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| Main Author: | |
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| Format: | Recurso digital |
| Language: | English |
| Published: |
Zenodo
2026
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| Subjects: | |
| Online Access: | https://doi.org/10.5281/zenodo.19376121 |
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Table of Contents:
- <p>This paper develops an exact causal, extremal, and decision-theoretic frontier for hidden-subgroup effect recovery in binary latent-stratum models under a potential-outcome framework. The setting is binary treatment, binary outcome, latent subgroup, and potential outcomes satisfying consistency and latent ignorability, so the target is the hidden-subgroup average treatment effect profile rather than an observational contrast.</p> <p>The impossibility theorem is proved in its strongest causal form: every interior observable 2x2 law of treatment and outcome admits one latent causal decomposition with all subgroup average treatment effects positive and another with all negative. The scale-robust corollary shows that no aggregate-only statistic — risk difference, risk ratio, odds ratio, logit coefficient, or any other collapse of the observable table — can uniformly certify the uniform sign of the hidden-subgroup causal effects. A minimax lower bound for n-sample sign testing is proved: any randomized test based only on observable treatment-outcome data has worst-case type I plus type II error summing to at least one, with a paired hypothesis pair attaining average error exactly one-half regardless of sample size.</p> <p>For an arbitrary latent variable, an exact causal decomposition of the marginal risk difference is derived, separating the treated-mixture average causal effect from a confounding-capacity term equal to the covariance between treatment propensity and baseline potential-outcome risk divided by the marginal treatment variance. Under oscillation bounds on treatment propensity spread and baseline risk spread, a sharp confounding radius is proved, and the constant one-quarter in the denominator is shown to be attained by an explicit two-point binary extremizer on the central interior strip. For the binary two-group common-effect model, the exact identified interval on the central strip is established, and exact boundary-truncation formulas are derived on the budget-saturating branch where positivity and outcome boundary constraints begin to bind. The full heterogeneous identified set in the plane of subgroup effects is characterized as a union of explicit parallelograms, with a two-variable semialgebraic kernel and cylindrical algebraic decomposition completing the elimination to a direct semialgebraic description.</p> <p>The paper then proves an extremal finite-support reduction theorem for the general latent-state proximal setting: for any finite family of functionals of the distribution of the causal effect profile, the sharp identified set is attained by finite-support latent laws. This converts the infinite-dimensional proximal problem into an exact finite-support extremal search. For one binary proxy with a known invertible channel matrix and proxy conditional independence, exact identification of all latent cell vectors and hence all subgroup causal effects is proved; under a singular known channel, identification fails and an explicit converse is given for the unknown-channel case. Under epsilon-level proxy misspecification or channel calibration error, recovery error scales at rate kappa(M) times epsilon, where kappa(M) is the condition number of the channel matrix.</p> <p>Finally, the paper formalizes the information hierarchy via Blackwell experiment comparison. The augmented experiment with a binary proxy strictly Blackwell-dominates the aggregate experiment for the latent-sign decision problem: on a paired least-favorable hypothesis pair, every aggregate-only metric is equivalent to the null experiment, while a single calibrated noisy proxy bit yields strictly smaller Bayes risk.</p>