Generalised Transportability via Causal Abstractions

Yorgos Felekis; Paris Giampouras; Fabio Massimo Zennaro; Theodoros Damoulas

cs.LGAug 16, 2026

Generalised Transportability via Causal Abstractions

Authors: Yorgos Felekis, Paris Giampouras, Fabio Massimo Zennaro, Theodoros Damoulas

Organizations: Department of Computer Science University of Warwick Coventry, CV4 7AL, UK · Department of Informatics University of Bergen Bergen, N-5020, NO · Departments of Computer Science & Statistics University of Warwick Coventry, CV4 7AL, UK

Abstract

Transporting a causal conclusion from a source study population to a target one is a fundamental problem in causal inference. The theory of transportability provides a criterion for when this is possible: given experimental data from the source and observational data from the target, it determines whether a target query is identifiable and does so completely; i.e. if the query can be transported, the criterion finds the exact formula. However, it works one query at a time and returns an expression rather than the value itself. It is also silent in two practically important regimes: when the query is not transportable and when no target data exist at all. To tackle both, we take a model-level perspective grounded in Causal Abstraction theory. Source and target share variables, graph, and interventions, differing only at a known set of mechanisms, which makes transportability a special case of same-level abstraction. Thus, instead of asking whether one query transports, we ask whether a single map aligns the source and target across their interventional behaviour. We characterise when such a map exists in both the Markovian and semi-Markovian settings; when it does, every target query transports at once. Our main contribution lies in the approximate case. When no exact map exists, the best approximate one still yields certified query intervals, recasting abstraction error as a quantitative notion of approximate transportability. We formulate model-level transport as distributionally robust optimisation over mechanism and environment perturbations of the unseen target and derive certificates for both challenging regimes: bounds for non-transportable queries, and guarantees under target-agnostic settings. We evaluate our framework on synthetic Markovian and semi-Markovian benchmarks and a real ecological dataset, and we show that the certified intervals bracket the true interventional query.

Figures & tables

Panel of Figure 1 — Figure 1: From query-level verdicts to model-level certified transportability. Left: Classical Transportability. A query estimated in a source population (blue) need not stay valid in a target (orange) where some nodes’ mechanisms shift. From the source domain we are given observational and interventional data $(\mathcal{D}_{s}^{\operatorname{o}},\mathcal{D}_{s}^{\operatorname{i}})$ , generated by the laws $\mathbb{P}_{s}$ and $\mathbb{P}_{s}^{\operatorname{do}(\cdot)}$ (interventions marked by ), together with target observational data $\mathcal{D}_{t}^{\operatorname{o}}$ from law $\mathbb{P}_{t}$ , entailed by the source and target SCMs $\mathcal{M}_{s},\mathcal{M}_{t}$ , which share graph and interventions and differ only at the shifted mechanism ( $Y$ ). One asks whether $Q_{t}=\mathbb{E}_{\mathbb{P}_{t}}[Y\mid do(X)]$ is identifiable. The do-calculus returns a binary verdict ( Yes : exact formula; No : none), leaving no answer in two regimes: ( $\mathcal{R}_{1}$ ) $Q_{t}$ is not identifiable; ( $\mathcal{R}_{2}$ ) no target data exist, so even a transportable formula cannot be evaluated. Right: Generalised Transportability via Causal Abstractions. As source and target share variables, graph, and interventions and differ only at the shifted mechanisms, we model transport through a causal abstraction map $\tau$ , with ambiguity concentrated around the shifted mechanisms. TraCA: We learn $\tau$ via a min-max problem that pushes $\mathbb{P}_{s}$ toward a neighbourhood $\mathcal{A}(\varepsilon)$ of admissible targets and hedges against the worst. Because $\tau$ is only approximate, the transported value $Q_{\tau\#s}=\mathbb{E}_{\tau_{\#}\mathbb{P}_{s}}[Y\mid do(X)]$ need not coincide with the true target value. Instead, it anchors the certified target interval $[\ell,L]$ , shown in red, which contains $Q_{t}^{\star}$ for every admissible target.

