11.1 Three data regimes, three outputs

Let \(V=\{ X_1,\ldots ,X_d\} \) be observed variables and let \(L\) denote unobserved variables. The meaning of a geometric score depends first on how the regimes were produced.

Known targets.

A reference sample and interventional samples are observed, with target set \(I_e\subseteq V\) supplied for each regime. BRIDGE returns a candidate mask, field-calibration diagnostics, and closure scores.

Unknown targets.

Regimes are labeled but their targets are unknown. The graph–target pair is generally identifiable only up to \(\Psi \)-Markov equivalence; the natural population output is a \(\Psi \)-partial ancestral graph (\(\Psi \)-PAG) and invariant target information [ Jaber et al. , 2020 ] .

General domains.

Environment labels need not correspond to interventions on a shared structural causal model. Without additional assumptions, the output is a stability or transport diagnostic.

These regimes cannot be pooled under one “causal discovery” claim. With latent variables, a partial or maximal ancestral graph (PAG or MAG) may be the proper estimand; a directed acyclic graph (DAG) selected over observed nodes is only a restricted visible approximation [ Spirtes et al. , 2000 , Pearl , 2009a ] .