Causal graph fusion across incomplete cohorts and causal steering of tabular foundation models

Yanke Li; Dhiyaa Al Jorf; Ayoube Ighissou; Anke Scheel-Sailer; Robert Riener; Diego Paez-Granados

cs.LGNov 6, 2026

Causal graph fusion across incomplete cohorts and causal steering of tabular foundation models

Authors: Yanke Li, Dhiyaa Al Jorf, Ayoube Ighissou, Anke Scheel-Sailer, Robert Riener, Diego Paez-Granados

Organizations: Spinal Cord Artificial Intelligence (SCAI) Lab, ETH Zürich, Zurich, Switzerland · Digital Healthcare and Rehabilitation, Swiss Paraplegic Research, Nottwil, Switzerland · Department of Physics, Universit`a degli Studi di Milano-Bicocca, Milan, Italy · Swiss Paraplegic Center, Nottwil, Switzerland ·

Abstract

Clinical data are split across incomplete cohorts with partially overlapping variables. Causal discovery under missing data and multi-source integration have developed separately, and tabular foundation models cannot take a causal graph without retraining. We make two contributions. First, causal graph fusion learns a partial ancestral graph on the shared variables, fuses it into each cohort's graph under temporal background knowledge and lets Markov blankets guide re-imputation, so missing data are handled inside discovery. Second, causal attention steering compiles the learned graph into the attention of a frozen tabular foundation model at inference. Against known ground truth, fusion dominates pooled and per-cohort baselines in every missingness and heterogeneity condition tested (p<0.001), and steering improves discrimination under distribution shift (p<10^{-5}), a gain that a random graph does not produce. In two spinal cord injury cohorts, fusion orients age, albumin and hemoglobin as ancestors of length of stay, a set no single cohort orients in full, and yields compact, temporally ordered predictor sets that match or exceed automatic feature selection on four of five outcomes.

Figures & tables

Figure 1: Causal graph fusion: from incomplete, partially overlapping cohorts to one causal graph and its downstream uses. Individual incomplete datasets are aligned and pooled on shared variables. A pooled-shared partial ancestral graph yields cross-cohort causal evidence, which, together with temporal-tier expert knowledge, constrains per-cohort PAG learning, with Markov-blanket re-imputation iterated until convergence. The outputs (imputed per-cohort datasets and a final causal graph) feed downstream uses: model-agnostic prediction, where the target’s Markov blanket selects features for predictors such as TabPFN, XGBoost, and random forests, and hypothesis generation for intervention planning over the learned graph.

Figure 2: Fusion recovers more accurate graphs and better-preserved distributions than discovery without fusion, under every missingness mechanism. (a) Structural recovery: graph F1 of the learned structure against ground truth as a function of graph size, under MAR, MCAR, and MNAR (blue: fusion; grey: non-fusion; lines are means over sample sizes 200–1000, bands

\pm 1

standard deviation). (b) Distribution preservation: KS distance between original and imputed values (lower is better) by missingness mechanism and their average; bars are means with

\pm 1

standard deviation across graph sizes. The fusion framework achieves higher F1 and lower KS across all mechanisms.

		Missingness level
Data type	Method	None	MCAR 10%	MCAR 20%	MCAR 30%	MNAR
$H_{0}$ : Type-I error (%; target $\approx$ 5%)
Syn. linear	GCM	2	0	10	0	0
	CRT-TabPFN	80	80	80	80	50
	MV-FisherZ	0	0	0	0	0
Syn. nonlinear	GCM	10	0	0	0	30

Table 1: Rejection rates for the three CI tests across data types and missingness conditions (

n=250

B=99

\alpha=0.05

; values read from Figure S6 in Supplementary). Bold: outside target range (Type-I

>15\%

; power

\leq 70\%

Figure 3: Imputation is a prerequisite for cross-cohort causal discovery: raw-data conditional-independence tests versus the fused pooled-shared graph for length of stay (

n=516

; the union of HAPI and PPIS restricted to variables observed in both cohorts). All three sub-panels share one layout. Each variable is placed in the tier in which it is measured (admission

\rightarrow

discharge), and node positions are identical throughout, so the panels can be read against one another. The outcome (coral) is length of stay, blue circles are its recovered Markov-blanket members, and grey circles are the other shared variables. (a) Two sub-panels on raw (unimputed) data, GCM-TabPFN (left) and CRT-TabPFN (right). Native-NaN conditional-independence tests recover almost nothing. With GCM-TabPFN as the FCI oracle only two variables attach to length of stay (Age, Hemoglobin), and with CRT-TabPFN it is isolated entirely (

\lvert\mathrm{MB}\rvert=0

). Only edges incident to length of stay are drawn, and the GCM graph contains 3–6 edges in total. Per-cohort results show the same pattern across all six HAPI and PPIS cohorts. (b) After imputation and fusion, the pooled-shared analysis orients three definite directed edges into length of stay, from Age, Albumin, and Hemoglobin, under the temporal-tier constraint (an admission-time variable cannot be caused by a discharge outcome). Each cohort alone orients only part of this set (Supplementary Table S27 ). Edge meanings (directed ancestral relation under PAG assumptions, shared hidden common cause, unresolved association) follow the plain-language legend, as in Figure 10 .

Figure 4: Information-fusion trajectory: each fusion stage builds the Markov blanket, and the predictive gain holds for every model family. Rows: HAPI (top) and PPIS (bottom). Left : length-of-stay prediction quality (5-fold cross-validated

R^{2}

) at each of the four fusion stages for five model families identified in the legend. Capped bars are 95% bootstrap confidence intervals, with points dodged along the stage axis so overlapping intervals stay legible. Right : the corresponding Markov-blanket composition, one row per variable and one column per stage; a filled cell marks membership at that stage, so growth and turnover are visible. Stage 1 uses only the three pooled-shared directed-ancestor variables of LoS (Age, Albumin, Hemoglobin); Stages 2–4 progressively add per-cohort discovery, temporal-tier constraints, and the pooled-shared constraints that complete the fusion at the deployable

\alpha=0.05

. The single largest predictive gain in both cohorts comes from Stage 1

\rightarrow

Stage 2 and holds for every model family (

+0.19

+0.40

R^{2}

on HAPI;

+0.49

+0.58

on PPIS), so the structural gain is not an artifact of any single learner. The compact cross-cohort claim becomes predictive only once embedded in a richer per-cohort blanket. At Stage 4 TabPFN leads on both cohorts (

R^{2}=0.718

HAPI,

0.735

PPIS; values printed in the panels). These per-stage blankets are the trajectory’s own progression and differ from the deployed

\alpha=0.05

benchmark blanket (HAPI 21, PPIS 34). Relaxing the threshold to

\alpha=0.10

is model-dependent, and deployable attention steering adds no material gain on these tier-saturated final blankets; both are reported separately (Supplementary Tables S19 , S7 , S6 ).

Figure 5: PAG-guided attention steering of TabPFN (schematic; illustrative feature set). (1) An input table is processed in context by (2) a frozen, prior-fitted TabPFN network, from which the self-attention logits $S_{\ell}$ are read at an intermediate layer. (3) The learned partial ancestral graph is compiled into a mask $M$ . Reachability in its ancestral map marks attention entries to boost (between ancestrally related features) or suppress (across pairs with no causal path), and the directed, bidirected and circle edge marks map to the corresponding rule. The mask is added to the pre-softmax attention logits, $\tilde{A}_{\ell}=\mathrm{softmax}(S_{\ell}+\gamma M)$ , and (4) the steered forward pass re-weights which feature tokens the prediction reads from. The six-feature example (Age, BMI, Glucose, Insulin, blood pressure, pedigree) illustrates the mechanism generically and is not a spinal-cord-injury result. The quantitative clinical effects of steering are reported in Figures 7 – 9 and Supplementary Tables S7 and S6 . Any pretrained model that exposes self-attention over feature and target tokens can be steered the same way; TabPFN v2 is the instance we evaluate here.

Figure 6: Attention steering as a deployable, shift-matched robustness tool. (a) Deployable per-fold (

\gamma

-selected)

y\!\to\!x

steering significantly improves interventional accuracy on synthetic ground truth, where a distribution shift is present (144 structural causal models), but stays within noise on the observational clinical cohorts, where no explicit shift is available against which to select

\gamma

. The measured shift is covariate-only (domain-classifier AUROC

\approx 0.68

with stable

P(y\mid x)

). (b) On synthetic ground truth, the better causal target depends on the shift type, with causal ancestors winning under concept (mechanism) shift and the Markov blanket under covariate shift. (c) On PPIS length of stay the benefit is causally specific. Steering the full feature pool toward the causal Markov blanket helps, whereas equally-sized mutual-information, random-forest, or random subsets do not, and ancestor-steering is harmful (the covariate-shift regime, where the Markov blanket is the correct target). Panels correspond to the analyses in Supplementary Tables S8 and S7 .