Panel of Figure 2 — Figure 2: Selection diagrams . Each panel shows one source-target pair: the two SCMs $\mathcal{M}_{s}$ and $\mathcal{M}_{t}$ share endogenous variables, causal graph, and intervention semantics, and differ only at the shifted nodes (orange), where the mechanism, the exogenous noise, or both may vary across domains. A square selection node $\sigma_{(\cdot)}$ marks each. (a) A semi-Markovian example: the bidirected arc $X\leftrightarrow Y$ encodes an unobserved confounder, and the shifted effect node $Y$ varies in its mechanism $f_{Y}$ (the coefficient on its parent $X$ ) and/or its exogenous noise, while the mechanism of $X$ is invariant. (b) A Markovian example with two shifted nodes: the outcome $Y$ shifts through its mechanism $f_{Y}$ , the coefficients on its parents $Z$ and $X$ and/or its exogenous noise, while the root $Z$ , having no parents, shifts only through its exogenous law $f_{Z}(U_{Z})$ ; the mechanism of $X$ is invariant.

Panel of Figure 3 — Figure 3: The hierarchy of source-target consistency notions. Constraint strength increases downward along the spine (the narrower the box, the more constraining the notion) and, orthogonally, from the $\mathcal{Q}$ -restricted relaxation toward the full-family notions on the left. (O) and (PI) hold for any pair of models; the real constraint begins at interventional consistency (IC). Constructive interventional consistency (CICk/CICm) is the strongest and the central object of Section 4 . In an orthogonal dimension, $\mathcal{Q}$ -restricted consistency (right) relaxes any of the spine notions to hold only on designated query marginals; it is informative precisely at IC and Constructive IC, since O and PI already hold for every model pair and their relaxations are therefore vacuous.

Panel of Figure 4 — Figure 4: Recursive global assembly (Theorem 8 ). Schematic illustration of the induction behind the theorem on a chain $X_{1}\to X_{2}\to X_{3}$ . At stage $i$ , the lifted kernel $\bar{K}_{i}$ acts on coordinate $i$ while the others are held fixed. Nodes shaded in blue still carry the source-side law, nodes shaded in orange already agree with the target law, and the red outline marks the coordinate updated at the current stage. The partially transformed laws $R_{i}^{(\iota)}$ satisfy the induction invariant that, after stage $i$ , the prefix $(X_{1},\dots,X_{i})$ agrees with the target interventional law $P_{t}^{(\iota)}$ ; by stage $d$ , the full target joint is recovered.

Panel of Figure 5 — Figure 5: Computation of the intervention-query error. From the source observational law $P_{s}$ , two pathways are compared: (a) intervene with $\iota$ on the source, then map to the target via $K$ (yielding $P_{s}^{(\iota)}K$ ); and (b) map via $K$ , then apply $\iota$ (yielding $P_{t}^{(\iota)}$ ). Their gap is the full-joint error $e_{K}^{(\iota)}$ for the unconstrained regime $O_{\iota}=[d]\setminus J_{\iota}$ . Projecting both laws onto the query coordinates via $\pi_{O_{\iota}}$ ( $O_{\iota}\subset[d]\setminus J_{\iota}$ ) instead gives the pointwise error $e_{K}^{(\iota,O_{\iota})}$ ; averaging this over all pairs in $\mathcal{Q}$ yields the $\mathcal{Q}$ -restricted transport error $\mathcal{E}_{\mathcal{Q}}$ .

Panel of Figure 6 — Figure 6: Two uses of distributional robustness. (a) Classical DRO and (b) TraCA share a minimax form, but the ambiguity set $\mathbb{B}_{\varepsilon}(\cdot)$ (dashed ball) plays different roles. In Classical DRO (internal validity): the set is centered on the empirical source $\widehat{\mathbb{P}}_{s}$ and hedges against sampling error ; the true law $\mathbb{P}_{s}^{\star}$ is assumed to lie in it, and the minimax is evaluated at the worst case $\mathbb{P}_{s}^{\mathrm{wc}}$ . In TraCA (external validity): the set is centered on the trusted source $\mathbb{P}_{s}^{\star}$ , and each admissible target perturbs it by a mechanism and an environment shift; the true target $\mathbb{P}_{t}^{\star}$ is assumed to lie in it, and $\tau$ is evaluated against the worst case $\mathbb{P}_{t}^{\mathrm{wc}}$ . Markers: center, worst case, true law.

