SCM.Do

Weak union for full conditional independence. Let M be a structural causal model and let μ be a finite measure on the full random state (observed together with latent variables). Suppose X, Y ∪ W, and Z ∪ W are all sets of M's random variables, and that, under μ, X is conditionally independent of Y ∪ W given Z. Then X is conditionally independent of Y given Z ∪ W.

Contraction for full conditional independence. Let M be a structural causal model and let μ be a finite measure on the full random state. Suppose X, Y, W, and Z are all sets of M's random variables, and that, under μ, X is conditionally independent of Y given Z ∪ W and X is conditionally independent of W given Z. Then X is conditionally independent of Y ∪ W given Z.

Decomposition. Let X, Y, W, Z be finite node sets of the structural causal model M, with X and the union of Y and W contained in the observed nodes, and Z contained in the observed nodes, under a finite measure μ on the observed values. If X is conditionally independent of the union of Y and W given Z, then X is conditionally independent of Y given Z.

Weak union. Let X, Y, W, Z be finite node sets of the structural causal model M, with X and the union of Y and W contained in the observed nodes, and Z contained in the observed nodes, under a finite measure μ on the observed values. If X is conditionally independent of the union of Y and W given Z, then X is conditionally independent of Y given the union of Z and W.

Contraction. Fix subsets X, Y, W, Z of the node set of the structural causal model M, with X, Y, W, and Z each contained in the observed nodes, and let μ be a finite measure on the observed values. If X is conditionally independent of Y given the union of Z and W and X is conditionally independent of W given Z, then X is conditionally independent of the union of Y and W given Z.

Full Local Markov Property. If v is an observed node of the model, then under the joint distribution over all random (observed and latent) coordinates at fixed value s, the v-coordinate is conditionally independent of its non-descendants — restricted to random nodes — given all of its parents, including any latent parents, likewise restricted to random nodes.

Latent Local Markov Property. If a is a latent (unobserved) node of the model, then under the joint distribution over all random coordinates at fixed value s, the a-coordinate is unconditionally independent of its non-descendants — restricted to random nodes — i.e. conditionally independent given the empty conditioning set.

Concatenation d-separation for the latent cutset. Suppose W consists of observed nodes and F consists of fixed nodes, and that the target set Y is d-separated from the randomised do-block Zr given W together with F. Then the latent cutset cutsetLatent Y (Zr ∪ W) — the latent roots reaching Y along a directed path whose interior avoids Zr ∪ W — is likewise d-separated from Zr given the same W ∪ F.

Cross-model concatenation d-separation for the latent cutset. Suppose that, in the intervened model M.fixSet Z _ _, the target set Y is d-separated from the randomised do-block Zr = Z.image .random given the adjustment set W together with the post-intervention fixed block, where W consists of observed nodes and no w ∈ W is, in the base graph M.dag, a descendant of any treatment's random copy .random D (D ∈ Z) — backdoor criterion (i). Then, in the base model M, the latent cutset cutsetLatent Y (Zr ∪ W) is d-separated from Zr given W ∪ M.fixed.

Rule 1: Insertion/deletion of observations (single-SCM form). On any structural causal model M', suppose Y, Z, and W are all observed nodes of M', and that Y is d-separated from Z given W together with the fixed nodes M'.fixed, in the split graph M'.dag. Then, at any fixed-value point s, Y and Z are conditionally independent given W under the observational kernel M'.obsKernel s.

**Rule 3*: insertion/deletion of actions (simplified joint-marginal form).** Fix a structural causal model M' and a treatment set Z for which each member's random copy is already observed in M' and each member's fixed copy is not yet among M''s fixed nodes, together with outcome and conditioning sets Y and W, both observed in M'. Suppose no node of Y ∪ W is, in the model intervened on Z, a descendant of the fixed copy of any variable in Z. For any post-intervention fixed value, the joint law of (Y, W) under the intervened model equals the joint law of (Y, W) under the base model M' at the corresponding pre-intervention fixed value.

Full Global Markov with fixed-node conditioning shadow. If X, Y, and Z_rand are sets of nodes drawn from the model's random (observed and latent) nodes and Z_fix is a set of the model's fixed (intervened) nodes, and X is d-separated from Y by the union Z_rand ∪ Z_fix in the model's causal graph, then under the joint distribution over all random coordinates at fixed value s, the X-coordinates and the Y-coordinates are conditionally independent given only the Z_rand-coordinates — the fixed nodes contribute to the graphical separation but, since their values are already pinned by s, drop out of the probabilistic conditioning set.

Full Global Markov Property. If X, Y, and Z are sets of nodes drawn from the model's random (observed and latent) nodes and X is d-separated from Y by Z in the model's causal graph, then under the joint distribution over all random coordinates at fixed value s, the X-coordinates and the Y-coordinates are conditionally independent given the Z-coordinates.

The bridge, restated. For any structural causal model M and any point s of its fixed values, the joint distribution of M's random values under s is a global I-map of M's own DAG — this is exactly the global Markov property full_globalMarkov.

Global Markov Property. If X, Y, and Z are sets of observed nodes and X is d-separated from Y by Z in the model's full causal graph, which also includes any latent nodes, then under the observational distribution — the law of the observed coordinates alone, at fixed value s — the X-coordinates and Y-coordinates are conditionally independent given the Z-coordinates.

Global Markov with fixed-node conditioning. If X, Y, and Z_obs are sets of observed nodes and Z_fix is a set of the model's fixed (intervened) nodes, and X is d-separated from Y by the union Z_obs ∪ Z_fix in the model's full causal graph, then under the observational distribution at fixed value s, the X-coordinates and Y-coordinates are conditionally independent given only the Z_obs-coordinates — the fixed nodes contribute to the graphical separation but, since their values are already pinned by s, drop out of the probabilistic conditioning set.

Rule 2, a.e. in the treatment value (product form + positivity). Consider intervening on the finite set of names Z, where each name's random copy is already observed in the base model and its fixed copy is not yet part of the base model's fixed coordinates, with outcome set Y and conditioning set W contained in the observed variables, together with the random copies of Z and their union with W. Assume in the post-intervention SWIG DAG, Y is d-separated from the random copies of Z given W together with the post-intervention fixed set, no fixed copy of a name in Z is an ancestor of any node of W in the post-intervention DAG, no random copy of a name in Z is an ancestor of any node of W in the original DAG, and positivity: the pushforward of the product of the treatment and conditioning marginals under the fill map is absolutely continuous with respect to the base model's law on Z.random ∪ W. Then for almost every pair (t, w) under that product measure, the W-conditional kernel of the model intervened at treatment value t, evaluated together with w, restricted to Y, equals the base model's conditional distribution of Y given Z.random ∪ W evaluated at the combined point (t, w).

**Rule 3* core — intervention on non-ancestors of T is irrelevant.** This is the simplified joint non-ancestor transport used by the library's Rule 3* interface. Fix a do-set Z of nodes whose random copies are observed in the base model and whose fixed nodes have not already been intervened on and a target block T of observed variables. If none of the fixed copies of Z's nodes is an ancestor, in the intervention SWIG graph, of any node in T, then the T-marginal law of the intervened model equals the T-marginal law of the base model evaluated at the corresponding fixed values.