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LIT-1334: Causal Consistency of Structural Equation Models

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StatusActive — read 2026-10-21 (NOTE-1037) in full, from arXiv:1707.00819v1; worth reading for the definition of an exact transformation between structural equation models, with its order-preserving map on interventions, and for its three families of exact coarse-grainings: marginalisation (THEORY-848), averaging under equal column sums (THEORY-847) and equilibria. Read it knowing that exactness is equality of push-forward laws for a declared intervention set, which Beckers and Halpern show to be too permissive and which can drop a variable that decides how a system responds; that only perfect interventions are modelled; and that the coefficient of its Theorem 11, as extracted from this copy, appears to lack a factor n.
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Tagscausality
Cited as “paper” byNOTE-1037 — Rubenstein et al. — Causal consistency of structural equation models
Cited as “source” byTHEORY-847 — In a linear structural equation model whose exogenous micro-causes W₁, …, W_n drive effects Z = AW + F, with interventions that set all of W, all of Z or both, averaging each block is an exact transformation onto Ẑ = (an/m)Ŵ + F̂ if and only if every column of A sums to the same a, the column-equitability of the block of A from W to Z; the transformation is exact only for those whole-block interventions, THEORY-848 — In an acyclic structural equation model, marginalising a variable that no declared intervention sets is an exact transformation, provided the coarse model's exogenous noises may be dependent; so exactness does not require the dropped variable to be causally idle, and it can drop a variable that decides how its children respond to an intervention that is declared, THEORY-849 — For deterministic causal models with a countable set of allowed interventions, a high-level model reproduces the coarse-grained interventional law of a low-level one for every distribution on contexts exactly when some map τ_U on contexts commutes with every allowed intervention, τ(M_L(u, i)) = M_H(τ_U(u), ω(i)); exactness for one distribution is strictly weaker, since a well-chosen distribution lets unrelated models match, and τ-abstraction and strong abstraction are strictly stronger; the context map is only required to exist, so high-level exogenous variables may carry what the coarse state drops

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Rubenstein, P. K., Weichwald, S., Bongers, S., Mooij, J. M., Janzing, D., Grosse-Wentrup, M. & Schölkopf, B. (2017). Causal Consistency of Structural Equation Models. Proceedings of the 33rd Conference on Uncertainty in Artificial Intelligence (UAI 2017), reissued as PMLR R15: 151–160. Preprint: arXiv:1707.00819 (4 July 2017).

Read in full (NOTE-1037).

Why this is here

A25: it “formalizes transformations between models that agree on intervention effects”, the first of the three steps (exact transformation, then abstraction, then approximation) A25 gave before saying governance-response sufficiency is not a new general theory of causal abstraction. Bears on CLAIM-407 and CLAIM-409.

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