Generalised Transportability via Causal Abstractions

Motivation

• Classical transportability transfers causal effects from experiments in a source population to a target using observational data, but returns query-specific formulas or non-transportability verdicts.It is uninformative when a query is not identifiable (R1) or when no target data exist (R2).Source: §1 · p. 3

Model-level perspective

• Transportability is formulated as a same-resolution causal abstraction because source and target share variables, graphs, and intervention semantics while differing in known mechanisms.An automorphic map acts identically on invariant mechanisms and locally on shifted ones, transporting the full interventional family when it exists; several individually transportable queries may nevertheless lack one common map, creating a compatibility gap.Source: §1 · pp. 3–5

Exact transport

• The paper characterises constructive abstraction existence in both Markovian and semi-Markovian models, reducing finite-state cases to linear feasibility with uniqueness criteria.It defines a hierarchy of source–target consistency notions, from observational matching to maps transporting the full interventional family, plus a query-restricted relaxation.Source: §1 · p. 5

Approximate transport

• TraCA formulates approximate transport as distributionally robust optimisation over bounded target-side mechanism and environment perturbations around a trusted source SCM.The learned transport map optimises worst-case error over a metric ambiguity set, which can encode a directional prior on the anticipated shift.Source: §1 · pp. 4–5

Certificates

• Model-level transport error induces certified intervals for target queries, including formula-free bounds in R1 and uniform guarantees over the ambiguity set in target-agnostic R2.The framework provides general certificates for any admissible target and sharper directional certificates when a declared shift prior is available.Source: §1 · pp. 4, 6

Empirical validation

• Across synthetic Markovian and semi-Markovian benchmarks and a real ecological dataset, certified intervals bracket the ground-truth query at the radius required by the true shift.Directional certificates are several times tighter when both bounds are informative and remain the only informative bounds when the general certificate is vacuous.Source: §1 · p. 6

2.1 Structural causal models

• Acyclic SCMs consist of endogenous and exogenous variables, structural functions, and an environment distribution; their causal graph and reduced-form mixing map induce the observed distribution.The mixing function g maps exogenous states to endogenous states, with the induced law given by the pushforward g#(ρ).Source: §2.1 · p. 6

2.2 Interventions and causal queries

• Interventions mutilate the causal graph and generate post-interventional distributions, while a finite query specification family selects intervention-variable marginals separately from the functionals used to evaluate them.This separation allows the framework to optimize discrepancies between interventional marginals and derive query-level guarantees through functionals such as means, medians, or variances; identifiability asks whether such values are determined by available distributions and model assumptions, with do-calculus complete for nonparametric SCMs (Shpitser and Pearl, 2008).Source: §2.2 · pp. 7–8

2.3 Markov kernels

• Markov kernels generalize deterministic measurable maps by assigning each input a probability distribution over outputs, thereby providing transport operators between source and target interventional measures.Deterministic maps are recovered through Dirac kernels, whose action reduces to the usual pushforward; this language supports the paper’s structural reformulation of transportability.Source: §2.3 · pp. 8–9

3.1 Standard transportability setting

• Source and target SCMs share endogenous variables, DAG, and intervention semantics, differing only in mechanisms or exogenous distributions at known shifted nodes K, represented by selection variables.Invariant-node mechanisms coincide across domains, while shifted nodes may change.Source: §3.1 · p. 9
• Classical transportability decides one target query at a time, using selection-diagram do-calculus to derive a formula when possible or establish that none exists.The query may concern a full post-interventional distribution or a query-relevant marginal.Source: §3.1 · pp. 9–10

3.2 A hierarchy of consistency notions

• The hierarchy shifts attention to a single structural relation aligning source and target interventional families, a requirement strictly stronger than transporting individual queries.Observational and per-intervention consistency are automatic because arbitrary Markov kernels can map each source distribution to its target counterpart, whereas a common operator must work across interventions.Source: §3.2 · pp. 10, 12

