Contribution
It supplies the approximate notion Beckers and Halpern (LIT-1316) left as future work, and with it a distance between causal models that compares their answers to interventions rather than their graphs. The choice of what counts as close is left to the user, on purpose. “We maintain that, in general, high-level models in science are only approximate abstractions.”
Key insight
Approximate abstraction measures, context by context and intervention by intervention, the gap between two routes to a high-level state: run the low-level model and coarse-grain, or lift the context and the intervention and run the high-level model. Everything about what matters is in the metric on high-level states and in which interventions and contexts are counted, worst case or on average.
Assumptions
- Finite signatures; deterministic recursive causal models with allowed interventions (Definitions 2.1–2.2); probabilistic models add a distribution on contexts.
- τ-consistency (Definition 2.4): τ surjective, I_H = ω_τ(I_L), and |R_L(U_L)| ≥ |R_H(U_H)|.
- A metric d_V on each model’s endogenous states, “typically straightforward to define” and left unspecified.
- Theorem 5.5’s appendix statement adds that U_L and I_L are finite.
Key results
- Definition 2.5. τ-abstraction: τ-consistent, and a surjective τ_U with τ(M_L(u⃗_L, X⃗←x⃗)) = M_H(τ_U(u⃗_L), ω_τ(X⃗←x⃗)) for every context and allowed intervention. Definition 2.6: constructive when τ is a product over a partition of V_L and I_L = I_L
- Definition 3.1. For two similar models (same signature), d_max(M₁, M₂) = max over allowed interventions and contexts of d_V(M₁(u⃗, X⃗←x⃗), M₂(u⃗, X⃗←x⃗)).
- Definition 3.2. For τ-consistent (M_L, M_H), d_τ(M_L, M_H) = min over surjective τ_U of max over X⃗←x⃗ ∈ I_L and u⃗_L of d_{V_H}(τ(M_L(u⃗_L, X⃗←x⃗)), M_H(τ_U(u⃗_L), ω_τ(X⃗←x⃗))). M_H is a τ-α approximate abstraction when d_τ ≤ α. “We take the minimum over all functions τ_U because the function that lifts the low-level contexts up to the high-level contexts does not play a major role.”
- Propositions 3.3–3.4. d_τ = 0 iff τ-abstraction; with τ the identity on similar models, d_τ is d_max.
- §4. Probabilistic d_max averages the distance over contexts (Definition 4.1); d_β takes the probability of a distance of at least β (Definition 4.2). Probabilistic τ-abstraction (Definition 4.3) is equality of push-forward distributions. Definition 4.4’s d_τ is the expected distance over Pr_L, worst case over interventions, minimized over τ_U with τ_U(Pr_L) = Pr_H. An intervention distribution Pr_I (Definition 4.7) puts probability on each low-level implementation X⃗←x⃗ of a high-level intervention Y⃗←y⃗, positive exactly when ω_τ(X⃗←x⃗) = Y⃗←y⃗; d_τ^{Pr_I} (Definition 4.8) is worst case over high-level interventions and expected over contexts and implementations. The motivation is Spirtes and Scheines’ ambiguous manipulations: setting a room to 70°F without superheating one corner.
- Comparison with SID. Peters and Bühlmann’s structural intervention distance counts pairs (X, Y) whose interventional distributions differ; the authors want arbitrary joint interventions, a size of the difference, and models that are not similar.
- Propositions 5.1–5.2. A τ-abstraction followed by a d_max-α approximation at the high level is a τ-α approximate abstraction. An α-approximation at the low level followed by a τ-abstraction is a τ-kα one, with k the worst-case ratio of high-level to low-level distances.
- Examples 5.3–5.4 and Theorems 5.5–5.6. An approximate abstraction need not be an exact abstraction of any model similar to M_L (Example 5.3, with τ(a, b, c) = (a ⊕ b, b ⊕ c) and distance √2), nor, even when constructive, an approximation of an exact abstraction of M_L (Example 5.4). Deciding either decomposition is in nondeterministic polynomial time in |I_L| × |R_L(U_L)| (Theorem 5.5). For constructive τ an M_L′ similar to M_L, of which M_H is an exact abstraction, can be found in polynomial time (Theorem 5.6).
