Contribution
It takes the question “when is a high-level causal model a faithful abstraction of a low-level one?” from a single definition to a ladder of definitions, each closing a loophole the previous one leaves, with an example for each loophole. It identifies the two sources of permissiveness in exact transformation: “because RW+ consider probabilistic causal models, some differences that are intuitively significant are overlooked by considering just the right distributions. Second, besides a function that maps low-level states to high-level states, RW+ define a separate mapping of interventions that can mask what we view as essential differences”. It characterizes the first strengthening exactly (Theorem 3.8).
Key insight
Making exactness hold for every distribution on contexts turns a statement about distributions into a statement about individual contexts: there must be a map τ_U on contexts such that, for every context u and allowed intervention i, coarse-graining the low-level outcome gives the high-level outcome in context τ_U(u). The map τ_U is existential. The high-level model may have exogenous variables of its own, and they may carry whatever low-level detail the endogenous coarse-graining τ drops.
Assumptions
- Deterministic, recursive (acyclic) causal models M = (S, F, I) with signature (U, V, R), a set I of allowed perfect interventions X⃗ ← x⃗ with the natural order, and, for the probabilistic notions, a probability on contexts R(U). Recursiveness is “only to simplify the exposition”; footnote 1 says unique solvability would do.
- Rubenstein et al.’s unique-exogenous-variable assumption is shown to be harmless in the probabilistic setting (Lemma 2.4) and is not assumed otherwise.
- Proposition 3.6 needs I_L countable.
Key results
- Definition 3.1. Rubenstein et al.’s exact (τ-ω)-transformation, restated: Pr_H^{ω(Y⃗←y⃗)} = τ(Pr_L^{Y⃗←y⃗}) for every allowed low-level intervention.
- Example 3.4. Two unrelated models, each with only the empty intervention and a point-mass distribution, are exact transformations of each other. “a high-level model M_H can be arbitrary in contexts that do not lead to settings v⃗_H that have positive probability for some allowed low-level intervention.”
- Definition 3.5 (uniform transformation). For every Pr_L there is a Pr_H making the pair exact.
- Proposition 3.6 and Theorem 3.8. With I_L countable, M_H is a uniform (τ-ω)-transformation of M_L iff there is τ_U : R_L(U_L) → R_H(U_H) compatible with τ: τ(M_L(u⃗_L, Y⃗←y⃗)) = M_H(τ_U(u⃗_L), ω(Y⃗←y⃗)) for all u⃗_L and allowed interventions; and then Pr_H can be taken to be τ_U(Pr_L). Theorem 3.9: uniform transformations compose.
- Examples 3.10–3.11. Uniformity still lets M₁ and M₂ be abstractions of each other when ω is chosen freely (X₁ ← x₁ mapped to (X₁, X₂) ← (x₁, x₁)), and lets the high level be more detailed than the low (a new variable X* that, with Pr(U* = 0) = 0, switches every equation to arbitrary behaviour only off the support).
- Definition 3.12 (ω_τ). τ fixes the map on interventions: ω_τ(X⃗←x⃗) = Y⃗←y⃗ when τ(Rst(V_L, x⃗)) = Rst(V_H, y⃗), and is undefined otherwise. Definition 3.13 (τ-abstraction): τ surjective, a surjective τ_U compatible with τ, and I_H = ω_τ(I_L). Proposition 3.14: a τ-abstraction is a uniform (τ, ω_τ)-transformation.
- Definition 3.15 (strong τ-abstraction). I_H is every high-level intervention and I_L is every low-level intervention on which ω_τ is defined. Example 3.16: energy from (velocity, height, mass) is not a strong abstraction, because M ← m maps to the empty high-level intervention (every energy pair is still reachable) while changing the outcome; dropping such interventions gives a τ-abstraction. Example 3.17: counts of black pixels in the upper and left halves can only be set jointly; Example 3.18: τ(x₁, x₂, x₃) = (x₁ ∨ x₃, x₂ ∨ x₃) sends both X₃ ← 0 and the empty intervention to the empty one, with different effects.
