Contribution
It gives the first general definition, for structural equation models, of when two models at different levels of detail are causal models of the same system: they must agree on the effect of every declared intervention, and the correspondence between interventions must respect how interventions compose. Marginalisation, micro-to-macro averaging and the passage from a dynamical process to its equilibrium are shown to be instances. The authors single out the partial order on interventions as the new idea: “An important novel idea of this paper is to explicitly make use of a natural ordering on the set of interventions.”
Key insight
A coarse model is causally consistent with a fine one relative to a list of interventions, not absolutely. “Observe that in each of the following examples, the particular set of interventions considered is important. If we were to allow larger sets of interventions 𝑋 in the SEM 𝑋, the transformations given would not be exact.” The test compares distributions: the coarse model must reproduce the push-forward, under the coarse-graining, of each interventional distribution of the fine one. So it asks whether a macro model predicts what macro observers would see under each intervention, not whether the macro state carries everything that determines a particular system’s response.
Assumptions
- SEMs in a wide sense (Definition 1): structural equations Xᵢ = fᵢ(X, Eᵢ), a declared subset of perfect interventions with the natural partial order (i ≤ j iff j intervenes on a superset of i’s variables, setting them to the same values), and an exogenous law ℙ_E. Exogenous variables may be dependent, cycles are allowed, and the only condition is a unique solution under every declared intervention with ℙ_E-probability one.
- Only perfect interventions do(X_S = x_S). A change to a mechanism that is not the fixing of a variable is not modelled.
- Theorem 10 needs acyclicity. Theorem 11 needs a linear SEM in which the W’s are exogenous and Z = AW + F, interventions that set all of W, all of Z or both, and equal column sums of A. Theorem 12 needs X_{t+1} = AX_t + E_t with A a contraction and the same noise at every time, E_t = E, and interventions that hold a variable fixed at all times.
Key results
- Definition 3 (exact transformation). M_Y is an exact τ-transformation of M_X if there is a surjective, order-preserving ω from M_X’s interventions to M_Y’s with ℙ^i_{τ(X)} = ℙ_Y^{do(ω(i))} for every i, the left side being the push-forward of ℙ_X^{do(i)} by τ. ω need not be injective: “this would imply that ω is injective, and hence also bijective. This would constrain the ways in which M_Y can be ‘simpler’ than M_X.”
- Lemmas 4–5. Relabelling is exact; exact transformations compose (ω_ZX = ω_ZY ∘ ω_YX).
- Theorem 6. Order preservation makes the square of interventions commute: if i ≤ j then ω(i) ≤ ω(j), so an intervention performed after another at the fine level has a counterpart performable after its counterpart at the coarse level.
- Examples 7–8. Without order preservation, an observational distribution can be matched by an interventional one (ω(∅) = do(Y₁ = 0)), and even with ω(∅) = ∅ a composite intervention can map to one that conflicts with its parts.
- Theorem 9. Marginalising childless variables is exact.
- Theorem 10. In an acyclic SEM, marginalising variables no declared intervention touches is exact, with the marginalised variable substituted into its children and its noise merged into theirs. This needs dependent noise: “marginalising out a common parent node will in general result in its children having dependent exogenous variables.”
- Theorem 11 (micro to macro). With W exogenous, Z = AW + F, every column of A summing to a, and interventions setting all of W, all of Z or both, the averages (Ŵ, Ẑ) form an exact transformation, with Ẑ = (an/m)Ŵ + F̂ (the coefficient as the record computes it; see Limitations). The equal-column-sum condition is also necessary, the record’s re-derivation in THEORY-847.
- Theorem 12 (equilibrium). For a contracting linear process with constant noise, the limit map is an exact transformation onto the cyclic SEM Yⁱ = Σ_{j≠i} A_ij Yʲ/(1 − A_ii) + Fⁱ/(1 − A_ii), with interventions that hold variables fixed for all time mapped to interventions on their limits.
- §1.1 and §6. Total cholesterol as a cause of heart disease is the motivating failure: diets raising LDL or HDL raise TC by the same amount with opposite effects, so TC is not an exact transformation of (LDL, HDL). The authors read Spirtes and Scheines’ ambiguous manipulations as “arising due to the application of an inexact transformation”.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Exact transformation, with ω surjective and order-preserving, is a sound formalisation of causal consistency between levels | moderate | Definition 3, Theorem 6, Examples 7–8 for why each condition is needed; no argument that it is not too permissive (Beckers and Halpern later show it is) |
| C2 | Marginalising childless or never-intervened variables is exact | strong | Theorems 9–10, proved in Appendix B |
| C3 | Averaging linear micro-variables is exact when every column of A has the same sum | strong | Theorem 11, proved in Appendix C; necessity re-derived here |
| C4 | The equilibrium of a contracting linear process with constant noise is an exact transformation onto a possibly cyclic SEM, which gives cyclic SEMs a causal reading | strong within the assumptions | Theorem 12 and Lemmas 13–15 |
| C5 | Exactness is relative to the declared interventions; enlarging the set can destroy it | strong as a remark | stated in §5 for every example; Theorem 11’s restriction to whole-block interventions is an instance |
| C6 | Only when a macro model is an exact transformation are macro causal statements such as “LDL causes heart disease” meaningful | weak | asserted in §4.4 (“Only if this is the case, causal statements … are meaningful”); not argued |
Concepts
- Perfect intervention do(X_S = x_S): replace the structural equations of X_S by constants.