Figure 7: HAPI length of stay (

n=250

): the 21-variable constrained graph set scores higher than every automatic feature selector and is the simplest model within one standard error of the best. Higher

R^{2}

is better. (a) Best out-of-fold performance achievable with each feature-selection strategy, our constrained FCI Markov blanket against the automatic baselines. Each row names the feature count

n

and the model that achieved it, since a best-of-

N

sweep is run per strategy and the winner is not always the same learner. Dots are point estimates, whiskers 95% bootstrap confidence intervals (300 resamples); the causal Markov blanket is shown in bold. (b) Marginal Shapley importance (mean

|\mathrm{SV}|

under mean-masking) for the constrained FCI Markov blanket, without steering and with Target-Boost (

\gamma=2.0

); temporal causal ancestors of the target are in blue, and only the twelve strongest features are shown, with the full membership in Supplementary Table S11 . (c) Out-of-fold

R^{2}

against number of features for every evaluated method (blue: our Markov-blanket / steered variants; grey: baselines). The ring marks the simplest-optimal model, the fewest-feature method whose performance is within one standard error of the best (one-SE rule), matching the “simplest within 1 SE” label printed in the panel. The SE is recovered from the 95% bootstrap interval as

(\mathrm{hi}-\mathrm{lo})/(2\times 1.96)

. Panels (a)–(c) carry the same meaning in Figure 8 . All metrics are held-out cross-validated with the feature sets fixed; patient-grouped estimates, which reflect unseen patients, are in Supplementary Table S13 .

Figure 8: PPIS length of stay (

n=266

): the constrained graph set matches the best automatic feature selectors. Higher

R^{2}

is better; panels (a)–(c) as defined in Figure 7 . The 34-variable graph set reaches

R^{2}=0.746

(XGBoost;

0.710

with TabPFN), level with mutual-information and random-forest selection (

0.749

0.748

) and within

0.033

of all 87 variables (

0.779

). Because the blanket includes during-stay variables, this is an explanatory model of the realized stay; models restricted to admission or pre-surgery information are in Supplementary Table S12 . Every member is already a temporal ancestor of length of stay, so steering within the set leaves it unchanged. Steering the full feature pool toward the Markov blanket helps instead (Figure 6 ).

Figure 9: PPIS post-surgical outcomes: the constrained graph set is the best feature set for secondary PI complication and ties the best for urinary tract infection. One row per outcome: urinary tract infection during stay (

n_{+}=77/266

), osteomyelitis (

n_{+}=114/266

), and secondary PI complication (

n_{+}=62/266

). (a) Out-of-fold ROC curve for each feature-selection strategy, with the constrained graph set highlighted; the legend ranks the strategies and gives AUROC with its 95% bootstrap confidence interval, and the dotted diagonal is chance. All curves share one model (TabPFN v2), so the panel isolates the feature set from model choice. The data-driven selectors are re-fitted within each training fold, so their feature counts are per-fold medians. (b) Mean

|\mathrm{SV}|

Shapley importance for that model on the constrained graph set, without steering (grey bars) and with Target-Boost (blue bars); feature names printed in blue are temporal (tier-earlier) causal ancestors of the target. Only the eight strongest features are shown; the blankets hold 50 features for UTI and secondary PI complication and six for osteomyelitis, with the full membership in Supplementary Table S11 . Osteomyelitis is the one outcome where automatic selection leads, its dominant predictors being non-ancestors of the target.

Figure 10: Temporally-organized causal graph for secondary PI complication (PPIS; depth-2 neighborhood of the outcome,

\lvert\mathcal{N}\rvert=10

of a 50-member Markov blanket). The learned graph reads left to right along the clinical timeline. Each circle is a clinical variable, placed in the column for when it is measured (before/at admission

\rightarrow

beginning of stay

\rightarrow

during stay

\rightarrow

end of stay). The blue circles are the outcome’s depth-2 temporally-available causal neighborhood (Methods). The prediction model itself conditions on the full 50-feature Markov blanket of this outcome (its total-conditioning, temporally-restricted m-separating set in the constrained

\alpha=0.05

fused PAG). To keep the graph legible we draw only the 10 blanket members lying within two steps of the outcome, so every node shown is one of the 50 blanket features; the complete 50-member list is in Supplementary Table S11 . The coral circle is the outcome. Edges follow partial-ancestral-graph semantics. A solid dark arrow

A\rightarrow B

denotes a directed (ancestral) relation under FCI/PAG assumptions, with

A

a possibly indirect cause of

B

. A purple double-headed arrow

A\leftrightarrow B

denotes latent confounding by a shared hidden common cause, and a light grey line denotes an association whose direction the data could not resolve. Edges incident to the outcome are drawn boldly; predictor–predictor links are faded to background context. This figure is the reading key for the per-target causal-neighborhood graphs of the other outcomes (Supplementary Figures S15 – S18 ).

Dataset	Clinical setting	Cohort composition	$n$	Representative variable domains	Main analytic role
HAPI	Hospital-acquired pressure injury in inpatient/rehabilitation care	Three related cohorts, including HAPI and non-HAPI patients, with partial variable overlap	250	Demographics, neurological severity, complications, laboratory variables, functional status, nutrition, and pressure-injury-related indicators	Modeling upstream pressure-injury risk and acquisition mechanisms across rehabilitation cohorts.
PPIS	Post-pressure-injury surgery	Single post-surgical cohort with broader longitudinal clinical detail	266	Wound characteristics, demographics, SCI descriptors, perioperative or treatment variables, laboratory variables, comorbidities, surgeries, functional measures, and outcomes	Modeling downstream severity, treatment, complications, and recovery after established pressure injury.

Table 2: Overview of the clinical datasets used in the fusion analysis.

Appendix figures & tables51 assets

Supplementary material from the paper’s appendix.

Appendix

Layer	CLF	REG
0	0.524	0.486
1	0.412	0.185 †
2–3	0.44–0.45	0.47–0.60
4	0.645	0.906
5	0.894	0.898
8	0.933	0.962

Appendix

Table S1: Head-averaged AUROC for

Y

-token parent detection by layer (parent vs. non-parent, 300 linear-Gaussian SCMs). Bold : values

\geq 0.89

(strong parent detection); the shaded late layers lie in the high-detection band used for PAG-guided attention steering (L9–11 for the classification models; the regression models steer all layers). The phase transition at layer 4 is visible in both CLF and REG modes; the below-chance dip at layer 1 for REG reflects a transient anti-causal intermediate representation.

Figure S1: Layer-by-layer AUROC profiles for linear-Gaussian (solid) and nonlinear tanh (dashed) SCMs. Six panels show Y-token head-averaged AUROC (CLF and REG), Y-token multi-head probe AUROC (CLF and REG), and X-to-X multi-head probe AUROC (CLF and REG). The phase transition at layers 4–5 and the flat X-to-X probe are mechanism-independent, with linear and nonlinear SCMs producing the same qualitative structure.

Figure S2: Multi-head probe AUROC (solid) versus per-head average AUROC (dashed) by layer, for the classification (CLF, left) and regression (REG, right) tasks; within each panel the

Y

-token probe is dark blue and the

X

-to-

X

probe light blue. Linear SCMs. The gap between the lines is the signature of distributed causal encoding, in which the joint 6-head vector detects causal edges that no single head detects reliably. Identical patterns hold for nonlinear SCMs (Supplementary Figure).

Panel of Figure 16 — Figure S3: Left: mean $Y$ -token attention for three node categories (non-ancestor, indirect ancestor, direct parent) across all 12 layers (linear SCMs). The model progressively separates direct parents from all other nodes; by layer 8 indirect ancestors are indistinguishable from non-ancestors. Right: $Y$ -token attention versus signed direct edge weight $w_{g\to Y}$ for direct parents at late layers. The V-shaped scatter confirms sign invariance. The model encodes $|w|$ (causal magnitude) but not the sign of the causal effect.

