A Characterization of Latent Variable Causal Models Consistent with Observational Data

Causal reasoning from observational data becomes significantly challenging when the underlying causal structure is only partially known. In this work, we study the the equivalence class of structural causal models (SCMs) with latent variables sharing the same set of conditional independencies, as represented by a partial ancestral graph (PAG). Specifically, we provide a new characterization of SCM-induced causal diagrams in this equivalence class using differentiable parameters. Building on this characterization, we introduce a new model class called PAG-constrained neural causal models ($\mathcal{P}$-NCMs) which parameterize this equivalence class of SCMs. We prove that $\mathcal{P}$-NCMs are expressive enough to represent counterfactual distributions induced by any SCM in the class, while remaining consistent with the structural constraints shared across all its members. Finally, we demonstrate how the differentiable parameterization of this model class enables causal inference under Markov equivalence by reducing counterfactual partial identification to an optimization problem over $\mathcal{P}$-NCMs. We establish the theoretical soundness of this approach and validate its performance on simulations.