- Definition 3.19 (constructive abstraction). A strong τ-abstraction in which τ is a product of maps on the cells of a partition of V_L, one cell per high-level variable plus one marginalised away. Rubenstein et al.’s three examples are constructive; that every strong abstraction is constructive under “a few minor technical conditions” is conjectured.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Exact transformation is too permissive: suitable distributions or intervention maps let models that differ essentially be exact transformations of each other | strong | Examples 3.4, 3.10, 3.11 |
| C2 | Exactness for every distribution is equivalent to a context map commuting with every allowed intervention | strong, given countable I_L | Proposition 3.6 and Theorem 3.8, proved; re-derived in THEORY-849 |
| C3 | Letting τ determine the map on interventions, and requiring τ surjective, removes the remaining counterexamples | moderate | Definitions 3.12–3.13; checked against Examples 3.10–3.11 |
| C4 | Strong abstraction is the clean relation between basic causal models, but some useful abstractions are only τ-abstractions | strong by example | Examples 3.16–3.18 |
| C5 | Combining micro-variables into macro-variables is constructive abstraction, and all of Rubenstein et al.’s examples are constructive | moderate | asserted for their examples; shown in detail for Example 3.3 only |
| C6 | Every strong abstraction is constructive up to minor conditions | weak | conjecture, “we have not proved this result yet” |
Concepts
- Context: a setting u⃗ of all exogenous variables. A deterministic model maps (context, intervention) to a state, M(u⃗, X⃗←x⃗).
- Allowed interventions I: those “we care about for some reason or other”, for example “the interventions that involve variables and values that are under our control”.
- Compatible context map τ_U: commutes with every allowed intervention as in Theorem 3.8.
- Rst(V, x⃗): the states of V that agree with x⃗ on the intervened variables; ω_τ sends X⃗←x⃗ to the high-level intervention whose Rst is the image of this one.
Connections
It builds directly on Rubenstein et al. (LIT-1334, NOTE-1037), whose Definition 3 is its starting point, and on Halpern (2016) and Halpern and Pearl (2005) for the models. It cites Chalupka, Eberhardt and Perona (2015, 2016) and Iwasaki and Simon (1994) for clustering micro-variables. Its “approximate abstraction” future work is LIT-1310 (NOTE-1045). LIT-038’s status note sends a reader wanting causal abstraction here, and this reading confirms the pointer.
Bearing on the record
Governance-response sufficiency and the τ-abstraction ladder (CLAIM-407, CLAIM-409, TERM-067). What follows is the record’s derivation; the paper does not treat governance or a deficiency.
- The exact relation. Take the low-level context to be the pre-intervention micro state S (with any further noise independent of S), the outcome Y, the allowed interventions the declared governance family I, and ω the identity on I. In the deterministic case, governance-response sufficiency of T = f(S) is exactly Theorem 3.8(b)‘s compatibility condition, compared on Y, with the context map required to be T itself: Y^(i)(s) = g_i(T(s)) for every s and i, that is, τ(M_L(s, i)) = M_H(τ_U(s), i) with τ_U = T. In the probabilistic case it is the kernel form: P(Y^(i) | S = s) = G_i(· | T(s)), a high-level model whose context is T and whose remaining noise is independent of S given T.
- So it is not a special case of exact transformation, uniform transformation or τ-abstraction, nor they of it. It is stronger in one respect: Theorem 3.8 only asks that some τ_U exist, and τ_U may encode what T drops into high-level exogenous variables. It is weaker in others: no high-level structural model need exist beyond the kernels g_i or G_i, nothing is required of composed interventions, only one outcome is compared, and (for Γ) only states of positive probability and the interventions ρ charges.
