Causal discovery, the problem of inferring the direction of causality, is generally ill-posed. We use the language of structural causal models (SCM) to show that assuming that the causal relations are acyclic and invariant across multiple environments (e.g., the way minimum wage affects employment rate is stable across different geographical regions), \textit{only} two auxiliary environments are sufficient to infer the causal graph for arbitrary nonlinear mechanisms. Moreover, we demonstrate that this implies identifiability of the SCM functional mechanisms: as a corollary, we show that \textit{two} auxiliary environments are sufficient to guarantee correct counterfactual inference. We empirically support our theoretical results on synthetic data.
Structural causal models provide a unified semantics for interventions and counterfactuals, but most identifiability results rely on restrictive assumptions like global monotonicity, which are often violated in embodied interaction, where the same exogenous perturbation can induce opposite responses under different contact contexts. We ask what structure still suffices once global monotonicity is dropped. We introduce non-monotone triangular structural causal models (NM-TM-SCM), which retain triangular recursion but replace global monotonicity with mechanism-wise invertibility and context-independent inverse transport. We prove that these conditions are equivalent to exogenous isomorphism and imply complete counterfactual identifiability, and we give a counterexample showing that local invertibility alone is insufficient. We instantiate the theory in CausalInverter, with triangular invertible layers, orientation gates, and transport-stability regularization. On synthetic non-monotonic mechanisms, the structural bias yields systematic counterfactual gains as non-monotonicity increases. On MuJoCo Door, our model achieves perfect event-level counterfactual recovery, lowers continuous angle error relative to a Transformer baseline, and delivers substantially more stable recovery than Transformer and conditional-flow predictors. On MuJoCo Push, where non-monotonicity is weaker, the same low-data predictors remain competitive or better, consistent with a bias-variance boundary. These results identify a broader identifiable regime between globally monotone triangular models and unconstrained black-box world models.
Causal abstractions formalize when a high-level structural causal model (SCM) captures the interventional behavior of a lower-level SCM. Existing applications of this notion largely follow a hypothesis-testing paradigm: an expert proposes a candidate high-level model and then evaluates if the low-level system implements it. We study the complementary problem of learning a high-level model directly from low-level measurements. Our contributions leverage hypotheses from low-rank causal discovery, and can be summarized as follows: (1) we show that observations generated by a low-rank graph induce latents that form a causal abstraction, (2) we provide identifiability results about these latents, and (3) we propose a practical objective to learn this high-level SCM.
Causal discovery methods such as LiNGAM identify causal structure from observational data by assuming mutually independent disturbances. This assumption is fragile: shared volatility, common scale effects, or other forms of dependence can cause the methods to recover the wrong causal order, even with infinite data. We introduce the Linear Mean-Independent Acyclic Model (LiMIAM), which replaces full independence with weaker one-sided mean-independence restrictions on the disturbances. Under finite-order consequences of these restrictions, source nodes are generically identifiable, and hence a compatible causal order can be recovered recursively. Our proof is constructive and leads to DirectLiMIAM, a sequential residual-based algorithm for causal discovery under dependent noise. In simulations with mean-independent but dependent disturbances, DirectLiMIAM outperforms LiNGAM methods. A large-scale empirical application to the oil market highlights the implausibility of the independence assumption and the ability of DirectLiMIAM to recover a realistic causal ordering, from policy to production and from prices to inflation.