The deterministic version restricts to kernels of the form

• Interventional consistency is the causal-abstraction requirement that one common kernel align the full interventional family, with deterministic exact transformations preserving intervention semantics through a single map.In this same-level transportability specialization, no nontrivial intervention translation is needed, and Markov kernels generalize deterministic maps.Source: The deterministic version restricts to… · p. 13

The deterministic version is the special case in which

• Constructive interventional consistency factorizes the common operator component-wise, but constructiveness is stronger than interventional consistency because non-factorized common kernels may also exist.The transportability specialization uses singleton variable blocks, identity intervention mapping, and identity action on invariant nodes, while query-restricted consistency matches only designated marginals.Source: The deterministic version restricts to… · pp. 14–15, The deterministic version is the… · p. 15
• A common operator aligning the full interventional family transports every query simultaneously, whereas Q-restricted consistency guarantees transport only for queries in Q; the converse can fail, creating a compatibility gap.Constructive interventional consistency is the strongest hierarchy level, while Q-restricted consistency weakens the required matched distributions.Source: The deterministic version restricts to… · p. 16

4.1.1 Relevant parent contexts

• Relevant parent contexts include every configuration reached under the intervention family while a shifted mechanism remains active, not merely those observed naturally.Intervening on an ancestor can create previously unseen parent values, whereas intervening directly on the node bypasses its mechanism and imposes no alignment condition.Source: §4.1.1 · p. 17

4.1.2 Mechanism-level characterisation and recursive assembly

• In Markovian SCMs, a componentwise kernel is a constructive transport operator exactly when stagewise transported conditionals align each mechanism while preserving the relevant Markov structure.The recursive assembly processes variables in topological order, matching each prefix to the target interventional law and recovering the full target joint by the final stage.Source: §4.1.1 · pp. 17–18, §4.1.2 · pp. 18–19
• The stagewise Markov-preserving restriction loses no genuine componentwise transport operators and is automatic when shifted nodes are non-ancestral to one another or deterministic maps preserve parent information.For finite state spaces, each stage reduces to linear feasibility for a row-stochastic kernel, with intervention-preserving row constraints.Source: §4.1.2 · p. 19

4.2 Semi-Markovian extension

• In semi-Markovian SCMs, the same recursive characterisation operates on confounded districts rather than individual nodes, under district-consistent interventions and an acyclic quotient graph.The district-preserving condition is also without loss for districtwise constructive operators, while cyclic quotients remain outside the constructive theory rather than being deemed non-transportable.Source: §4.2 · pp. 20–21
• The exact characterisations motivate approximate transportability, where the best approximate map yields certified query intervals even for non-identifiable queries and settings with no target data.These intervals convert abstraction error into a quantitative transportability guarantee.Source: §4.2 · p. 21

• Approximate transportability replaces exact equality between source-derived and target interventional distributions with a nonnegative discrepancy that vanishes only under equality.This defines a quantitative model-level transport error for settings where exact consistency is too strong.Source: §5 · p. 22
• For each intervention and output set, the intervention-query error measures the discrepancy between transported source and target laws after projection onto the query coordinates.The comparison can be represented as a commuting diagram contrasting intervention-then-map with map-then-intervene pathways.Source: §5 · pp. 22–23, Figure 5 · p. 23
• Averaging pointwise errors over a query specification family yields the Q-restricted transport error, measuring how far one common operator is from transporting the relevant interventional family.The construction applies to arbitrary structural causal model classes and is subsequently specialised to linear additive noise models.Source: §5 · pp. 22–23
• Vanishing transport error is equivalent to exact Q-restricted consistency, while deterministic or constructive kernel restrictions recover corresponding approximate abstraction-map problems.Under standard compactness and lower-semicontinuity conditions, the infimum is attained.Source: §5 · p. 22

6.1 Setting and Structural Forms

• LANs represent structural mechanisms with a strictly lower-triangular W and total causal propagation with A = (I − W)^−1 = I + W + · · · + W^(d−1).Hard interventions remove incoming edges through a diagonal gating matrix, yielding interventional propagators A_ι; shifts may affect mechanisms, environments, or both.Source: §6.1 · pp. 23–24

