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

• URL: http://arxiv.org/abs/2608.11954v1
• Date: Wed, 12 Aug 2026 11:39:50 GMT
• Title: Causal inference for group-contaminated structured outcomes: observable quotients, lossless reduction and exact randomization inference
• Abstract summary: We study the unrestricted observation model X = .<n>Y(A) and characterize its observable information.<n>A quotient-faithful reconstruction theorem shows quotient reduction is sufficient for the full transformed experiment.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: 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 = Γ . 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.

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