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

arXiv: https://arxiv.org/abs/2608.11954 · HTML: https://mlsanity.com/paper/2608.11954/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 27 references that are in the map.
- First posted on arXiv 12 Aug 2026.

## Builds on (most cited first, 5 of 5)

- [RxRx1: A Dataset for Evaluating Experimental Batch Correction Methods](https://mlsanity.com/paper/2301.05768/) (2023)
- [Geodesic Causal Inference](https://mlsanity.com/paper/2406.19604/) (2024)
- [Doubly robust estimation of causal effects for random object outcomes with continuous treatments](https://mlsanity.com/paper/2506.22754/) (2025)
- [Kernel-based estimators for functional causal effects](https://mlsanity.com/paper/2503.05024/) (2025)
- [Absolute average and median treatment effects as causal estimands on metric spaces](https://mlsanity.com/paper/2407.03726/) (2024)

## Abstract (arXiv)

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 = Gamma . 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.