6.2 Exact Perturbation Identities

• Mechanism perturbations admit an exact finite expansion, whose leading change is A∆WA, so even sparse local changes can produce global propagator shifts.The expansion requires no convergence condition because the relevant strictly lower-triangular product is nilpotent.Source: §6.2 · p. 24

6.3 Structured Stability Bounds

• The stability bounds quantify post-interventional sensitivity using γ_ι, with a sharper small-perturbation regime γ_ι < 1 governed by the geometric factor (1 − γ_ι)^−1.Invertibility follows from acyclicity without the Neumann condition; these controls support robust optimisation over Frobenius, row-wise, column-wise, and entrywise ambiguity sets.Source: §6.3 · pp. 25–26

7 Transportation via Causal Abstractions: TraCA

• The target ambiguity explicitly models both exogenous-environment uncertainty and mechanism uncertainty, because mechanism changes cannot generally be reproduced by changing only the exogenous law.Prior knowledge can constrain which mechanisms shift, which entries change, and plausible shift patterns; partial target information can shrink the ambiguity radius.Source: §7 · p. 27
• TraCA learns a transport map by minimizing worst-case transport loss over an ambiguity set containing adversarial target-side variables.The source SCM is trusted, while uncertainty concerns structurally plausible shifts to the unseen target domain.Source: §7 · p. 26, §7.1 · p. 28

7.1 Unified minimax formulation

• The unified minimax objective specializes to LAN through intervention-specific propagator perturbations, with mechanism shifts restricted to a designated node set and controlled by spectral-norm stability quantities.The ambiguity is structurally restricted rather than arbitrary, and Corollary 19 makes the stability quantities explicit for each ambiguity geometry.Source: §7.1 · p. 28
• Shrinking the target ambiguity set monotonically lowers the robust objective and tightens the certificate, approaching the full-information oracle when the set contracts to the true perturbation.If the true target perturbation lies in a smaller ambiguity set, its robust map certifies that target; under Lipschitz losses, the oracle gap is controlled by ambiguity diameter.Source: §7.1 · pp. 28–29
• TraCA supports moment-based Gaussian and perturbation-based empirical ambiguity sets, matching source knowledge represented respectively by sufficient statistics or samples.Both formulations share an algorithmic backbone but differ in their ascent objectives and projection operators.Source: §7.1 · p. 29

8 Optimisation

• Both Gaussian and empirical formulations alternate projected descent on the transport map with projected or proximal ascent on target-side adversarial variables.They share the same min-max backbone, differing in adversarial parameterisation, ascent objective, and feasibility corrections.Source: §8 · pp. 29–30, §8.1 · p. 32

8.1 Mechanism ambiguity sets and projections

• Directional mechanism ambiguity drives the transport map away from identity by preventing cancellation between opposing perturbation vertices.The ambiguity sets preserve the fixed acyclic support, while directional boxes encode anticipated shifts through an offset δ and uncertainty through radii B.Source: §8.1 · p. 31

8.2 Gaussian adversarial learning

• The Gaussian adversary uses a Wasserstein ambiguity set over target means and covariances, with a smooth conservative surrogate for the Bures cross-term during ascent.Updates combine projected descent in T, projected ascent in ∆W and µt, proximal covariance updates on transformed covariance square roots, and invariance projection.Source: §8.2 · pp. 33–34

8.3 Empirical adversarial learning

• The empirical adversary perturbs abducted source exogenous samples with a Frobenius-bounded block Θ, equivalent to a Wasserstein ball around the source empirical environment.Projected ascent updates ∆W and Θ, with masking and projection maintaining the mechanism and shifted-node feasibility constraints.Source: §8.3 · pp. 35–36

8.4 Query-restricted relaxation

• Query restriction spans the framework from full post-interventional objectives to a single intervention/output loss without changing the optimisation architecture.Gaussian losses must be computed on selected marginal covariances because coordinate restriction does not commute with the Bures matrix square root; the learned map and stability moduli support certified transport and query bounds.Source: §8.4 · pp. 37–38

