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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

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Properties4
StatusActive
Tagscausality
SourceLIT-1334 — Causal Consistency of Structural Equation Models
SupportsCLAIM-399 — Causal abstraction in the senses of Rubenstein et al., Beckers and Halpern, and Beckers, Eberhardt and Halpern does not secure governance-response sufficiency: exact transformation, τ-abstraction and zero approximate-abstraction distance all accept the hidden-bit coarse-graining, which loses ρ(h = 1) bits of the response to h, because each lets high-level noise carry what the coarse state drops; for one outcome and the identity correspondence, sufficiency is their compatibility condition with the high-level context fixed to the coarse state, and strong abstraction rejects the case only through interventions on the hidden parameter, CLAIM-407 — A coarse-graining exactly informationally closed under a process's ordinary dynamics need not preserve its responses to an intervention on the mechanism: for a declared intervention distribution ρ, independent of the state, the deficiency Γ = I(S; Y^(I) | T(S), I) can be positive, as in the hidden governance bit, where it is ρ(h = 1) bits; and Γ = 0 exactly when, for every intervention ρ charges, the response kernels are constant on T's fibres over the states of positive probability

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Source

Rubenstein et al. (2017), LIT-1334, read in full in NOTE-1037: Theorem 10 and its proof in Appendix B. Exact transformation is their Definition 3: a map τ on states and a surjective, order-preserving ω on the declared interventions with τ(ℙ_X^{do(i)}) = ℙ_Y^{do(ω(i))} for every i. The corollary and the example are the record’s.

What was actually shown

Theorem 10, re-derived. Let M_X be acyclic, and let X₁ be set by no declared intervention. By acyclicity its equation is X₁ = f₁(X_pa(1), E₁) and X₁ is not an ancestor of its own parents. Build M_Y by deleting X₁’s equation and, in each child Xᵢ’s equation, replacing X₁ by f₁(X_pa(1), E₁), so that Xᵢ = fᵢ(f₁(X_pa(1), E₁), X_pa(i)∖{1}, Eᵢ); give each child the noise (Eᵢ, E₁), jointly distributed as under ℙ_E. Keep the declared interventions, with X₁ deleted from none of them because it appears in none, and take ω the identity and τ the projection that drops X₁. Under any declared i, the solution of M_Y is the solution of M_X with X₁ removed, since every occurrence of X₁ has been replaced by the expression that M_X^{do(i)} would give it (X₁ is not intervened on, so that expression is in force). So τ(ℙ_X^{do(i)}) = ℙ_Y^{do(i)}. Iterating removes any set of such variables. The noises of two children of X₁ now share E₁: “marginalising out a common parent node will in general result in its children having dependent exogenous variables”. Definition 1 allows it, and the theorem fails without it.

The corollary. Nothing in the proof asks what X₁ does. It may be the variable that decides how its children respond to a declared intervention. The hidden bit (CASE-125) is an instance. Its SEM: K = E_K, Y_t = E_t, H = E_H, Y_{t+1} = (1 − H)E_{t+1} + HK, with E_K, E_t, E_{t+1} fair bits, independent, and declared interventions ∅, do(H = 0), do(H = 1).

  • K is never intervened on, so marginalising it is exact. The coarse SEM is Y_{t+1} = (1 − H)E′₁ + HE′₂, with E′₁ = E_{t+1} and E′₂ = K. Under do(H = 1), (Y_t, Y_{t+1}) is a pair of independent fair bits at both levels; under do(H = 0) likewise.
  • Over a held do(H = 1) at several steps, Y_{t+1} = Y_{t+2} = K: the coarse model reproduces it with one noise E′₂ shared across the steps, which is the dependent noise the theorem needs.
  • Yet the coarse state Y_t determines nothing of the response to h: Γ(Y_t) = ρ(h = 1) bits (CLAIM-407), since P(Y_{t+1} | S = (k, y), do(H = 1)) is the point mass at k, which differs between states with the same Y_t.

What could have failed: exactness could have required the dropped variable to be causally idle, as Theorem 9 requires it to be childless. Theorem 10 shows that being unintervened is enough.

What this does not say

  • Not that marginalising an intervened variable is exact. Declaring do(K = 1) breaks it: under it H stays a fair bit and P(Y_{t+1} = 1) = ½ · ½ + ½ · 1 = ¾, while the marginalised model gives ½ under ∅ and fixes H or Y_{t+1} under any intervention it has. Beckers and Halpern’s strong abstraction rejects the coarse-graining for this reason (NOTE-1032, Bearing on the record, item 4).
  • Not that the coarse model keeps the response of a given system. Exactness is about push-forward laws. Governance-response sufficiency (TERM-067) is about the response kernels of individual states, and it fails here.
  • Not cyclic models. Theorem 10 assumes acyclicity; the paper points to Bongers et al. for the cyclic case, with further conditions.
  • Not independent-noise SEMs. With the usual requirement of independent exogenous variables, the theorem is false.

Connections

  • CLAIM-407 and CLAIM-350. The distributional criterion this theorem satisfies is not the fibre-wise factorization they state; the hidden bit separates them.
  • THEORY-847. The other exact transformation in the same paper, with the opposite lesson: there, enlarging the declared interventions breaks exactness; here, the declared interventions are what let the dropped variable be anything.
  • THEORY-820. Under h = 0, Y_t is informationally closed; this theorem adds that dropping K is exact as a causal model for interventions on H. Neither keeps the response of a given state.

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