- Natural partial order on interventions: i ≤ j when j sets a superset of i’s variables to the same values; it “corresponds to the ability to compose physical implementations of interventions”.
- Poset of distributions 𝒫_X: the family ℙ_X^{do(i)} indexed by the declared interventions, ordered as they are.
- Exact τ-transformation: Definition 3 above.
Connections
It builds on Bongers, Peters, Schölkopf and Mooij (2016) for marginalisation and cycles, on Chalupka, Perona and Eberhardt (2015, 2016) for learned macro variables, on Iwasaki and Simon (1994) and Simon and Ando (1961) for aggregation, on Hoel, Albantakis and Tononi (2013), and on Spirtes and Scheines (2004) for ambiguous manipulations. Beckers and Halpern (LIT-1316, read in NOTE-1032) take Definition 3 as the weakest of their notions and show it too permissive; Beckers, Eberhardt and Halpern (LIT-1310, NOTE-1045) carry out the “softening” this paper’s §6 proposes.
Bearing on the record
- CLAIM-407, CASE-125 and TERM-067. Exact transformation does not contain the hidden-governance-bit point, and accepts the coarse-graining the case rejects. Write the case as an acyclic SEM: K = E_K, Y_t = E_t, H = E_H, Y_{t+1} = (1 − H)E_{t+1} + HK, with the declared interventions ∅, do(H = 0) and do(H = 1). K is never intervened on, so by Theorem 10 marginalising it is exact: the coarse SEM Y_{t+1} = (1 − H)E′₁ + HE′₂, with fair noises, reproduces the law of (Y_t, H, Y_{t+1}) under every declared intervention. Over several steps under a held do(H = 1), the marginalised K reappears as noise shared across time, which Definition 1 allows. Yet Γ(Y_t) = ρ(h = 1) bits. The reason is that Definition 3 compares push-forward distributions, while governance-response sufficiency compares the response kernels of individual states, P(Y^(i) | S = s), across the fibres of T. A merged difference that only redistributes the response among states with the same coarse value is invisible to the first and is exactly what Γ measures. Rubenstein et al.’s cholesterol case is the nearest anticipation of the qualitative point (a macro variable merging micro-states whose interventional consequences differ), and their remark that enlarging the intervention set breaks exactness is CLAIM-407’s relativity to I. The coexistence of exact closure under the ordinary dynamics with that failure is not in this paper.
- TERM-043. Its D_causal, Σ_a π(a) d(T_# P_o(Y | do(a)), P_t(Y′ | do(τ(a)))), is Definition 3’s condition made graded: with π of full support, D_causal = 0 exactly when the push-forward of each source interventional law equals the target’s under the corresponding intervention. It drops the requirement that the correspondence on interventions be surjective and order-preserving, which is what Theorem 6 and Examples 7–8 show to be needed for composed interventions. By the same token D_causal is zero for the hidden bit, where Γ is not, so TERM-067’s governance-response sufficiency is not D_causal with τ the identity, as that term’s “Why it narrows TERM-043” says. See NOTE-1032 for the exact relation.
- THEORY-755 and THEORY-733. Theorem 11’s condition is THEORY-755’s column-equitability, for the one pair of blocks (rows of Z, columns of W) through which the SEM couples them: Σ_{i∈Z} A_ij is the same for every j in W. THEORY-755 states it for a dynamical x(t+1) = Ax and the vector of block sums; here it is a one-shot SEM, averages rather than sums, and a causal rather than a dynamical criterion. The algebra is the same QA = ĀQ. The join is the record’s: neither paper cites the other’s literature.
- TERM-065. Its ℐ is this paper’s declared intervention set, and C5 is the regime-relativity the term records, proved for one notion of adequacy.
- CLAIM-350. Its causal bullet’s “intervention-consistent coarse-graining for a declared family of interventions” is, in the literature, this paper’s exact transformation; but it is a distributional notion, not the fibre-wise factorization CLAIM-350 proves.
No instruction for machine-learning practice; nothing for the anthology.
Limitations
- Theorem 11’s coefficient. By direct computation, (1/m)Σᵢ Σⱼ A_ij Wⱼ = (a/m)Σⱼ Wⱼ = (an/m)Ŵ. In this copy the extracted text of both the statement and its proof shows the coefficient of Ŵ as “a” over “m”; whether the printed text lacks n or the extraction lost it could not be told. Beckers and Halpern’s restatement (their Example 3.3) has an/m, which agrees with the computation.
- Sufficiency only. Theorem 11 states the column condition as sufficient. It is also necessary for the averaging τ with these interventions (THEORY-847); the paper does not say so.
- Permissive. Nothing in Definition 3 rules out matches won by the choice of distribution or of a sparse intervention set. Beckers and Halpern’s Example 3.4 makes two unrelated models exact transformations of each other.
- Perfect interventions only, and no counterfactuals, which the authors name as not discussed.
- No choice among exact transformations, and no approximation; the §6 “softening” is future work.
- Theorem 12’s constant noise (E_t = E for all t) is a strong modelling assumption, adopted to give cyclic SEMs a reading.
- C6 is asserted, not argued.
Open questions
- Which coarse-grainings are exact for every intervention a class of physical implementations allows, rather than for a chosen list?
- For governance in particular: when a mechanism change is modelled as a perfect intervention on a regime variable H, which coarse-grainings of the pre-intervention state are exact transformations and governance-response sufficient at once? The marginalisation of K shows the first does not imply the second.