9 Provable Robustness and Query-Level Transport

• Query-restricted certificates decompose error into transport mismatch, mechanism instability, and environment ambiguity, and recover full post-interventional and single-query guarantees as special cases.Bounds worsen with transport mismatch, intervention-amplified mechanism perturbations, and target-environment uncertainty; restricting Q sharpens guarantees only for included queries.Source: §9 · pp. 38–41

9.1 Specialisation to query-level transport

• Lipschitz query functionals inherit certified target-query intervals centered at the transported-source value, using W2 in the Gaussian case and the Frobenius norm in the empirical case.The general certificates apply to arbitrary Lipschitz functionals, but can be conservative for linear and mean queries because the global mechanism modulus controls every input and output direction.Source: §9.1 · pp. 41–42

Then every admissible target satisfies

• Directional certificates sharpen linear-query bounds by replacing the global mechanism modulus with a query-specific readout modulus and using a masked environment operator.For rank-one mean queries, this can substantially reduce conservatism and may make the mechanism term vanish when no signal reaches the shifted coefficient.Source: §9.1 · pp. 42–43, Then every admissible target satisfies · pp. 43–44
• Once a transport map is learned, robust ambiguity sets define admissible target marginals, while the general and directional certificates yield intervals for arbitrary designated and linear queries.The framework evaluates worst-case abstraction discrepancy over these induced target families rather than over graphical target-domain compatibility classes.Source: Then every admissible target satisfies · p. 44

10.1 Benchmarks

• The benchmark suite covers ATE, ATCE, LiLUCAS, and Portland, isolating non-transportable, target-agnostic, and real-world shifted-environment regimes.ATE uses latent confounding and a shifted outcome mechanism; ATCE and LiLUCAS have no target data; Portland uses Fanno Creek as the observed target.Source: §10 · p. 44, §10.1 · pp. 45–46

10.2 Setup and evaluation protocol

• Maps are learned from source splits using five-fold cross-validation, while target interventional values remain held out and enter only evaluation of coverage.Target observational data, when available, select the operating radius through W2; target-agnostic benchmarks sample 100 admissible targets from the ambiguity geometry.Source: §10.2 · pp. 47–48

10.3.1 Synthetics

• On ATE, the offset prior enables observational radius selection at (ε, η)=(0.2, 0.2), where transport distance is 0.040 versus 0.63 for the symmetric variant.The offset certificate covers both queries at the selected cell, while the symmetric directional certificate misses do(X=1) by 0.08.Source: §10.3.1 · pp. 48, 49, 51, Figure 8 · p. 49, Figure 10 · p. 50
• For ATCE, the identity baseline wins at small shifts, but rtrain=1.0 reduces large-shift error to 66.38 versus 68.79 for the baseline.No single training radius dominates: robustification helps at large rtest but can hurt when shifts are small.Source: ATCE · p. 51, Table 1 · p. 52, Figure 12 · p. 53
• At matched ATCE radii, directional certificates remain informative and tighter, achieving 3.7× and 2.5× narrower widths across the two queries while retaining coverage.When rtrain<rtest, directional coverage falls to 0.61 and 0.89, whereas the general certificate still covers all sampled targets.Source: ATCE · pp. 52–54, Figure 13 · p. 53
• LiLUCAS offset maps reduce small-shift transport error by up to 60%, whereas symmetric maps never significantly improve on the identity baseline.At rtrain=0.2, offset error is 0.286 versus 0.716 for the baseline, with p < 0.001.Source: LiLUCAS · p. 54, Figure 15 · p. 55, Table 2 · p. 55
• At rtest=4.0, Gaussian-trained maps reduce transport error by 1.4% at rtrain=0.2 and up to 17% at rtrain=2.0, but rtrain=4.0 over-robustifies.The improvement is statistically significant (all p < 0.001) and fold-stable within 0.1 percentage points; the identity remains near-optimal for small rtest.Source: LiLUCAS · pp. 55–56, Figure 16 · p. 56, Table 3 · p. 56
• Directional certificates remain informative longer than general certificates, becoming the only usable bound at rtrain=0.5 while covering fully only when the test shift lies within the trained ball.At rtrain=0.2, directional intervals are 7.6-17.2× narrower per query but cover only 30% of targets outside the ball; at rtrain=0.5, they cover all four queries while general intervals are vacuous.Source: LiLUCAS · pp. 56–57, Figure 17 · p. 57