- The hidden bit passes every rung up to τ-abstraction. In CASE-125, with K an endogenous variable fixed by U_K and interventions on H only, τ dropping K is a τ-abstraction: take τ_U(u) = (u_t, u_H, (u_K, u_{t+1})), and let the high level read Y_{t+1} from the first or second component of its noise according to H. With K exogenous instead (a “hidden mechanism parameter”, as A16 calls it), τ is the identity on endogenous variables and the model is trivially a strong, indeed constructive, abstraction of itself. Either way Γ(Y_t) = ρ(h = 1) bits.
- Strong abstraction catches it only through interventions on K. If K is endogenous and every intervention ω_τ translates is allowed, then K ← k is allowed and ω_τ(K ← k) is the empty intervention, since every high-level state is still reachable; likewise (K ← k, H ← 1) maps to H ← 1. For a context with u_K = 0, H ← 1 must give Y_{t+1} = 0 and (K ← 1, H ← 1) must give Y_{t+1} = 1 from the same high-level context and the same high-level intervention, which no τ_U can do. That is Example 3.16’s pattern (mass) and Example 3.18’s (X₃), not the governance point: it fails because the parameter can be set, not because states that the coarse description merges respond differently to the governance intervention.
- What the ladder already contains. The relativity to a declared intervention set (Rubenstein et al., and here I_L); the support caveat (Example 3.4 and 3.11’s behaviour off the support, which is CLAIM-407’s “for every intervention ρ charges … over the states of positive probability”, and the case ρ(h = 1) = 0); and a macro variable merging micro-states with different interventional consequences (Rubenstein et al.’s cholesterol case). What it does not contain: exact informational closure under the ordinary dynamics together with failure for a mechanism intervention, a state-conditional deficiency, or Γ’s monotonicity under refinement (CLAIM-409); Theorem 3.9’s composition is a different property.
The consolidator may want this filed as a demonstrated CLAIM; the text is proposed in the batch report.
- TERM-043. Its D_causal is a graded form of Definition 3.1 (see NOTE-1037), so it inherits Example 3.4’s permissiveness and, like Definition 3.1, is zero for the hidden bit.
- CLAIM-350. Its fibre-wise factorization is the deterministic shape of item 1, with T the coarse map and the judgment the response; the paper’s compatibility condition is the same shape one level up, on contexts.
- THEORY-243. Interchange-intervention tests of a network against a causal model are tests of a constructive abstraction’s commuting condition on chosen inputs; THEORY-243’s identifiability gap is a case where two alignments both satisfy it.
- CLAIM-122. Its supervisory test (“intervene on the governance relations”) is a declared I in the paper’s sense; the paper’s ladder says that whether a coarse model passes depends on whether the test’s own interventions, or also interventions on the governance parameters, are allowed.
- TERM-065. Allowed interventions are one coordinate of the regime; the ladder shows that how the interventions are fixed (chosen freely, fixed by τ, or all that τ can translate) is a further choice the regime must state.
No instruction for machine-learning practice; nothing for the anthology. The paper names explainable AI as a motivation in one sentence.
Limitations
- Theorem 3.8 omits a hypothesis its proof uses. The (a)⇒(b) direction goes through Proposition 3.6, which assumes I_L countable; the theorem is stated without it.
- Slips. In Example 3.11, “regardless of how the equations in M₄ are defined if X* = 1” should read X* = 0, the case where they are arbitrary. The appendix proof headed “Proof of Theorem 3.3” proves Proposition 3.14.
- C5 is shown for one example. Only Example 3.3 is worked as a constructive abstraction; the other two of Rubenstein et al.’s are asserted.
- Deterministic and exact. Probabilities enter only through Definition 3.1 and uniformity; approximation is left to future work.
- Perfect interventions only. A change of mechanism must be modelled as setting a variable.
- The context map is unconstrained beyond surjectivity (and, in the sequel paper, a cardinality bound), which is what lets high-level exogenous variables hold low-level detail.
Open questions
- C6: is every strong abstraction constructive, and under which conditions?
- Which notion of abstraction results if τ_U is required to factor through τ, so that the high-level context holds nothing the coarse state does not? Item 1 above says that for one outcome and identity ω it is governance-response sufficiency.