$Y$	Hidden $C$	INT	AUROC obs	AUROC int	Drop
P38	PKA (direct parent)	b2cAMP	0.689	0.459	$-$ 0.230
Raf	PKC (direct parent)	b2cAMP	0.748	0.571	$-$ 0.177
PIP3	Plcg (only parent)	b2cAMP	0.602	0.504	$-$ 0.098

Appendix

Table S2: Sachs baseline AUROC under observational vs. interventional distribution. Hidden confounders excluded from features; training on observational data only.

Scenario	PAM type	$\gamma^{*}$	$\Delta$ int	Int (steered)	$\Delta$ obs	Obs
P38/PKA/b2cAMP	boost, ancestor	$+20$	$+0.092$	0.551	$+0.002$	0.691
Raf/PKC/b2cAMP	full-sym, ancestor	$+5$	$+0.023$	0.594	$-0.001$	0.747
Erk/PKC/PMA	ancestor	$-10$	$+0.018$	0.460	$+0.005$	0.703
PIP3/Plcg (ctrl)	— (PAM=0)	—	$0.000$	0.504	$0.000$	0.602

Appendix

Table S3: Best boost-steering result per Sachs scenario (oracle

\gamma

, layers L9–11). Delta is AUROC int (steered) minus AUROC int (baseline). Obs delta shows cost on the training distribution. Bold : best result. Shaded row: negative control (PAM=0, expected zero gain). Beyond the three baseline scenarios (Table S2 ), Erk/PKC/PMA is reported here as a fourth, shifted-ancestor case.

Figure S4: Left: baseline vs. steered AUROC for the P38/PKA/b2cAMP scenario (observational and interventional). Right: AUROC vs.

\gamma

sweep for boost and FF. Boost peaks at

\gamma=+20

(AUROC int

0.551

); FF saturates quickly and underperforms boost on this single-ancestor problem.

Task	Condition	$\Delta$ int (mean)	$\Delta$ obs (mean)	Win rate
CLF (AUROC)	Boost ( $y\!\to\!x$ ), L9–11	$+0.010$	$\approx 0.000$	57%
CLF (AUROC)	FF ( $x\!\to\!x$ ), L9–11	$+0.047$	$-0.013$	91%
REG ( $R^{2}$ )	Boost (y-to-x), L9–11	$+0.098$	$\approx 0.000$	47%
REG ( $R^{2}$ )	FF (x-to-x), L9–11	$+0.675$	$-0.006$	71%

Appendix

Table S4: Synthetic diversity steering under an oracle

\gamma

(selected per repetition on the interventional test set, which gives a non-deployable upper bound): mean interventional metric improvement and win rate across all mechanisms and topologies. Under this test-set-tuned oracle the aggressive feature–feature (FF) mode scores highest, but the advantage is an artefact of the oracle. At a mechanistically fair fixed strength FF is the least effective mode (Table S9 ), and on the clinical cohorts FF never helps and is harmful at strength (Table S7 ). The deployable recommendation is therefore the direct

y\!\to\!x

(boost) path; both modes are reported here for completeness.

Model	Layers	Base	FF	$\Delta$ AUROC (mean)	Win rate
TabPFN v2	L9–11 (symmetric)	$0.718$	$0.711$	$-0.007$	42.5%
TabPFN v2.5	L18–23 (boost)	$0.725$	$0.727$	$+0.002$	45.0%

Appendix

Table S5: Cross-model steering under a deployable fixed-

\gamma

protocol. FF steering (

\gamma=2.0

, no per-repetition tuning) versus the unsteered baseline on interventional AUROC, across the same synthetic grid, for two independently pretrained foundation models. Both means are within noise of zero (per-model

\Delta

standard deviation

\approx 0.025

over 40 repetitions); per-repetition deltas are positively correlated across the two models (

r=0.35

), indicating the operator engages the same causal structure in both.

Target	Metric	Baseline	Fixed $\gamma{=}2.0$ ( $\Delta$ )	Nested $\gamma$ ( $\Delta$ )
PPIS length of stay	$R^{2}$	$0.656$	$0.685$ ( $+0.029$ )	$0.665$ ( $+0.009$ )
HAPI length of stay	$R^{2}$	$0.761$	$0.741$ ( $-0.020$ )	$0.757$ ( $-0.004$ )
UTI during stay	AUROC	$0.894$	$0.894$ ( $+0.000$ )	$0.900$ ( $+0.006$ )
Osteomyelitis	AUROC	$0.902$	$0.900$ ( $-0.002$ )	$0.901$ ( $-0.001$ )
Secondary PI complication	AUROC	$0.752$	$0.742$ ( $-0.010$ )	$0.746$ ( $-0.006$ )

Appendix

Table S6: Leak-free nested per-fold

\gamma

selection for

y\!\to\!x

MB-Boost steering (the effective mode), applied to TabPFN on the full leakage-safe feature pool and steered toward the Markov blanket, versus a fixed

\gamma=2.0

and the unsteered baseline. Metric is

R^{2}

(length of stay; kernel-matched log-target base) or AUROC (binary outcomes; baselines are the full-pool model, e.g. osteomyelitis

0.902

). For each outer fold,

\gamma

is chosen by an inner 3-fold cross-validation on the training data and applied to the held-out fold; no test-fold information is used. Nested selection is conservative and stays within noise of the baseline on every target, damping both the one fixed-strength gain (PPIS length of stay,

+0.029\to+0.009

) and the small fixed-strength losses elsewhere. The PPIS length-of-stay row uses patient-grouped outer folds with a kernel-matched

\gamma=0

baseline (

R^{2}=0.656

); the HAPI length-of-stay row and the three binary rows are episode-level. Under the episode-level protocol of Figure 8 the same MB-Boost configuration gives

-0.014

on PPIS length of stay.

		MB-Boost (steer $\to$ MB)			Target-Boost (steer $\to$ ancestors)			FF
Target	Metric	$y\!\to\!x$	$x\!\to\!y$	$y{+}x$	$y\!\to\!x$	$x\!\to\!y$	$y{+}x$	$x\!\to\!x$
UTI during stay	AUROC	$+0.000$	$+0.001$	$-0.000$	$+0.001$	$+0.001$	$+0.002$	$-0.012$
Osteomyelitis	AUROC	$-0.002$	$+0.000$	$-0.002$	$-0.000$	$+0.000$	$+0.000$	$-0.006$
Secondary PI complication	AUROC	$-0.010$	$-0.007$	$-0.021^{*}$	—	—	—	$-0.019$
HAPI length of stay	$R^{2}$	$-0.020^{*}$	$-0.015$	$-0.032^{*}$	$-0.012^{*}$	$-0.019^{*}$	$-0.041^{*}$	$-0.091^{*}$
PPIS length of stay	$R^{2}$	$\mathbf{+0.029^{*}}$	$+0.017$	$+0.029$	$-0.017$	$-0.005$	$-0.005$	$-0.237^{*}$

Appendix

Table S7: Complete clinical steering grid: deployable (

\gamma=2.0

) metric delta versus the kernel-matched (

\gamma{=}0

) baseline for every steering-target

\times

mode combination, plus feature–feature (FF) steering. Metric is

R^{2}

(length of stay) or AUROC (binary outcomes); paired bootstrap of

2{,}000

resamples; ∗ marks a 95% CI that excludes zero. Bold : the only significant positive cell in the grid. FF and the combined mode degrade further at higher strength (see text); a single positive gain (

y\!\to\!x

MB-Boost on PPIS length of stay) survives to

+0.044

\gamma{=}5

. “—”: secondary PI complication has no directed PAG ancestors in the leakage-safe pool, so ancestor-steering is undefined there.

Validation regime	mean $\Delta$ AUROC	win rate	paired $p$
Matched shift (upper bound)	$+0.029$	69%	$1.5\times 10^{-12}$
Decoupled shift (deployable)	$\mathbf{+0.011}$	44%	$\mathbf{1.4\times 10^{-6}}$

Appendix

Table S8: Deployable steering on synthetic ground truth (144 SCMs; interventional test). Per-fold

\gamma

selection with the effective

y\!\to\!x

(target-boost) operator on a structure-aware shifted validation set without test labels, versus the unsteered baseline, under two validation regimes. The validation shift either matches the deployment shift (optimistic upper bound) or is decoupled from it (deployable case). Paired Wilcoxon across SCMs.

Figure S5: Interventional (left columns) and observational (right columns) metric deltas as a function of the fraction of causal structure revealed to the steerer. Rows: classifier AUROC, classifier AUPRC, regressor

R^{2}

, regressor MSE. Boost (near-black, dashed) scales approximately linearly with revealed structure, reaching

+0.15

R^{2}

delta at 100% for regression. FF (blue, solid) saturates at 25% reveal, recovering most of its gain from a small fraction of the graph. Both methods show a small but growing observational cost as more structure is revealed (right column).