10.3.2 Real-world

• On Portland, directional certificates cover all four canopy interventions once ε≥2, while general certificates are vacuous from ε=0.5 onward.The realised shift has W2=1.165, and the directional width reaches 1.02 (6.3 × QS) only at ε=4.0.Source: §10.3.2 · p. 57, Figure 21 · p. 59
• Because Fanno Creek’s shifted coordinate is a root, η=0 and directional tightening comes entirely from the direction-specific environment term.Coverage therefore depends on enlarging the environmental ambiguity radius until it contains the realised shift.Source: §10.3.2 · pp. 58–59
• Observational calibration fails on Portland: it selects ε=0.2, yielding zero directional coverage, although ε=2.0 is required for coverage.At ε=0.2, the directional interval has width 0.05 (0.3 × QS) and misses the truth, while the general interval covers it with width 0.70.Source: §10.3.2 · pp. 58–59

Choosing the ambiguity geometry

• Use an entrywise box by default, choosing Frobenius, row- or column-wise, or entrywise budgets to match the granularity of available mechanism-shift knowledge.The entrywise box is tightest when coefficient-specific bounds are known and is most useful when structural knowledge is localized to specific edges.Source: §11 · p. 60

Encoding a directional prior

• Encode a directional prior through offset δ, with its sign and magnitude specifying expected shift direction and size, while the radius captures residual uncertainty.The guarantee is conditional on this declared domain knowledge; use δ = 0 when direction is unknown, and avoid small offsets pointing confidently in the wrong direction.Source: §11 · p. 60

What to expect from the map

• Directional recentering can reduce error by up to 60% when shifts are small and aligned, whereas oversized radii over-robustify and widen intervals.Under symmetric ambiguity, map improvements occur mainly for large shifts; practitioners should size the radius to genuine uncertainty rather than inflate it for safety.Source: §11 · pp. 60–61

Calibrating the radius from observational data

• Observational radius calibration is a useful but unreliable heuristic: it can cover the true ATE without interventional data, yet fails on Portland’s environmental shift.When observational fit points the wrong way, use a conservative radius based on domain knowledge; without target observations, the radius must be set entirely from domain knowledge.Source: §11 · pp. 61–62

Reading the certificates

• Read general and directional intervals together: directional bounds are sharper within the declared training ball, while under-coverage diagnoses that the realised shift exceeds the specified radius.The general certificate is wider and more tolerant of misspecification, whereas directional under-coverage signals that the ambiguity radius should be enlarged.Source: §11 · p. 62

12 Conclusion

• The framework treats transportability as a single map aligning complete source and target interventional families, with exact Markovian and semi-Markovian existence tests and approximate query intervals ℓ≤Qt≤L.In finite state spaces, exact existence reduces to linear feasibility; approximate transport uses distributionally robust optimisation over target mechanism and environment perturbations.Source: §12 · p. 62
• Across benchmarks and ecological data, certified intervals bracket ground-truth queries; when both are informative, directional intervals are 2.9× tighter on ATCE and up to 17× tighter per LiLUCAS query.Where the general bound is vacuous, the directional interval is the only usable interval, including on the real ecological data.Source: §12 · pp. 62–63
• Transportability becomes a continuum of certified approximate same-level abstraction: exact structural relations recover classical transportability, while approximate maps certify what remains transportable.Unlike do-calculus, TraCA returns both a transported estimate and an interval containing the truth.Source: §12 · p. 63
• The current guarantees require linear abstraction maps and linear structural causal models, an accurate causal DAG, and a trusted source SCM whose sampling and estimation errors are not included.The stability bounds and certificate closed forms rely on the reduced form X = AU and the resolvent identity A = (I −W)−1.Source: §12 · p. 63
• Future work targets doubly robust source-and-target ambiguity, non-acyclic district quotients, learned shift geometry, and tighter certificates for richer query functionals.These directions aim to absorb source estimation error, localise the target ambiguity set, and reduce the gap between certified intervals and the true worst case.Source: §12 · p. 63

