Source
Rubenstein, Weichwald, Bongers, Mooij, Janzing, Grosse-Wentrup and Schölkopf (2017), LIT-1334, read in full in NOTE-1037: Theorem 11 and its proof in Appendix C. Beckers and Halpern (LIT-1316, NOTE-1032) restate it as their Example 3.3, with the coefficient an/m. Exact transformation is Rubenstein et al.’s Definition 3.
What was actually shown
The micro model: Wᵢ = Eᵢ for i ≤ n, Z_k = Σⱼ A_kj Wⱼ + F_k for k ≤ m, with (E, F) of any joint law, and declared interventions ∅, do(W = w), do(Z = z) and do(W = w, Z = z) for all w ∈ ℝⁿ, z ∈ ℝᵐ. τ maps (W, Z) to (Ŵ, Ẑ) = ((1/n)ΣWᵢ, (1/m)ΣZ_k). Write c_j = Σ_k A_kj, the column sums.
The macro law under each intervention. Under ∅ or do(W = w), Ẑ = (1/m)Σ_k Σⱼ A_kj Wⱼ + F̂ = (1/m)Σⱼ c_j Wⱼ + F̂, with F̂ = (1/m)ΣF_k.
If c_j = a for every j, then (1/m)Σⱼ c_j Wⱼ = (a/m)ΣⱼWⱼ = (an/m)Ŵ. So the macro SEM Ŵ = Ê, Ẑ = (an/m)Ŵ + F̂, with Ê = (1/n)ΣEᵢ and F̂ jointly distributed as the averages are, and ω sending do(W = w) to do(Ŵ = w̄), do(Z = z) to do(Ẑ = z̄) and the joint intervention to the joint one, reproduces the push-forward law under every declared intervention. ω is surjective and order-preserving (an order embedding, as the paper says). Under do(Z = z) the macro variable Ẑ is the constant z̄ at both levels, and Ŵ keeps its law. Checked numerically for n = 4, m = 3, a = 0.7 and a random w: the micro sum and (an/m)w̄ agreed to machine precision, while (a/m)w̄ did not.
Only if. Suppose the averaging τ is exact for these interventions, with Ê and F̂ not almost surely constant. Under do(W = w) the push-forward puts Ŵ at the point w̄ and gives Ẑ the law of (1/m)c·w + F̂, which is not a point mass. Under ∅ or do(Ẑ = ·) the macro model gives Ŵ the law of Ê, not a point mass, and under a joint intervention Ẑ would be a point mass. So ω(do(W = w)) must be do(Ŵ = w̄), and the macro law of Ẑ under it depends on w only through w̄. Hence the law of (1/m)c·w + F̂ is the same for any two w with the same mean. Two translates of one law are equal only if the shifts are equal, so c·w = c·w′ whenever w̄ = w̄′, that is, c is orthogonal to every vector whose entries sum to zero. That forces c = a·1: equal column sums.
Column-equitability. With Q the 2 × (n + m) matrix of block indicators, the condition is THEORY-755’s for the pair of blocks (Z, W): Σ_{k∈Z} A_kj is the same for every j in W. The W block receives nothing and Z nothing from itself, so every other block pair is trivially equitable.
Why only whole-block interventions. do(W₁ = w₁) alone leaves Ŵ with the law of (w₁ + Σ_{j≥2}Eⱼ)/n, which is neither the point mass of a macro intervention on Ŵ nor, unless E₁ is itself a point mass at w₁, the law of Ê. No ω can match it, so adding it to the declared set breaks exactness. This is the paper’s general remark: “If we were to allow larger sets of interventions 𝑋 in the SEM 𝑋, the transformations given would not be exact.”
What could have failed: the averages could have needed a condition on the noise as well, or on rows of A. They need only the column sums, because the noise passes through the averaging unchanged.
What this does not say
- Not about interventions on single micro-variables, or on proper subsets of a block, which the transformation does not survive.
- Not that the macro model predicts any one micro-variable, or any one system’s response given its micro state: exactness is equality of push-forward laws (NOTE-1037, Bearing on the record).
- Not about nonlinear models or approximate aggregation. Nearly equal column sums give no bound here.
- Not dynamics. It is a one-shot SEM; THEORY-755 has the dynamical version for block sums, and Theorem 12 of the same paper treats equilibria.
Connections
- THEORY-755. The same algebra, QA = ĀQ for a partition, as a causal rather than dynamical criterion. The join is the record’s.
- THEORY-733. The dynamical-independence criterion LA = BL is the linear-map generalization of both.
- CLAIM-350. Its causal bullet’s “intervention-consistent coarse-graining for a declared family of interventions” is exact transformation; this is a worked instance with the declared family explicit.