Mode	v2 $\gamma{=}2$	v2 $\gamma{=}20$	2.5 $\gamma{=}2$	2.5 $\gamma{=}20$
$y\!\to\!x$ (target-boost)	$\mathbf{+0.021}$	$\mathbf{+0.032}$	$+0.003$	$-0.006$
$x\!\to\!y$ (transpose)	$+0.004$	$+0.032$	$+0.005$	$+0.007$
$y\!\to\!x\,{+}\,x\!\to\!y$ (combined)	$+0.030$	$+0.080$	$+0.003$	$-0.025$
$x\!\to\!x$ (feature–feature)	$-0.008$	$+0.014$	$-0.001$	$-0.014$

Appendix

Table S9: Synthetic interventional

\Delta

AUROC by steering mode, TabPFN v2 vs. 2.5 (kernel-matched baseline, 72 SCM repetitions).

y\!\to\!x

= target-boost (the deployable mode);

x\!\to\!y

= transpose;

y\!\to\!x\,{+}\,x\!\to\!y

= combined;

x\!\to\!x

= feature–feature. The effect is present on v2 and absent on 2.5.

Layer	$Y$ -tok CLF	$Y$ -tok REG	X-to-X CLF	X-to-X REG
0	0.620	0.608	0.683	0.654
3	—	—	0.676	0.663
6	—	—	0.640	0.642
11	0.889	0.973	0.602	0.645

Appendix

Table S10: Multi-head logistic regression probe AUROC (5-fold CV) for

Y

-token parent detection and X-to-X causal edge detection at selected layers. The

Y

-token probe rises monotonically (layer 0 → 11); the X-to-X probe is flat, confirming causal structure is present from the first layer but does not concentrate in later layers.

Figure S6: Rejection rates of native-NaN conditional-independence tests under missingness (

n=250

B=99

). Top row (H0, calibration): Type-I error rate, target

\approx 5\%

(dashed line). Bottom row (H1, power): target

>70\%

(dotted line). Columns show synthetic linear, synthetic nonlinear, and pooled real Bayesian-network data for GCM-TabPFN, CRT-TabPFN, and MV-FisherZ. On real discrete/BN data all three tests have low power (

\leq 30\%

), while CRT-TabPFN and MV-FisherZ additionally inflate Type-I error on nonlinear nulls. This is the primary motivation for imputation before causal discovery; the summary values in Table 1 are read from this figure.

Figure S7: Imputation accuracy across missingness mechanisms. Root-mean-squared error of the imputed against the hidden values for the fusion framework (blue) versus the non-fusion baseline (grey), by missingness mechanism and their average; lower is better. Bars are means over the synthetic grid with

\pm 1

standard deviation whiskers. Fusion yields lower imputation error under every mechanism.

Figure S8: Synthetic cross-cohort structure recovery. Recovery of the shared (invariant) skeleton over the overlapping variables, versus per-variable missing fraction ( A , C ; heterogeneity fixed at

k=3

) and versus cross-cohort heterogeneity

k

( B , D ; missingness fixed at

0.3

). Top row F1, bottom row precision; bands are

\pm 1

standard error over 50 random DAGs. Fusion

\cap

(both cohorts) has the highest precision throughout; fusion

\cup

(either cohort) has the best F1 of any method, significantly beating both the per-cohort average and the fair pool-on-shared baseline in every missingness and heterogeneity condition (paired Wilcoxon

p<10^{-3}

); the naive union-pool is dominated on both metrics; and listwise deletion (no imputation) collapses at high missingness.

Figure S9: Robustness of the fusion precision advantage at the heterogeneous reference setting (

k=3

, missingness

0.3

). Precision ( E ) and F1 ( F ) for both fusion operating points versus the fair pool-on-shared and naive union-pool baselines under MCAR, MAR and MNAR missingness, Laplace noise, polynomial and RBF functional forms, and sparse (

p=0.08

) and dense (

p=0.20

) graphs. Bars are means with

\pm 1

standard-error whiskers over 50 random DAGs. Fusion

\cap

retains the highest precision under every variant, with the margin narrowing only for the RBF-nonlinear data that violate the Fisher-

z

test’s assumptions.

Figure S10: Marginal Shapley feature importance for TabPFN (baseline, Target-Boost, FF at fixed

\gamma=2.0

) on HAPI length-of-stay across all feature sets. Blue bars: temporal ancestors of the target; grey: non-ancestors. All HAPI LoS features are temporal ancestors, so steering produces no ancestor/non-ancestor ratio change (tier saturation confirmed).

Figure S11: Marginal Shapley feature importance for PPIS length-of-stay across all feature sets. Every PPIS length-of-stay feature set is tier-saturated, with all selected features temporal ancestors of the target (RF top-10: 10/10 ancestors; FCI MB: 34/34). The ancestor/non-ancestor ratio is thus undefined and steering produces no ratio change, confirming the softmax-invariance (tier-saturation) argument. FF steering likewise yields no leak-free

R^{2}

gain on the RF top-10 set (fixed

\gamma=2.0

-0.010

versus the unsteered model).

Figure S12: Marginal Shapley feature importance for PPIS UTI during stay across all feature sets. On the constrained FCI Markov blanket panel (50 features, 41 ancestors), the ancestor/non-ancestor attribution ratio is

2.40

at baseline. Target-Boost raises it to

2.84

(

+18\%

), more than the feature–feature control (FF:

2.72

+13\%

), so the effective

y\!\to\!x

mode concentrates attribution most on the causal ancestors.

Figure S13: Marginal Shapley feature importance for PPIS osteomyelitis across all feature sets. On the constrained FCI Markov blanket panel (6 features, 5 ancestors), the two modes move the ancestor/non-ancestor attribution ratio in opposite directions. Target-Boost raises it from

2.73

4.24

(

+55\%

), whereas the feature–feature control lowers it (FF:

2.08

-24\%

), so the effective

y\!\to\!x

mode concentrates attribution on the causal ancestors.

Figure S14: Marginal Shapley feature importance for PPIS post-surgical complication across all feature sets. On the non-deployed RF top-10 set, the feature–feature (FF) control produces the largest attribution shift (ancestor/non-ancestor ratio

0.21\rightarrow 0.45

+114\%

), yet this redistribution does not translate into a predictive gain, and the leak-free steering effect is within noise. On the deployed constrained graph set, Target-Boost, the effective mode, most concentrates attribution on the ancestors (

0.74\rightarrow 1.01

+37\%

, versus FF

+6\%

). Duration of Immobilization, a non-ancestor, dominates the baseline.

Figure S15: Temporally-organized causal graph for HAPI length of stay (full Markov blanket,

\lvert\mathrm{MB}\rvert=21

\alpha=0.05

). Reading conventions as in Figure 10 .

Figure S16: Temporally-organized causal graph for PPIS length of stay (full Markov blanket,

\lvert\mathrm{MB}\rvert=34

\alpha=0.05

). For this large blanket only the edges incident to the outcome are drawn; predictor–predictor associations are omitted for legibility.

Figure S17: Temporally-organized causal graph for PPIS urinary tract infection during stay (depth-2 neighborhood of the outcome,

\lvert\mathcal{N}\rvert=6

of a 50-member Markov blanket, shown for legibility;

\alpha=0.05

). Reading conventions as in Figure 10 .

Figure S18: Temporally-organized causal graph for PPIS osteomyelitis (full Markov blanket,

\lvert\mathrm{MB}\rvert=6

\alpha=0.05

; the compact deployed blanket). Reading conventions as in Figure 10 .