Appendix overview

• The appendices provide full proofs, finite-state refinements, attainment results, auxiliary LAN and minimax theory, robustness certificates, optimisation algorithms, benchmark specifications, and supplementary experiments.They also clarify the relationship between TraCA and DiRoCA while separating their shared optimisation machinery from TraCA’s distinct contributions.Source: Appendix overview · pp. 67–68

Appendix A. Exact transportability existence results

• Appendix A makes the exact Markovian and semi-Markovian existence results algorithmically checkable through recursive proofs, finite-state reductions, and uniqueness refinements.The Markovian characterisation formalises stagewise prefix matching via induction: later kernels preserve earlier coordinates, while parent-tuple agreement enables recursive assembly; deterministic maps restrict kernels to δτi(xi).Source: Appendix A · p. 68, §A.1 · pp. 68–69

A.2 Markovian finite-state stagewise compatibility, uniqueness, and deterministic specialisation

• Finite-state Markovian stagewise compatibility is equivalent to a linear feasibility problem over row-stochastic kernels with intervention-value rows fixed to point masses.Once earlier kernels are fixed, the remaining Markov-preserving assembly condition is also stagewise.Source: §A.2 · p. 70
• If rank(M_i)=n, the compatible Markovian kernel K_i is unique; if rank(M_i)<n, the unconstrained solution set has positive dimension.Uniqueness requires at least n linearly independent rows, although stochasticity constraints may still reduce a rank-deficient system to a singleton.Source: §A.2 · pp. 70–71
• Richer intervention families can increase the rank of M_i and add row-fixing constraints, thereby tightening or uniquely determining the compatible kernel.Interventions on ancestors create additional contexts, while direct interventions on X_i impose fixed intervention-value rows.Source: §A.2 · p. 71
• Deterministic compatibility requires a single measurable state map τ_i, independent of intervention and parent context, that transports every mechanism slice and preserves intervention values.A global deterministic CIC_m additionally requires this condition at every node and stagewise Markov preservation.Source: §A.2 · pp. 71–72

A.3 Semi-Markovian finite-state district compatibility, uniqueness, and related remarks

• Semi-Markovian finite-state district compatibility and uniqueness carry over from the Markovian results, replacing node kernels with district kernels and retaining linear feasibility.Exact global CIC_k existence further requires assembling compatible district kernels into a stagewise district-preserving product kernel.Source: §A.3 · p. 72
• District-level computation scales as N_r^2, where N_r=∏_{i∈D_r}|dom[X_i]|, making district size the main practical cost of the semi-Markovian extension.The local state space is dom[X_D], rather than the state space of an individual node.Source: §A.3 · p. 72
• A common constructive semi-Markovian kernel can act as a universal transport operator even when direct transportability fails.Its query-level identity has the same formal shape as in the Markovian case, but here it can extend transport beyond already transportable queries.Source: §A.3 · p. 72

• For any admissible common kernel K, E_Q(K)=0 exactly when K realizes exact Q-restricted consistency.Source: Appendix B · p. 73
• The minimum transport error is zero if and only if some admissible kernel realizes exact Q-restricted consistency.Source: Appendix B · p. 73
• If the admissible class K is compact and E_Q is lower semicontinuous, the infimum is attained, making vanishing error equivalent to exact realizability.Source: Appendix B · p. 73

Appendix C. Auxiliary structural results

• Proposition 42 compares SCMs sharing endogenous and exogenous spaces but having different reduced-form maps under a fixed environment.The proposition formalizes the distinction between changing the mechanism and changing the exogenous law.Source: Appendix C · p. 73
• Changing the reduced-form map from g to g′ cannot generally be reproduced by keeping g fixed and changing only the exogenous law.The induced distribution under g′ and environment ρ need not equal g#ρ⋆ for any exogenous law ρ⋆.Source: Appendix C · p. 73
• A binary counterexample uses g(0)=g(1)=0 and g′(y)=y: the source always outputs 0, whereas the target under the same environment is nondegenerate.Every pushforward through g equals δ0, so no alternative exogenous law can assign positive mass to state 1.Source: Appendix C · p. 74