Panel of Figure 7 — Figure 7: Causal graphs of the four benchmarks. Shaded nodes are shifted across domains; the dashed arc in (a) is latent confounding.

Figure 8: ATE: observational transport distance (left column) and certificate widths (center: general, right: directional) over the $(\varepsilon,\eta)$ grid, for the symmetric (top) and offset (bottom) variants. The observational distance uses no interventional information; the selected cell $(\varepsilon{=}0.2,\,\eta{=}0.2)$ , boxed, sits at $\mathcal{W}_{2}^{2}=0.040$ under the offset variant.

Figure 9: ATE: certificate width along the diagonal

r_{\mathrm{train}}=\varepsilon{=}\eta

for both variants and both certificates (log scale). The horizontal line is the vacuity threshold (

10\times\text{QS}=10.5

). Note that in the left panel the two directional curves, symmetric (solid, circles) and offset (dashed, squares), coincide exactly and are drawn on top of one another: the intervention sets

X

to zero, so the shifted

X\to Y

coefficient contributes nothing to this query and the offset prior has nothing to act on; the width is then set by the environment term alone. This is the vanishing-mechanism case of Remark 31 .

Figure 10: ATE: certified intervals at the observationally selected cell

r_{\mathrm{train}}{=}0.2

, for both variants and both queries. Under the symmetric variant the directional certificate misses

\mathbb{E}[Y\mid do(X{=}1)]

by a gap of

0.08

(annotated); the offset variant covers both queries.

Figure 11: ATE: directional misspecification. The target is fixed and the offset prior

\delta

varies, at

r_{\mathrm{train}}=\varepsilon{=}\eta{=}0.2

. Shading marks the

\delta

for which the box

[\delta{-}\eta,\,\delta{+}\eta]

contains the true shift

\beta_{t}-\beta_{s}=0.5

. Left: transport error relative to the identity (log scale); the minimum falls at

\delta=0.3

, where the learned coefficient

\tau_{YX}\approx 0.465

is closest to the truth on the grid. Right: directional width, following

4(|\delta|{+}\eta)+2\eta

(dashed) on

do(X{=}1)

and constant at

0.415

do(X{=}0)

. Coverage fails only at

\delta=0

(red).

Figure 12: ATCE: transport error against

r_{\mathrm{test}}

, one line per

r_{\mathrm{train}}

(log scale). The identity baseline (dashed) grows monotonically. The lines separate most clearly at moderate

r_{\mathrm{test}}

; at

r_{\mathrm{test}}{=}4.0

the errors are large and their differences small in relative terms, so the large-shift comparison is quantified in Table 1 .

	$r_{\mathrm{test}}=0.2$	$r_{\mathrm{test}}=4.0$
Method	Error	Error
Baseline ( $\tau{=}I$ )	$\mathbf{0.044\pm 0.033}$	$68.794\pm 71.240$
$r_{\mathrm{train}}=0.2$	$0.070\pm 0.068$	$68.213\pm 71.017$
$r_{\mathrm{train}}=0.5$	$0.236\pm 0.152$	$67.426\pm 70.739$
$r_{\mathrm{train}}=1.0$	$0.935\pm 0.314$	$\mathbf{66.383\pm 70.464}$
$r_{\mathrm{train}}=2.0$	$4.487\pm 0.689$	$65.928\pm 70.981$

Table 1: DRO crossover on ATCE: transport error (mean

\pm

std) at small (

r_{\mathrm{test}}{=}0.2

) and large (

r_{\mathrm{test}}{=}4.0

) perturbation magnitudes. For each perturbation seed

k

, we average the error over

5

folds, yielding

n{=}100

paired observations; the pairing unit is

k

, so the folds do not inflate the sample. Paired

t

-tests compare each

r_{\mathrm{train}}

against the baseline (

\tau{=}I

). At small

r_{\mathrm{test}}

, the baseline wins significantly; i.e. no robustification is needed. At large

r_{\mathrm{test}}

, TraCA wins significantly.