Target (cohort, $\alpha$ , $\lvert\mathrm{MB}\rvert$ )	Temporally-available Markov-blanket members
HAPI LoS
( $\alpha=0.05$ , 21)	AIS A; Age at SCI; Albumin; First rehabilitation; Hemoglobin; Hypotensive episode; PI history number; PI history treatment; PI_onset; SCI to Admission time; SCIM; Sedative medications; Surgeries; current PI time since 1st last PI; current PI time since 2nd last PI; current PI time since 3rd last PI; current PI time since 5th last PI; history PI location 2; history PI location 3; history PI location 5; history PI location 6
PPIS LoS
( $\alpha=0.05$ , 34)	AB before Flap Surgery; AB_1; AB_2; AIS; Administration of AB; Age at SCI; Albumin; Antibiotics; Bacteria_2; CRP; Complication; Congestive_Heart_Failure; DM; Duration of Antibiotics; Duration of surgery (Min); Education; Ferritin; Hemoglobin; Hypoxemia_Episode; ICU Stay; INR; Localization of PI; Mechanical Ventilation; Number of AB; Number of Bacteria; Number of PI (sakral, ischial, trochanteric); PI Stage; Psychiatric Diagnosis; Rheumatoid_Arthritis; Time between Surgeries; Total Protein; Type of Surgery; Vit B12; deep PI at other location
PPIS UTI
( $\alpha=0.05$ , 50)	AB before Flap Surgery; AB for other Acute Condition; Acidum Folicum; Aetiology; Age at Admission; Age at SCI; AIS; Albumin; Alcohol Consumption; Amputation; Bloodpressure_Systolic_Entry; Body_Temperature_Abnormal; BP_Episode_Abnormal; Calcium; Complication; Congestive_Heart_Failure; CRP; deep PI at other location; DM; Duration of Immobilization; Duration of surgery (Min); Education; Employment; Gastroenteritis_During_Stay; Hemoglobin; Hypertony; ISNCSCI; Level of injury; Localization of PI; Mechanical Ventilation; Number of PI (sakral, ischial, trochanteric); OSAS; Osteoarthritis; PI at admission; PI Stage; Pneumonia; Previous OM; Psychiatric Diagnosis; Recurrence; Rheumatoid_Arthritis; Sex; Smoking; Social Status; Spasticity; Total Protein; Type of Residence; Type of Surgery; Vascular comorbidity; Vit B12; Years post Injury

Appendix

Table S11: Full temporally-available causal Markov blanket at

\alpha=0.05

(total-conditioning m-separation, confounders retained; restricted to variables measured no later than the target). For every target this is the complete deployed Markov blanket used for prediction. For legibility, the per-target graphs (Figures 10 and S15 – S18 ) draw only the depth-2 causal neighborhood of the outcome, a subset of these members. The reduction is largest for UTI and secondary PI complication, whose blankets have

50

features each. The header column lists (cohort,

\alpha

\lvert\mathrm{MB}\rvert

), the full blanket size.

Cohort	Decision time	$n$ feat.	Gradient-boosted $R^{2}$	TabPFN $R^{2}$
HAPI	Admission-only	11	0.646	0.690
HAPI	Pre-surgery	13	0.663	0.719
HAPI	Retrospective (all)	21	0.684	0.745
PPIS	Admission-only	12	0.312	0.341
PPIS	Pre-surgery	21	0.362	0.310
PPIS	Retrospective (all)	34	0.746	0.710

Appendix

Table S12: Decision-time-stratified length-of-stay prediction (

R^{2}

, 5-fold cross-validation). Each length-of-stay feature set is split by temporal tier into admission-only, pre-surgery (adds beginning-of-stay laboratory values), and retrospective (the full Markov blanket, adds during-stay variables). On PPIS the strong retrospective result collapses to

R^{2}\approx 0.31

–

0.34

with TabPFN (

0.31

–

0.36

across learners) once restricted to information available at or before surgery, whereas HAPI length of stay is already well predicted at admission. The retrospective PPIS model is explanatory rather than prospective.

Target	Metric	Episode-level	Patient-grouped
PPIS length of stay	$R^{2}$	0.710	0.625
UTI during stay	AUROC	0.894	0.778
Osteomyelitis	AUROC	0.848	0.807
Secondary PI complication	AUROC	0.776	0.783

Appendix

Table S13: Patient-grouped versus episode-level cross-validation on PPIS. The 266 episodes come from 174 patients. Patient-grouped CV (GroupKFold / StratifiedGroupKFold on patient ID) prevents any patient from appearing in both training and validation folds and is the primary PPIS validation here. Metrics are for the TabPFN predictor on each target’s deployed causal Markov blanket (no steering). Grouping lowers performance by

0.04

–

0.12

for length of stay, UTI and osteomyelitis, so the episode-level column is an optimistic upper bound. Secondary PI complication is essentially unchanged.

Target	Metric	Causal MB	RF top-10	MI top-20	Lasso CV	All
PPIS length of stay	$R^{2}$	0.674	0.640	0.628	0.622	0.656
HAPI length of stay	$R^{2}$	0.706	0.718	0.722	0.749	0.761
UTI during stay	AUROC	0.765	0.742	0.761	0.785	0.743
Osteomyelitis	AUROC	0.783	0.818	0.734	0.789	0.804
Secondary PI complication	AUROC	0.690	0.739	0.760	0.654	0.742

Appendix

Table S14: Generalizability of the discovery procedure. A central claim of the method is that the causal-fusion procedure recovers predictive feature sets from new data, not merely that a fixed model predicts held-out patients (the deployment question, Table S13 ). We test this with a fully nested cross-validation. Inside each outer fold, every method’s feature set is re-derived from the training data only (per-cohort FCI at

\alpha=0.05

for the constrained FCI Markov blanket), a single fixed TabPFN is fit without best-of-

N

model selection, and the held-out fold is predicted. PPIS uses patient-grouped folds. Nesting removes the full-data selection advantage that every method enjoys in the episode-level benchmark, so all numbers fall relative to the main-text figures. Under nesting, the re-discovered constrained graph set ranks first on PPIS length of stay (

R^{2}=0.674

, ahead of every automatic set) and second on UTI (

0.765

AUROC, behind Lasso CV at

0.785

), evidence that the fusion procedure recovers predictive causal structure from new data without overfitting the development cohort. It trails on osteomyelitis, secondary PI complication, and HAPI length of stay, consistent with the temporal-availability restriction excluding the during-stay infection and inflammatory markers that the automatic baselines are free to exploit. Bold: best (leak-corrected) per row. Metrics:

R^{2}

(length of stay), AUROC (binary); non-steered TabPFN throughout.

Target	Prev.	AUROC (base $\to$ Boost)	AUPRC	Brier (base $\to$ Boost)	Calib. slope
UTI	0.29	$0.894\to 0.891$	0.83	$0.106\to 0.110$	0.79
Osteomyelitis	0.43	$0.848\to 0.844$	0.81	$0.154\to 0.156$	0.97
Secondary PI complication	0.23	$0.776\to 0.767$	0.52	$0.144\to 0.146$	0.78

Appendix

Table S15: Calibration, average precision, and paired steering effect for the three PPIS binary outcomes (TabPFN on the deployed causal Markov blanket; base and Target-Boost-steered

y\!\to\!x

, fixed

\gamma=2.0

on independent data). Brier score, AUPRC and calibration slope are for the base predictor (slope

1

= perfect; AUPRC compares to prevalence).

\Delta

AUROC is the paired bootstrap difference (Target-Boost

-

base) with a 95% CI, computed leak-free (deployable fixed

\gamma

, no per-fold selection). On this cohort (

n=266

) the deployable steering effect on all three outcomes is small and not statistically distinguishable from zero. Every

\Delta

AUROC CI includes

0

, and the inert feature–feature (FF) control behaves similarly (Table S7 ). Steering’s causal specificity is therefore established on the synthetic benchmark (Table S8 ) and the random-graph/shuffled-label controls (Table S16 ), not on these clinical AUROC deltas.

Figure S19: Net-benefit decision curves of base and Target-Boost-steered (

y\!\to\!x

\gamma=2.0

) TabPFN on the deployed causal Markov blanket for the three PPIS binary outcomes. Net benefit is plotted against the threshold probability

p_{t}

; the dashed “treat all” line and the “treat none” baseline at

0

are the two default strategies, and the vertical dash-dotted line marks the outcome prevalence. The Target-Boost-steered predictor lies above both defaults across the full range for UTI and osteomyelitis and for

p_{t}\geq 0.15

for secondary PI complication, and meets or exceeds the base predictor throughout.

Target	Real ancestors $\Delta$ AUROC	Random-set $\Delta$ AUROC	Shuffled-label AUROC
UTI	$-0.003$	$+0.001$	0.49
Osteomyelitis	$-0.004$	$-0.003$	0.46
Secondary PI complication	$-0.010$	$-0.003$	0.45

Appendix

Table S16: Steering negative controls (PPIS binary outcomes; deployable Target-Boost,

y\!\to\!x

, fixed

\gamma=2.0

). Paired

\Delta

AUROC (steered

-

base) from steering toward the real graph ancestors versus a random same-size ancestor set, averaged over three random sets. Shuffled-label sanity check: base AUROC on permuted labels, expected

\approx 0.5

. On this cohort the deployable gains are small for both the real ancestors and the random sets and lie within the paired-bootstrap noise (Table S15 ); the real ancestor set does not exceed the random-set mean on any outcome. The clinical negative control is thus underpowered at this sample size, and causal specificity is established primarily on the synthetic benchmark with known ground truth (Table S8 ) and the PPIS length-of-stay specificity analysis (Table S7 ).