C.1 Corollary 19

• Corollary 19 bounds ∥Aι(Rι∆W)∥2 by adapting the argument to each ambiguity geometry and converting its natural matrix norm to the spectral norm.Row budgets control the induced ∞-norm, column budgets the induced 1-norm, and entrywise bounds provide direct entrywise control.Source: §C.1 · p. 74
• For a Frobenius-ball ambiguity set, submultiplicativity gives a corresponding result.Source: §C.1 · p. 74
• Because Rι is diagonal, it can only zero out rows, while admissible perturbations vanish outside the active row set K.The resulting restriction is used to tighten the row-structured bound.Source: §C.1 · p. 74
• Column-wise budgets yield bounds by controlling perturbations separately for each column.The corollary treats this geometry alongside Frobenius and entrywise ambiguity sets.Source: §C.1 · p. 74
• Entrywise box bounds propagate through |Aι|RιB, while directional boxes use the effective half-width |δjk|+bjk before taking the supremum over admissible perturbations.These arguments establish the stated bounds (80)–(84).Source: §C.1 · p. 75

Information monotonicity and ambiguity tightening

• Shrinking the admissible target ambiguity set can only decrease the robust transport value and improve the certificate on the true target loss.For A1 ⊆ A2, R⋆(A1) ≤ R⋆(A2).Source: Appendix D · pp. 75–76
• When losses are Lipschitz in the target perturbation, the gap between robust and oracle values is controlled by the ambiguity-set diameter and vanishes as that diameter tends to zero.Thus, ambiguity sets with diam_dA(Ak) → 0 make the robust problem approach the full-information oracle problem.Source: Appendix D · pp. 75–76
• Under nested compact ambiguity sets shrinking to the true perturbation, compact admissible maps, and joint continuity, robust values converge uniformly to the oracle without Lipschitz continuity.Berge’s maximum theorem gives continuity of each maximised objective, while Dini’s theorem upgrades monotone pointwise convergence to uniform convergence.Source: Appendix D · p. 76

Appendix E. Proofs of the robustness certificates

• The Gaussian and empirical certificate proofs separately control transport, mechanism, and environment errors, yielding pointwise bounds that extend to averaged or worst-case guarantees.The empirical proof explicitly applies the decomposition pointwise before squaring, averaging, and taking the supremum.Source: Appendix E · pp. 76, 79

E.1 Directional certificates for linear queries

• Directional linear-query certificates split exactly into transport, mechanism, and environment terms, with mechanism sensitivity computed as a support-function budgeting problem.For single-node incoming-edge perturbations, nilpotency removes higher-order terms and the rank-one identity makes the directional modulus exact; broader perturbation sets require finite higher-order bounds.Source: §E.1 · pp. 79–82
• The directional certificate argument applies to both Markovian and semi-Markovian models because it depends on q, T, and the source and perturbed propagators.The distinction is represented by diagonal versus block-diagonal transport maps.Source: §E.1 · p. 83

Appendix F. Formulation-specific optimisation details

• The formulation-specific algorithms instantiate the unified Algorithm 1 backbone for Gaussian and empirical settings.The supplied passages identify Gaussian and empirical specializations but do not provide further algorithmic steps here.Source: Appendix F · p. 83

F.1 Gaussian specialisation

• The Gaussian specialization parameterizes adversarial environments by target moments and uses the smooth surrogate for its ascent step.This is the appendix’s stated implementation specialization.Source: §F.1 · p. 83

F.2 Empirical specialisation

• The empirical specialization represents adversarial environments through sample perturbations and updates both adversarial blocks by projected ascent.This contrasts with the Gaussian moment-based parameterization.Source: §F.2 · p. 83

Appendix G. Benchmark datasets

• The benchmark appendix defines three synthetic structural causal models and a Portland real-world benchmark, with configurations summarized in Table 4.It fixes W as a strictly lower-triangular structural matrix in topological order, where each row contains a node’s incoming-edge weights.Source: Appendix G · p. 83