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | d_τ generalizes both exact abstraction and approximation between similar models | strong | Propositions 3.3–3.4, 4.5–4.6 (immediate from the definitions) |
| C2 | Errors compose: abstraction then approximation gives α; approximation then abstraction gives kα | strong | Propositions 5.1–5.2, proved; k can be undefined (see Limitations) |
| C3 | An approximate abstraction need not decompose into an exact abstraction and an approximation in either order | strong | Examples 5.3–5.4 |
| C4 | For constructive τ the low-level decomposition always exists and is found in polynomial time | strong | Theorem 5.6, proved |
| C5 | The intervention distribution is how knowledge of a high-level intervention’s mechanism enters an approximate abstraction | moderate | Definitions 4.7–4.8 and the heater example; no result uses it |
| C6 | High-level scientific models are in general only approximate abstractions | weak, as a thesis | the El Niño example (Chalupka et al. 2016) and the ideal gas law, asserted |
Concepts
- Similar models: same signature, differing only in their equations.
- τ-consistent pair: Definition 2.4.
- Intervention distribution Pr_I: Definition 4.7.
Connections
It builds on Beckers and Halpern (LIT-1316, NOTE-1032) and through them on Rubenstein et al. (LIT-1334, NOTE-1037), on Chalupka, Bischoff, Perona and Eberhardt (2016) for the El Niño macro variables, and compares itself with structural Hamming distance (Acid and de Campos 2003) and structural intervention distance (Peters and Bühlmann 2015).
Bearing on the record
- CLAIM-407 and TERM-067. The deficiency Γ is not d_τ for any choice of metric. d_τ minimizes over τ_U, so the high-level context may carry what the coarse endogenous state drops: for the hidden bit (CASE-125) the coarse-graining that drops K is an exact τ-abstraction when only H is intervened on (NOTE-1032, Bearing on the record, item 3), so d_τ = 0 while Γ(Y_t) = ρ(h = 1) bits. Three further differences: d_τ takes the worst case over interventions where Γ averages them by ρ; it is a metric distance between states where Γ is a conditional mutual information between the micro state and the response given the coarse state; and its probabilistic intervention distribution weights the low-level implementations of one high-level intervention, where ρ weights distinct governance interventions. A16’s warning that Γ values under different ρ are not on one scale has its counterpart here: d_τ is defined relative to the metric, and “the choice of distance function makes a crucial difference”.
- TERM-043. D_causal = Σ_a π(a) d(T_# P_o(Y | do(a)), P_t(Y′ | do(τ(a)))) sits between this paper’s notions. It averages over interventions by π, as none of them does; it compares push-forward distributions, as Definition 4.3 does, rather than context-wise outcomes, as Definitions 3.2 and 4.4 do. The paper’s composition results (C2) are a template for TERM-043’s componentwise comparison of transports composed in sequence, which the record has not stated.
- QUESTION-005. It asks how corresponding interventions can be made operational. The intervention distribution is one formal answer for the causal component: a high-level intervention corresponds to a declared distribution over its low-level implementations, positive only on those ω_τ sends to it. It does not say where the distribution comes from, which is the question’s worry.
- TERM-065. The metric d_V and the gender/height example (“One researcher might not be interested in gender … another researcher might care about gender and not about height”) are the regime’s query family and tolerance in another notation; d_τ ≤ α is Adequate(T; …, ε) with ε = α.
- CLAIM-409. Nothing here orders d_τ along refinements of τ; the monotonicity of Γ has no counterpart in this paper.
No instruction for machine-learning practice; nothing for the anthology.
Limitations
- k in Proposition 5.2 can be undefined. It is a maximum of a ratio whose denominator is d_{V_L} between M_L′ and M_L, zero wherever they agree; the statement does not say how such terms count.
- Slips. Example 5.3 concludes “Thus, M_H cannot be recursive” where the argument shows that no M_L′ can be. The caption of Fig. 4 speaks of “the distance between the expectations of these distributions”, while Definition 4.8 takes an expectation of distances. Proposition 5.2’s conclusion writes (M_L′, I_L′) for a model whose interventions are I_L.
- No guidance on the metric, deliberately. A small d_τ under one metric says nothing under another.
- Worst case over interventions, even in the probabilistic version, so a single rare intervention can dominate; the intervention distribution averages only within one high-level intervention.
- The context map is free (up to surjectivity and the cardinality bound), as in Beckers and Halpern.
- No estimation. Nothing is said about computing d_τ from data, and Theorem 5.5’s procedure is a guess-and-check over context maps.
Open questions
- With τ_U required to factor through τ, what is the graded notion, and is it Γ-like? For the hidden bit it would be positive where d_τ is zero.
- Does d_τ, or a variant, decrease along refinements of τ, as Γ does for a fixed target?