Causally sufficient perfect-oracle discovery regime

Let $V$ be a finite set of observed vertices, let $G$ be a directed acyclic graph on $V$, and let $(X_v{)}_{v\in V}$ be the endogenous variables of a finite recursive structural causal model whose observed set is all of $V$ and whose latent set is empty. The model is Markovian: its exogenous random elements are mutually independent. Require that the full-joint law $P$ of $(X_v{)}_{v\in V}$ has a topological probability-kernel factorization: for a topological order $v_1,\dots ,v_p$, its joint law is formed by composing probability kernels $K_{v_r}(x_{\mathrm{pa}(v_r)},d{x}_{v_r})$ that depend only on the parents of $v_r$. Also require that $P$ is faithful to $G$, meaning its conditional independences among vertex subvectors are exactly the d-separations in $G$. Assume AC for these probability interfaces.

A perfect conditional-independence oracle for this regime accepts any pairwise disjoint $A,B,S\subseteq V$ and returns “independent” exactly when $\sigma (X_A)$ and $\sigma (X_B)$ are conditionally independent given $\sigma (X_S)$. The causally sufficient perfect-oracle regime consists of the finite observed DAG, its Markovian recursive model and full-joint law, the faithfulness condition, and this exact oracle. No latent vertex is omitted from the graph supplied to the oracle.

Recursive structural causal models on finite DAGs

Independent random elements

The product sigma-algebra and its finite iterates

Measure kernel and probability kernel

DAG factorization implies the global Markov property

Faithfulness of a law to a DAG

Conditional independence given a sigma-algebra

Paths, active paths, and d-separation in a finite DAG

The Axiom of Choice