Identifying a Canonical Latent Causal Model from Multi-Environment Mixed Observations

Scientific variables are often observed through heterogeneous measurements rather than directly: some coordinates are continuously recorded after unknown monotone distortions, whereas others are ordinal or binary coarsenings. We study a linear Gaussian latent DAG shared across environments, with invariant measurement mechanisms and environment-varying independent structural-noise variances. We show that measurement invariance aligns latent marginal scales across environments and identifies a common-reference latent covariance family. When the structural noises have pairwise distinct cross-environment variance profiles, this family identifies a canonical latent structural causal model and the full latent DAG. The identified canonical model also determines hard-intervention laws in observed environments; for strictly increasing measurements that may be discontinuous, we further characterize the sharp set of physical measured responses and the exact condition for point identification. We develop a mixed-data covariance front end and a regularized coefficient estimator with fixed-dimensional finite-sample guarantees. Across eight graph topologies, three sample sizes, and strong and weak heterogeneity regimes, recovery improves with sample size and variance-profile separation; under weak separation, our estimator attains higher mean area under the precision–recall curve than same-front direct diagonalization and BACKSHIFT. A BRFSS 2015 application illustrates real-data behavior through held-out covariance fit and bootstrap edge-selection stability.