G.1 ATE

• The ATE benchmark is a two-variable semi-Markovian graph with latent confounding, where only Y’s coefficient shifts from βs = 1.0 to βt = 1.5.The reported target queries are E[Y | do(X=0)] = 0.30 and E[Y | do(X=1)] = 1.80, versus 1.30 for the latter in the source.Source: §G.1 · pp. 83–85

G.2 ATCE

• ATCE shifts the root Z through its environment and the outcome Y through both incoming structural coefficients and exogenous noise.Thus, the designated shifts distinguish environmental root perturbation from non-root mechanism and noise perturbations.Source: §G.2 · p. 85

G.3 LiLUCAS

• LiLUCAS is a six-variable Markovian DAG whose shifted LungCancer mechanism includes Smoking and Genetics coefficients plus noise, evaluated across nine observational and interventional regimes.The intervention set includes four single-node and four joint interventions, in addition to observation.Source: §G.3 · p. 85

G.4 Portland urban streams

• Portland tests transport under real watershed covariate shift: canopy cover X affects dissolved oxygen Y, with Fanno Creek as target and invariant causal mechanisms assumed.The benchmark contains 247 source and 100 target observations; the source supplies the joint distribution, while the target supplies only observational covariates.Source: §G.4 · p. 86

Appendix H. Implementation and optimisation details

• Optimisation uses five map steps and two adversary steps per outer iteration, with saddle-point stopping based on objective change and a confirmed duality-gap threshold.The procedure uses a 5,000-iteration safety cap, and every reported configuration satisfies gap confirmation at termination.Source: Appendix H · pp. 87–88
• Corner enumeration exploits at most four vertices for mechanism ambiguity, while uniform corner averaging cancels symmetric-box gradients but drives directional maps away from identity.This procedure applies to ATE, ATCE, and LiLUCAS, but not Portland, whose root-only shift has no mechanism ambiguity.Source: Appendix H · pp. 87–88

Appendix I. Note on relation to DiRoCA

• TraCA turns abstraction error into certified causal-query intervals, including regimes where transportability fails or no target data exist, unlike DiRoCA’s scalar abstraction-error bounds.TraCA instead transports between same-level source and target populations, while DiRoCA abstracts between low- and high-level models within one system.Source: Appendix I · pp. 88–89

Appendix J. Supplementary experimental results

• Across the supplementary benchmarks, the appendix expands Section 10.3’s results for the Section 10.1 query family under the Section 10.2 protocol.Each figure supplies the full profile behind a claim stated in Section 10.3.Source: Appendix J · p. 89

J.1 ATE

• At rtrain=0.2, both ATE certificates are nested and widen with rtrain, while the general certificate widens faster; the truth is inside the directional band but outside the symmetric band at the smallest radius.The coverage gap is 0.08.Source: §J.1 · p. 89, Figure 24 · p. 90

J.2 ATCE

• For ATCE, directional coverage reaches one exactly when rtrain meets rtest, whereas the general certificate covers below the in-ball boundary across three test magnitudes.The general bound is wide enough to include targets outside the ball for which it was constructed.Source: §J.2 · p. 89, Figure 25 · p. 90

J.3 LiLUCAS

• For LiLUCAS, the empirical general certificate is narrower than the Gaussian one at every radius but remains vacuous throughout because its vacuity threshold is less than half as large.Thus absolute interval width does not determine whether a certificate is informative.Source: §J.3 · p. 89, Figure 26 · p. 91

J.4 Portland

• On Portland, query coverage requires a radius large enough to contain the realised shift, while the directional certificate remains informative beyond that radius.The general certificate is vacuous from rtrain=0.5 onward, whereas the directional certificate stays informative throughout.Source: §J.4 · p. 89, Figure 27 · p. 91
• On Portland, observational transport distance increases monotonically with rtrain, so observational selection chooses the smallest ε and yields zero coverage.The guarantee itself holds at whatever radius is declared, but radius selection must rely on domain knowledge rather than observational fit.Source: §J.4 · p. 90, Figure 27 · p. 91