Target	Metric	Median	MICE	missForest
HAPI length of stay	$R^{2}$	0.745	0.650	0.646
PPIS length of stay	$R^{2}$	0.710	0.721	0.725
UTI during stay	AUROC	0.896	0.906	0.904
Osteomyelitis	AUROC	0.848	0.849	0.838
Secondary PI complication	AUROC	0.775	0.760	0.781

Appendix

Table S17: Imputation-method sensitivity. Primary metric (TabPFN on each target’s deployed causal Markov blanket; 5-fold CV) under the pipeline-default median/mode imputation, MICE (IterativeImputer with Bayesian ridge), and a missForest-like imputer (IterativeImputer with extremely randomized trees). Median-column values are this sensitivity run’s default-imputer result and match the main benchmark to within

0.002

. Most binary-outcome metrics are robust to the imputation method, and MICE/missForest even improve UTI slightly. HAPI length of stay is sensitive to the imputer (median

R^{2}=0.745

versus

\approx 0.65

under MICE/missForest), a non-negligible difference that is a caveat on the length-of-stay results. This table reports prediction-time imputation; graph-level imputation sensitivity (effect on the recovered edges and Markov-blanket sizes) is reported in Supplementary Table S21 .

		$\alpha=0.05$ (primary)		$\alpha=0.10$
Target	Metric	$\lvert\mathrm{MB}\rvert$	Score	$\lvert\mathrm{MB}\rvert$	Score
UTI during stay	AUROC	50	0.894	65	0.890
Osteomyelitis	AUROC	6	0.848	6	0.848
Secondary PI complication	AUROC	50	0.776	65	0.785
PPIS length of stay	$R^{2}$	34	0.710	79	0.783
HAPI length of stay	$R^{2}$	21	0.745	24	0.745

Appendix

Table S18: Significance-threshold (

\alpha

) sensitivity of the deployed causal Markov blanket. For each target, the leak-safe blanket size and TabPFN performance (AUROC for binary outcomes,

R^{2}

for length of stay; 5-fold CV) at

\alpha=0.05

(primary) and

\alpha=0.10

. Blankets are re-extracted through the full deployed pipeline at each threshold and remain temporally leak-free (zero tier violations). Relaxing

\alpha

enlarges every blanket except the well-separated osteomyelitis target, but only PPIS length of stay gains materially, by absorbing additional during-stay predictors. Pooled PPIS graph density:

158

edges /

28.5\%

bidirected at

\alpha=0.05

versus

205

37.1\%

\alpha=0.10

	HAPI length of stay		PPIS length of stay
Model family	$\alpha=0.05$ (S4)	$\alpha=0.10$ (S5)	$\alpha=0.05$ (S4)	$\alpha=0.10$ (S5)
Markov-blanket size	11	11	23	35
TabPFN	0.718	0.718	0.735	0.788
XGBoost	0.647	0.647	0.731	0.775
Random forest	0.676	0.676	0.726	0.753
Ridge	0.485	0.485	0.586	0.264

Appendix

Table S19: Relaxing the threshold beyond the deployable operating point: the fifth fusion stage. The information-fusion trajectory (Figure 4 ) ends at Stage 4, full fusion at the deployable

\alpha=0.05

. Here we report the additional Stage 5, which relaxes the per-cohort conditional-independence threshold to

\alpha=0.10

, for the same five model families and the same protocol (5-fold cross-validated

R^{2}

on the stage’s Markov blanket). On HAPI the relaxed threshold leaves the blanket unchanged (11 members), so every model is identical. On PPIS it admits 12 further variables (

23\rightarrow 35

), and the effect depends on the model. The flexible learners exploit them (TabPFN

0.735\rightarrow 0.788

, XGBoost

0.731\rightarrow 0.775

, random forest

0.726\rightarrow 0.753

) whereas the penalized linear models collapse (ridge

0.586\rightarrow 0.264

, lasso

0.571\rightarrow 0.272

), because the added collinearity cannot be absorbed by an

\ell_{2}

\ell_{1}

penalty at this sample size. This model-dependence, rather than any gain in the deployed pipeline, is why

\alpha=0.05

is retained as the primary operating point (Table S18 ) and why the relaxed stage is reported here, outside the main trajectory. These are the trajectory’s own per-stage blankets, distinct from the deployed leak-safe blankets of Table S18 , which grow

34\rightarrow 79

on PPIS and

21\rightarrow 24

on HAPI.

(a) $\alpha$ (LoS MB size)	HAPI	PPIS
$\alpha=0.01$	13	15
$\alpha=0.05$	21	34
$\alpha=0.10$	23	79
(b) Age harmonization (adjacency freq.)	Age–LoS	Alb–LoS	Hb–LoS
Age at SCI, both cohorts	1.00	0.37	1.00
Within-cohort $z$ -score	1.00	0.37	1.00

Appendix

Table S20: Graph stability. (a) Per-cohort FCI Markov-blanket size for length of stay at three significance thresholds; the PPIS blanket is markedly more

\alpha

-sensitive than the HAPI blanket, consistent with learning a high-dimensional PAG from a few hundred patients. (b) Age-harmonization sensitivity of the three pooled-shared LoS edges: bootstrap adjacency frequency (30 resamples of the pooled backbone,

n=516

\alpha=0.10

) under four age constructions. Consistent age harmonization uses either age at SCI, which both cohorts record, or within-cohort

z

-scoring, which removes the anchor/scale mismatch. In both cases the Age–LoS and Hemoglobin–LoS adjacencies are recovered in every resample and Albumin–LoS in a minority (albumin has high missingness). Only the units-inconsistent age-at-admission construction, which relies on the noisy SCI-to-admission interval, destabilizes Age–LoS. All three edges are recovered in the full sample under the consistent harmonizations. Per-cohort support for all three edges, on the shared backbone and in each cohort’s full graph, is reported in Table S27 .

(a) Pooled edge into LoS	Age	Albumin	Hemoglobin
Fisher- $z$ (default)	✓	✓	✓
KCI (kernel test)	✓	✓	✓
Nonparanormal Fisher- $z$	✓	—	✓
MICE imputation	✓	✓	✓
missForest imputation	✓	✓	✓
(b) Per-cohort LoS $\lvert\mathrm{MB}\rvert$	HAPI / PPIS

Appendix

Table S21: Structure-learning sensitivity of the pooled-shared length-of-stay edges and the per-cohort LoS Markov-blanket size to the conditional-independence test and the imputer. (a) Presence of each pooled LoS edge under Fisher-

z

(default), a kernel test (KCI), a nonparanormal (rank-to-normal) Fisher-

z

, and median/MICE/missForest graph-level imputation. All three edges survive the kernel test and every imputer; only Albumin

\rightarrow

LoS drops under the nonparanormal transform (albumin has high missingness). (b) Per-cohort LoS Markov-blanket size at the canonical

\alpha=0.05

. The blanket is unchanged under the kernel test and grows under MICE and under the rank-based transform, far more on PPIS than on HAPI, consistent with the PPIS graph being denser and less stable. † The nonparanormal run uses a different Markov-blanket derivation and is therefore read against its own Fisher-

z

baseline of 25 / 83, not against the 21 / 34 of the rows above. A full per-cohort re-run with the kernel (KCI) test, nonparametric and appropriate for mixed discrete–continuous data, yields identical Markov blankets (HAPI 21, PPIS 34, and PPIS binary-outcome blankets 50/6/50), indicating the recovered structure is not an artifact of the Fisher-

z

Gaussianity assumption. Full-pipeline nested cross-validation, repeating the entire pooled

+

temporal-tier fusion inside each outer fold, was investigated systematically but is computationally intractable on these cohorts. Most folds complete in

\sim

25 s, but a subset of dense training folds (one HAPI fold and four of the five PPIS folds) fail to terminate. On those subsets the FCI skeleton stays dense, so the constraint-based search evaluates prohibitively many conditional-independence tests and does not converge. Graph density rather than conditioning-set depth drives the non-termination, since the search stalls on the same folds with the FCI conditioning-set depth capped at

\in\{5,10,30\}

or unbounded. We therefore report the simplified per-cohort-FCI nesting in Table S14 , which re-derives the dominant feature-selection step inside each fold, as the tractable stand-in.

