0.004}$
Derma-Melanoma	$0.171{\scriptstyle\pm 0.009}/0.051{\scriptstyle\pm 0.002}$	$\mathbf{0.898}{\scriptstyle\pm 0.040}/0.053{\scriptstyle\pm 0.004}$	$0.795{\scriptstyle\pm 0.060}/0.046{\scriptstyle\pm 0.003}$

Appendix

Table 22: Effect of joint maximisation at

N=1500

and

\alpha=0.05

. Each cell reports

H_{1}/H_{0}

, with means

\pm

one standard error across 20 outer repetitions.

Two-Sample Testing

Two-sample testing develops statistical methods for deciding whether two sets of observations come from the same distribution.

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Semiparametric Efficient Test for Interpretable Distributional Treatment Effects

Houssam Zenati, Arthur Gretton

1Gatsby Computational Neuroscience Unit, University College London

Semiparametric Inference Causal Effect Estimation Kernel Methods Causal Inference Causal Interpretability Doubly Robust Estimation+1

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Distributional treatment effects can be invisible to means: a treatment may preserve average outcomes while changing tails, modes, dispersion, or rare-event probabilities. Kernel tests can detect discrepancies between interventional outcome laws, but global tests do not reveal where the laws differ. We propose DR-ME, to our knowledge the first semiparametrically efficient finite-location test for interpretable distributional treatment effects. DR-ME evaluates an interventional kernel witness at learned outcome locations, returning causal-discrepancy coordinates rather than only a global rejection. From observational data, we derive orthogonal doubly robust kernel features whose centered oracle form is the canonical gradient of this finite witness. For fixed locations, we characterize the local testing limit: DR-ME is chi-square calibrated under the null, has noncentral chi-square local power, and uses the covariance whitening that optimizes local signal-to-noise for discrepancies visible through the selected coordinates. This efficient local-power geometry yields a principled location-learning criterion, with sample splitting preserving post-selection validity. Experiments show near-nominal type-I error, competitive power against global doubly robust kernel tests, and interpretable learned locations that localize distributional effects in a semi-synthetic medical-imaging study.

Abstract

Two-Sample Testing

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