Figure 13: ATCE: certified intervals at three operating points. Left:

r_{\mathrm{train}}<r_{\mathrm{test}}

(out-of-ball). Center: matched. Right:

r_{\mathrm{train}}>r_{\mathrm{test}}

(in-ball with margin). Blue = general, orange = directional; each red point is one sampled target’s true query value, so coverage is the fraction falling inside the band.

Figure 14: ATCE: mean utilisation ratio

\bar{u}

at the same three operating points. The dashed line marks

\bar{u}{=}1

, the coverage boundary; higher is tighter, and a mean above the line indicates the average target lies outside the interval (out-of-ball regime).

Figure 15: LiLUCAS (empirical): transport error against

r_{\mathrm{test}}

for the symmetric (left) and offset (right) variants, one line per

r_{\mathrm{train}}

(log scale), against the identity baseline (dashed).

	Symmetric		Offset
$r_{\mathrm{train}}$	$r_{\mathrm{test}}{=}0.2$	$r_{\mathrm{test}}{=}4.0$	$r_{\mathrm{test}}{=}0.2$	$r_{\mathrm{test}}{=}4.0$
Baseline ( $\tau{=}I$ )	$\mathbf{0.034\pm 0.027}$	$\mathbf{11.715\pm 9.315}$	$0.716\pm 0.266$	$\mathbf{11.762\pm 10.353}$
$r_{\mathrm{train}}{=}0.2$	$0.046\pm 0.032$	$\mathbf{11.689\pm 9.312}$	$\mathbf{0.286\pm 0.085}$	$\mathbf{11.791\pm 9.379}$
$r_{\mathrm{train}}{=}0.5$	$0.105\pm 0.049$	$\mathbf{11.691\pm 9.323}$	$0.327\pm 0.052$	$\mathbf{11.923\pm 9.320}$
$r_{\mathrm{train}}{=}1.0$	$0.309\pm 0.084$	$\mathbf{11.800\pm 9.383}$	$0.496\pm 0.035$	$\mathbf{12.249\pm 9.334}$
$r_{\mathrm{train}}{=}2.0$	$1.108\pm 0.159$	$12.408\pm 9.658$	$1.222\pm 0.152$	$13.293\pm 9.793$

Table 2: LiLuCaS transport error under symmetric and offset selectors. Each cell shows mean

\pm

std over

n{=}100

perturbation seeds

k

(fold-averaged). Bold marks entries not significantly worse than the column minimum (paired

t

-test,

p>0.05

). The baselines differ because the offset selector centres its ambiguity set at

\delta{=}0.5

Figure 16: LiLUCAS (Gaussian): transport error against

r_{\mathrm{test}}

, one line per

r_{\mathrm{train}}

(log scale), against the identity baseline (dashed).

	$r_{\mathrm{test}}=0.2$	$r_{\mathrm{test}}=4.0$
Method	Error	Error
Baseline ( $\tau{=}I$ )	$\mathbf{0.083\pm 0.042}$	$25.916\pm 15.466$
$r_{\mathrm{train}}=0.2$	$\mathbf{0.085\pm 0.048}$	$25.560\pm 15.188$
$r_{\mathrm{train}}=0.5$	$0.128\pm 0.098$	$24.919\pm 14.666$
$r_{\mathrm{train}}=1.0$	$0.522\pm 0.271$	$23.437\pm 13.328$
$r_{\mathrm{train}}=2.0$	$5.551\pm 0.936$	$\mathbf{21.468\pm 10.324}$

Table 3: LiLuCaS (Gaussian): transport error at small and large test perturbations. Each cell shows mean

\pm

std over

n{=}100

perturbation seeds

k

(fold-averaged). Bold marks entries not significantly worse than the column minimum (paired

t

-test,

p>0.05

Figure 17: LiLUCAS (Gaussian): certified intervals at three operating points,

r_{\mathrm{test}}{=}0.5

fixed. Left (

r_{\mathrm{train}}{=}0.2

, out-of-ball): both informative, directional

7.6

17.2\times

tighter per query, but the directional interval covers only

30\%

as the shift exceeds the training radius. Center (matched): directional informative and fully covering, general vacuous. Right (margin): both vacuous. Blue = general, orange = directional, red = sampled targets.