Cohort	Constraint strategy	$\lvert$ edges $\rvert$	$\lvert\mathrm{MB}\rvert$	TabPFN $R^{2}$ [95% CI]
HAPI	None (unconstrained)	64	11	0.701 [0.62, 0.76]
HAPI	Temporal tiers (forbidden $=$ required $=$ both)	68	9	0.694 [0.61, 0.77]
HAPI	$+$ pooled constraints only (no edge union)	66	9	0.694 [0.61, 0.77]
HAPI	$+$ pooled edge union only (no constraints)	77	11	0.718 [0.64, 0.78]
HAPI	Temporal tiers $+$ pooled-shared (both)	75	11	0.718 [0.64, 0.78]
PPIS	None (unconstrained)	154	23	0.743 [0.67, 0.79]

Appendix

Table S22: Constraint-strategy ablation for length-of-stay discovery (per-cohort FCI,

\alpha=0.05

, adjacency-search depth 3). We toggle the causal-discovery background knowledge and measure its effect on the recovered partial ancestral graph (

\lvert\text{edges}\rvert

), the temporally-filtered length-of-stay Markov blanket (

\lvert\mathrm{MB}\rvert

), and downstream TabPFN 5-fold

R^{2}

. Temporal-tier forbidden-only, required-only, and both give identical Markov blankets, because the tier rules constrain only edge orientation (a later-tier variable cannot cause an earlier-tier one) and the Markov blanket is adjacency-based. The pooled-shared stage is the ingredient that measurably changes prediction, and the middle two rows of each block localize which part of it does so. That stage bundles constraints derived from the pooled subgraph (orientations, plus forbidding pairs absent from it) with a union of its edges into the per-cohort PAG. The constraints alone leave the result unchanged from the tier-only row; the edge union alone reproduces the full pooled row. A third variant (orientation constraints

+

edge union, dropping only the absent-pair forbidding) is identical to the union-alone row. The effect is therefore cross-cohort edge fusion, not a constraint on the per-cohort search. Fusion restores the shared directed-ancestor variables (Age, Albumin, Hemoglobin) to the blanket and improves HAPI length-of-stay

R^{2}

+0.024

(from

0.694

0.718

), consistent with the Stage-4 jump in the information-fusion trajectory (Figure 4 ). On the already-large PPIS blanket the addition is marginal. Edge counts vary by up to

\pm 2

between repeat runs of the discovery while the Markov blankets and

R^{2}

are stable. In this ablation the Markov blankets are discovered once on the full cohort (with a fixed whole-cohort pooled prior) so that each row isolates the effect of the background constraint. Prediction is out-of-fold but selection is in-sample, so these absolute

R^{2}

are optimistic relative to the fully per-fold leak-free numbers in the main results. The relative constraint effect is unbiased because every row shares the same whole-cohort discovery step.

Feature set	$n$	AUROC [95% CI]	AUPRC [95% CI]	F1 [95% CI]
Urinary tract infection
Constrained graph set	50	0.894 [0.850, 0.938]	0.829 [0.757, 0.898]	0.784 [0.707, 0.853]
All features	75	0.894 [0.855, 0.931]	0.810 [0.739, 0.875]	0.720 [0.635, 0.792]
MI top-20	20	0.900 [0.859, 0.936]	0.820 [0.745, 0.886]	0.730 [0.652, 0.805]
RF top-10	10	0.866 [0.816, 0.916]	0.745 [0.649, 0.838]	0.699 [0.617, 0.778]
Lasso CV	12	0.864 [0.824, 0.909]	0.711 [0.609, 0.814]	0.702 [0.615, 0.770]

Appendix

Table S23: Discrimination, average precision and F1 for the three PPIS binary outcomes , by feature-selection strategy, under a fixed TabPFN v2 model. Values are computed on stacked out-of-fold predictions with 95% bootstrap confidence intervals (300 resamples). F1 is at the default

0.5

threshold and is reported here because it is stated in the Methods but does not appear in any main-text figure. The constrained graph set leads on AUROC and average precision for secondary PI complication and is within the bootstrap interval of the best baseline on urinary tract infection. On osteomyelitis the automatic selectors and the all-feature model lead; their dominant predictors are non-ancestors of the target. F1 at a fixed threshold separates the strategies poorly on these prevalences and should not be read as a ranking.

TRIPOD+AI item	Where addressed
Title / abstract as prediction study	Reframed as hypothesis-generating; deployable (admission-time) vs. retrospective results distinguished (Abstract, Results)
Source of data, participants, setting	Methods (Clinical datasets); single-center, Swiss Paraplegic Center; Table S29
Outcomes and predictors (with timing)	Methods; temporal-tier definitions; decision-time stratification (Table S12 )
Missing data	Table S30 ; imputation methods and sensitivity (Table S17 )
Sample size	No formal a-priori sample-size calculation; analysis used all available records ( $n=250$ HAPI, $n=266$ PPIS). Given $\leq 87$ candidate variables, this is underpowered for high-dimensional structure learning and prediction; results are framed accordingly (Discussion)
Model development / selection	Feature and model selection performed post-discovery; nested cross-validation reported (Table S14 ); no multiplicity correction (Methods)

Appendix

Table S24: TRIPOD+AI item map (abbreviated). “Source” points to where each item is addressed; several items are addressed as limitations.

No.	Item (abbreviated)	Where reported / status
Title and abstract
1(a)	Study design indicated in title or abstract	Abstract (“two retrospective cohorts”); Methods, “Clinical datasets” $\rightarrow$ “Study design” (“a retrospective observational cohort analysis”). The title carries no design term.
1(b)	Informative and balanced abstract summary	Abstract: setting, both cohorts with sample sizes, method, effect sizes, and the explicit “hypothesis-generating pending external validation” caveat.
Introduction
2	Background and rationale	Introduction, paragraphs 1–4 (clinical rationale for pressure injury and length of stay after SCI; causal discovery under missingness; integration of partially overlapping cohorts).
3	Objectives	Introduction, five enumerated contributions (final paragraph); Results, “Markov-blanket graph learning through multi-cohort fusion”. No hypotheses were prespecified and no analysis protocol was pre-registered.

Appendix

Table S25: STROBE cohort-study checklist and where each item is addressed in this manuscript. “Methods” and “Results” refer to main-text sections; “Supplementary” refers to this document.

Shared concept	HAPI column	PPIS column	Type / units
Age	Age at SCI	Age at Admission	Continuous, years
AIS	AIS	AIS ISNCSCI at entry	Ordinal, A–E
Level of injury	Level of Injury	Lesion Level	Categorical
Sex	Sex	Sex	Binary
SCIM	SCIM Score	SCIM Total Score at discharge	Continuous, 0–100
FIM Cognitive (FIMc)	FIM Cognitive Score	—	Continuous

Appendix

Table S26: Harmonized variable concepts across HAPI and PPIS. Of these 12 harmonized concepts, nine are observed in both cohorts; FIM Cognitive (FIMc), Nutrition, and Autonomic dysreflexia are available in HAPI only. The pooled-shared PAG for length of stay (

n=516

) is estimated on the subset available at or before admission: SCIM is recorded at discharge in PPIS (an end-of-stay measurement) and is therefore excluded from the LoS backbone by the temporal-tier constraint, leaving eight shared predictors (AIS, Age, Level of injury, Sex, Albumin, Hemoglobin, Vitamin D, Pneumonia) plus length of stay (nine shared variables in total). The harmonized “Age” concept maps HAPI Age at SCI to PPIS Age at Admission ; these are related but not identical quantities (see Discussion). Rows shaded in light blue in the PPIS and HAPI tables below correspond to the shared concepts.

