Causal inference for group-contaminated structured outcomes: observable quotients, lossless reduction and exact randomization inference

This paper studies causal inference for structured outcomes (such as microscopy images) that are observed after an unknown, unit-specific transformation (group action). The core observation model is X = Γ · Y(A), where A is treatment, Y(a) is the intrinsic potential outcome, and Γ is an unobserved group element, with no independence assumptions imposed between Γ and treatment, covariates, or potential outcomes.

The paper develops a three-layer theoretical framework:

1. Observability: A target is uniformly recoverable exactly when it is constant on group orbits (Theorem 2.1). A Borel maximal invariant retains every measurable invariant target (Theorem 2.4). The interventional law of the maximal invariant is identified under standard causal assumptions (Theorem 2.5).

2. Statistical losslessness: Quotient reduction is sufficient for the full transformed experiment exactly when the conditional law of the raw observation given treatment, covariates and the quotient has a parameter-free version (Theorem 3.2). Conditional Haar contamination on a compact group yields Blackwell equivalence as a special case (Corollary 3.4).

3. Product versus diagonal actions: Componentwise canonicalization is appropriate for independent site-specific product actions but can discard relative cross-site information under shared diagonal actions (Proposition 4.1).

The paper also proves an approximate-contamination stability result (Theorem 5.1) bounding quotient-law Wasserstein error and the induced perturbation of population maximum mean discrepancy under explicit metric and kernel regularity.

For finite-support multichannel lattice images, the paper constructs a maximal invariant under integer translations and quarter turns (Theorem 6.1), combines its characteristic Gaussian kernel with a complete paired-swap test (Theorem 7.1), and retains the original simulations and RxRx1 HUVEC study.

Key empirical results: Under the sharp null, the quotient test rejected in 0.052 of simulation replicates (95% CI 0.028–0.087); at unit effect strength its power was 0.992. The primary RxRx1 contrast had an enumerated paired-swap p-value of 0.0078.

Improvements to AI systems:

1. Invariance-aware causal inference for imaging data: AI systems analyzing microscopy or biomedical images can now perform causal effect estimation that is provably robust to unknown, unit-specific transformations (e.g., rotation, translation, staining variability). The system can automatically identify and use a maximal invariant representation, ensuring that treatment effect estimates are unbiased even when image alignment or orientation is uncontrolled—without requiring manual preprocessing or registration.

2. Lossless dimension reduction for high-dimensional outcomes: AI systems can reduce raw high-dimensional outcomes (e.g., multi-channel lattice images) to a quotient space (invariant features) with a mathematical guarantee of no statistical information loss for causal queries, under specified conditions. This enables faster training and inference on large image datasets while preserving full causal fidelity, and it can detect when such reduction is not lossless (e.g., under shared diagonal group actions), preventing erroneous conclusions.

3. Robust hypothesis testing under distribution shift: The paired-swap test with enumerated p-values provides a finite-sample, distribution-free testing procedure. AI systems can now run causal hypothesis tests (e.g., does treatment alter cell morphology?) that are valid even when the group action (e.g., translation) is unknown and when the data has only moderate sample sizes—yielding calibrated Type I error (0.052 with 95% CI) and high power (0.992) at unit effect strength.

4. Stable metric learning under approximate contamination: The approximate-contamination stability result allows AI systems to compute maximum mean discrepancy (MMD) or Wasserstein distances between treatment groups on quotient spaces with explicit error bounds, even when the observed data is only approximately invariant (e.g., due to noise or partial alignment). This improves the reliability of generative model evaluation and distribution comparison in real-world imaging pipelines.

5. Automatic canonicalization with cross-site information preservation: AI systems can now decide when to apply componentwise canonicalization (e.g., per-image alignment) versus a shared global transformation. The system will avoid discarding relative cross-site information (e.g., spatial relationships between cells in a tissue) when the underlying group action is diagonal, leading to more accurate causal inference for multi-site or multi-channel experiments.

6. Explicit maximal invariant construction for discrete symmetries: For integer translations and quarter turns on lattice images, the system can construct a maximal invariant (Theorem 6.1) and combine it with a characteristic Gaussian kernel. This enables AI systems to perform kernel-based causal inference (e.g., kernel ridge regression or hypothesis testing) directly on invariant features, with theoretical guarantees of completeness—useful for automated drug screening or high-content imaging analysis.

What the improved AI system can do specifically:

• Given raw, unaligned microscopy images from a treatment/control experiment, it can output a valid causal effect estimate and a p-value, without any manual image registration, and with known Type I error control.

• It can automatically detect whether aligning images per-site is safe or harmful for the causal question, and choose the correct invariant representation accordingly.

• It can compute a robust distance between treatment groups (e.g., for generative model evaluation) with explicit error bounds even when images are only approximately transformed.

• It can run a complete, finite-sample paired-swap test on multichannel lattice images under translation/rotation symmetries, yielding exact p-values without asymptotic approximations.

Structured potential outcomes such as microscopy images may be recorded after an unknown, unit-specific transformation. If that transformation can depend on treatment, covariates or the intrinsic outcome, raw-coordinate analyses may mix biological effects with acquisition geometry. We study the unrestricted observation model X =. Y(A) and characterize its observable information: a target is uniformly recoverable exactly when it is constant on group orbits, while a Borel maximal invariant retains every measurable invariant target. We then distinguish observability from statistical losslessness. A quotient-faithful reconstruction theorem shows that quotient reduction is sufficient for the full transformed experiment exactly when the conditional law of the raw observation given treatment, covariates and the quotient has a parameter-free version. Conditional Haar contamination on a compact group yields Blackwell equivalence as a special case; it is not imposed in the main model. We also separate independent site-specific product actions from shared diagonal actions and show why componentwise canonicalization can discard relative cross-site information. Under explicit metric and kernel regularity, an approximate-contamination theorem bounds quotient-law Wasserstein error and the induced perturbation of population maximum mean discrepancy. For finite-support multichannel lattice images, we construct a maximal invariant under integer translations and quarter turns, combine its characteristic Gaussian kernel with a complete paired-swap test, and retain the original simulations and RxRx1 HUVEC study. Under the sharp null, the quotient test rejected in 0.052 of simulation replicates; at unit effect strength its power was 0.992. The primary RxRx1 contrast had an enumerated paired-swap p-value of 0.0078.