Figure 18: LiLUCAS (Gaussian): mean utilisation ratio at the three operating points of Figure 17 . At the out-of-ball point the directional utilisation exceeds one for two queries, consistent with the

30\%

coverage there.

Figure 19: LiLUCAS (empirical symmetric): certified intervals at three operating points,

r_{\mathrm{test}}{=}0.5

fixed. The directional certificate is informative and fully covering where the general is vacuous (left, center); both are vacuous at the margin (right).

Figure 20: LiLUCAS (empirical symmetric): mean utilisation ratio at the three operating points of Figure 19 .

Figure 21: Portland: coverage against

r_{\mathrm{train}}

. The dashed line marks the Wasserstein distance of the realised Fanno shift (

\mathcal{W}_{2}=1.165

); the green region indicates

\varepsilon\geq\mathcal{W}_{2}

, where the target lies inside the ball.

Figure 22: Portland: certified intervals at

r_{\mathrm{train}}{=}0.2

for four canopy interventions. Left: full view (general + directional). Right: directional zoom. The directional interval is tight but does not yet cover the truth at this radius.

Figure 23: Portland: certified intervals across

r_{\mathrm{train}}

for each canopy intervention. Blue = general, orange = directional, dashed = truth. The vertical line marks

r_{\mathrm{train}}{=}2.0

, where directional coverage reaches one for all queries.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

	Objective	Shifted	Budgeted coeff.	Radii swept	Target
ATE	Gaussian	$Y$	$w_{X\to Y}$	$(\varepsilon,\eta)$ grid	one real
					obs. only
ATCE	Gaussian	$Z$ , $Y$	$w_{Z\to Y}$ , $w_{X\to Y}$	$\varepsilon=\eta$	$K{=}100$
					sampled
LiLUCAS	Empirical + Gaussian	LC	$w_{\mathit{Sm}\to\mathit{LC}}$ , $w_{\mathit{Ge}\to\mathit{LC}}$	$\varepsilon=\eta$	$K{=}100$
					sampled

Appendix

Table 4: Benchmark configurations. Shifted nodes carry the mechanism and/or environment ambiguity; budgeted coefficients are the incoming edges of each shifted node, and there are none for Portland, since its shifted node is a root. Targets are either a single observed domain, from which only the observational distribution is used, or

K{=}100

shifts sampled from the ambiguity geometry.

Figure 24: ATE: nested certified intervals along the diagonal

r_{\mathrm{train}}=\varepsilon{=}\eta

, for both variants and both queries. Blue = general, orange = directional, dashed = truth. The band widths are read more precisely from Figure 9 .

Figure 25: ATCE (sub-family): coverage against

r_{\mathrm{train}}

at three fixed

r_{\mathrm{test}}

values, for the general (left) and directional (right) certificates. Directional coverage rises to one at

r_{\mathrm{train}}{=}r_{\mathrm{test}}

; the general certificate already covers below that boundary.

Panel of Figure 26 — Figure 26: LiLUCAS: certificate width against $r_{\mathrm{train}}$ (log scale), with the vacuity threshold $10\times\text{QS}$ . (a) Under the empirical objective the general certificate is vacuous across the entire grid; only the directional certificate is informative, and only at the two smallest radii. (b) Under the Gaussian objective the general certificate is vacuous from $r_{\mathrm{train}}{=}0.5$ onward, while the directional certificate stays roughly an order of magnitude narrower and remains informative longer. Note the thresholds differ ( $6.5$ and $14.6$ ): the empirical query scale is smaller, so a narrower interval can still be vacuous.

Panel of Figure 27 — Figure 27: Portland, against $r_{\mathrm{train}}$ . (a) Certificate width (log scale): the general certificate is vacuous from $r_{\mathrm{train}}{=}0.5$ onward, while the directional stays informative throughout. (b) Observational transport distance with fold-level error bars: it increases monotonically, so the observational selector returns the smallest $\varepsilon$ , at which coverage is zero.

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