Identifiability of Latent State Dynamics under Adaptive Multi-Environment Selection

Latent state models underpin many machine learning systems, but in practice data is rarely collected i.i.d. and is instead generated by an adaptive mechanism that depends on the latent state itself. This introduces a form of hidden confounding that standard causal representation learning (CRL) methods do not address. We propose the Latent Adaptive Selection Model (LASM), a structural causal framework for modeling latent dynamics under adaptive data collection. Unlike prior CRL settings with exogenous interventions, LASM captures endogenous selection, where policies both depend on and distort the latent state. Our main contributions are: (i) a geometric characterization of non-identifiable directions induced by adaptive policies via the Policy Selection Gradient Matrix; (ii) a sharp identifiability result showing that, under a logged-policy anchor and a policy diversity condition, the latent transition matrix is exactly recoverable across environments; (iii) a tight lower bound $E^{\star }=\lceil d/(k-1)\rceil$ on the number of environments required for identifiability; and (iv) the Multi-environment Adaptive Invariant Latent Estimator algorithm, which enforces latent frame anchoring via a logged-policy consistency objective and promotes diversity through a log-determinant regularizer. Experiments on synthetic data verify the predicted identifiability threshold, while real-world experiments on ASSISTments demonstrate that recovered skill-transition structures are highly reproducible across schools ($\rho =0.91$) only under multi-environment pooling, and collapse in single-environment settings.