A Semi-Analytical Inference Framework for Structural Causal Models

Structural causal models (SCMs) were conceived to formulate and answer causal questions. This paper shows that SCMs can also be used to formulate and answer teleological questions, concerning the intentions of a state-aware, goal-directed agent intervening in a causal system. We review limitations of previous approaches to modeling such agents, and then introduce intentional interventions, a new time-agnostic operator that induces a twin SCM we call a structural final model (SFM). SFMs treat observed values as the outcome of intentional interventions and relate them to the counterfactual conditions of those interventions (what would have happened had the agent not intervened). We show how SFMs can be used to empirically detect agents and to discover their intentions.

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), 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 two auxiliary environments are sufficient to guarantee correct counterfactual inference. We empirically support our theoretical results on synthetic data.

Counterfactual inference in Gaussian-process structural causal models (GP-SCMs) has been developed primarily for continuous endogenous variables, limiting applicability to causal graphs that contain discrete child nodes with continuous parents. We introduce a unified probabilistic framework for counterfactual inference with heterogeneous variable types by pairing GP predictors with explicit exogenous noise mechanisms. For discrete outcomes, we derive exact conditional noise-abduction procedures using a uniform threshold for binary variables, a Gumbel-max race for nominal categories, and a latent Gaussian cut-point model for ordinal ones. In each case, we propagate abducted noise through interventions while accounting for posterior uncertainty in the GP latent functions, and prove that the resulting mechanisms reproduce the fitted model's observational and interventional distributions. On synthetic SCMs with known ground-truth counterfactuals, we evaluate estimation accuracy, consistency, and robustness to coupling misspecification. A key finding is that applying a categorical coupling to ordinal data inflates counterfactual error roughly threefold even when observational fit remains comparable, and that this error does not diminish with more data. As the training set grows, the fitted structural equation converges to the truth while the counterfactual error flattens onto a floor. In the reverse direction, forcing a false order onto nominal data instead degrades the fitted equation itself. The choice of coupling must therefore be justified on structural grounds rather than read off the fit.