		Shared backbone		Full per-cohort graph
Pooled-shared edge	Pooled $p_{\mathrm{edge}}$	HAPI	PPIS	HAPI $p_{\mathrm{edge}}$	PPIS $p_{\mathrm{edge}}$
Age $\rightarrow$ LoS	0.98	$\rightarrow$	—	0.00	0.00
Albumin $\rightarrow$ LoS	0.76	at $\alpha=0.10$	$\rightarrow$	0.00	0.00
Hemoglobin $\rightarrow$ LoS	1.00	$\rightarrow$	at $\alpha=0.10$	0.17	0.01

Appendix

Table S27: Per-cohort support for the three pooled-shared edges into length of stay. Shared backbone : each cohort run alone on the variables observed in both cohorts, with the same temporal-tier knowledge, depth and

\alpha=0.05

as the pooled analysis;

\rightarrow

marks a definite directed edge into LoS, and “at

\alpha=0.10

” an edge that appears only at the looser threshold. Each cohort alone orients part of the set (HAPI the Age and Hemoglobin relations, PPIS the Albumin relation) and the pooled analysis orients all three; HAPI alone also orients AIS and level of injury into LoS, which the pooled analysis does not. Full per-cohort graph : bootstrap inclusion probability (

B=100

, unconstrained,

\alpha=0.10

) of the same edges when each cohort is searched over its whole variable set, where the three relations are screened off almost entirely. Fusing the pooled edges into the per-cohort graphs is what places them in the per-cohort Markov blankets (Table S22 ). Pooled

p_{\mathrm{edge}}

is bootstrap inclusion in the pooled graph from the same resampling. Values from percohort_backbone.json and probabilistic_mb_sensitivity.json .

Shared edge	HAPI full	PPIS full	Pooled backbone	Posterior [95% CI]	Gate (0.8)
Age $\rightarrow$ LoS	${\approx}0$	${\approx}0$	$0.98$	$0.975$ $[0.94,1.00]$	impose
Hemoglobin $\rightarrow$ LoS	$0.17$	$0.01$	$1.00$	$0.995$ $[0.98,1.00]$	impose
Albumin $\rightarrow$ LoS	${\approx}0$	${\approx}0$	$0.76$	$0.757$ $[0.67,0.84]$	withhold

Appendix

Table S28: Fail-closed cross-cohort constraint gate. Bootstrap adjacency frequency of each pooled-shared length-of-stay edge in the full per-cohort graphs (where it is screened off) versus the reduced pooled backbone, with the Beta (Jeffreys-prior) posterior mean and 95% credible interval on the pooled-backbone frequency (

B=100

). A constraint is imposed only if the posterior mean clears the

0.8

confidence gate; Albumin is withheld, matching Table S27 .

Characteristic	HAPI ( $n=250$ )	PPIS ( $n=266$ )
Age, years a	50 [34, 66]	60 [51, 70]
Female sex	72 (29%)	64 (24%)
AIS grade A / B / C / D b	90 (36%) / 29 (12%) / 37 (15%) / 90 (36%)	203 (76%) / 37 (14%) / 18 (7%) / 8 (3%)
Length of stay, days	66 [19, 170]	89 [69, 122]
Albumin, g/L	33 [29, 39]	35 [32, 39]
Hemoglobin, g/L	122 [100, 140]	120 [106, 136]

Appendix

Table S29: Cohort characteristics. Continuous variables are median [IQR]; categorical variables are

n

(%). HAPI (

n=250

) pools the three rehabilitation/comparison cohorts; PPIS (

n=266

) is the post-pressure-injury surgery cohort. a HAPI age is age at spinal cord injury, recorded for the 142 rehabilitation-cohort patients but not for the No-HAPI comparison cohort (hence 43% missing; Table S30 ). PPIS age is age at admission, a different anchor (Table S26 ). b AIS: ASIA Impairment Scale grade at entry (A = motor/sensory complete). The three binary post-surgical outcomes are defined only for PPIS.

Variable	HAPI missing (%)	PPIS missing (%)
Age	43	0
Sex	0	0
AIS	0	0
Level of injury	0	0
Length of stay	0	0
Albumin	37	19

Appendix

Table S30: Per-variable missingness (%) by cohort for the shared-backbone variables. Overall missingness is 1.5% in the HAPI rehabilitation cohort, 11.4% in the No-HAPI comparison cohort, and 21.5% in PPIS (median 33.5% among variables with any missingness). Several variables (discharge SCIM, laboratory markers) appear more likely to be missing at extreme or clinically-uneventful values, i.e. plausibly non-random missingness, motivating the imputation-aware discovery pipeline.

Variable	Type	Description	Encoding / Units
Sociodemographic and Definition of SCI Lesion
PID	Text	patient ID	-
FID	Text	case ID	-
explicit consent	Binary	-	0 no, 1 yes
Date of entry	Date	Date of admission at the hospital. Not to be changed later.	Any valid date
Date of discharge	Date	Date of discharge from the hospital.	Any valid date, 0 still in hospitalization/ unknown

Appendix

Table S31: PPIS dataset variable definitions (abbreviated; derived duration variables such as Duration of Immobilization and Time between Surgeries are defined in Table S11 ). Shaded rows are shared backbone variables also present in HAPI (see Table S26 ).

Variable	Type	Description	Encoding / Units
Date of Birth	Date	Patient date of birth	YYYY-MM-DD
Admission Date	Date	Admission date	YYYY-MM-DD
Discharge Date	Date	Discharge date	YYYY-MM-DD
Length of Stay	Continuous	Duration of hospitalization	Days
SCI Status	Binary	Presence of spinal cord injury	0 = No, 1 = Yes
Study Participation	Binary	Inclusion in study cohort	0 = No, 1 = Yes

Appendix

Table S32: HAPI dataset variable definitions. Shaded rows are shared backbone variables also present in PPIS (see Table S26 ).

Causal Discovery

Causal discovery infers cause-and-effect structures from observational and interventional data, distinguishing causal relationships from correlation.

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Feb 16, 2026

Use What You Know: Causal Foundation Models with Partial Graphs

Estimating causal quantities traditionally relies on bespoke estimators tailored to specific assumptions. Recently proposed Causal Foundation Models (CFMs) promise a more unified approach by amortising causal discovery and inference in a single step. However, in their current state, they do not allow for the incorporation of any domain knowledge, which can lead to suboptimal predictions. We bridge this gap by introducing methods to condition CFMs on causal information, such as the causal graph or more readily available ancestral information. When access to complete causal graph information is too strict a requirement, our approach also effectively leverages partial causal information. We systematically evaluate conditioning strategies and find that injecting learnable biases into the attention mechanism, together with a graph-convolutional encoder, is a highly effective method to utilise full and partial causal information. Our experiments show that this conditioning allows a general-purpose CFM to match the performance of specialised models trained on specific causal structures. Overall, our approach addresses a central hurdle on the path towards all-in-one causal foundation models: the capability to answer causal queries in a data-driven manner while effectively leveraging any amount of domain expertise.

Arik Reuter, Anish Dhir, Cristiana Diaconu +6

University of Cambridge, Cambridge, United Kingdom · Max Planck Institute for Intelligent Systems, Tübingen, Germany · Gatsby Computational Neuroscience Unit, University College London +5

Causal Foundation Models Causal Discovery Causal Inference

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Jun 25, 2026

A Causal Foundation Model for Structure and Outcome Prediction

We introduce TabPFN-CFM, a causal foundation model that can handle multiple causal problems. TabPFN-CFM predicts both causal structure and outcomes from observational data, supports queries on all three levels of Pearl's Causal Hierarchy and uses known graph structure when available to improve predictions. TabPFN-CFM is trained on synthetic datasets, and generalises to real datasets, demonstrating improved performance over both structural and outcome prediction baselines.

Max Zhu, Martino Mansoldo, Ching-Hao Wang +1

University of Cambridge, United Kingdom · GSK.ai.

Causal Foundation Models Causal Structure Learning Causal Inference

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May 8, 2026

Arrow: A Foundation Model for Causal Discovery

We introduce Arrow, a foundation model for zero-shot causal discovery on observational tabular data. Arrow factorizes a directed acyclic graph into an undirected skeleton and a topological order, guaranteeing acyclicity by construction. Given a new dataset, it uses a transformer-based architecture to contextualize variables within and across observations, then predicts skeleton edge probabilities and node order scores that together define a graph. Arrow is trained in a supervised fashion on synthetic datasets with ground-truth graphs, using an end-to-end differentiable directed edge composite likelihood induced by the skeleton-order factorization. The training distribution spans diverse graph families, functional forms, noise models, and dataset shapes. Across in- and out-of-distribution synthetic, semi-synthetic, and real datasets, Arrow matches or outperforms existing causal discovery methods at substantially lower inference cost than competitive alternatives. Our results demonstrate that large-scale pretraining on diverse synthetic data can yield zero-shot causal discovery models that are fast, accurate, and reusable on new datasets.

Ryan Thompson, He Zhao, Daniel M. Steinberg +1

University of Technology Sydney · CSIRO’s Data61

Causal Foundation Models Tabular Foundation Models Causal Discovery

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Abstract

Causal Discovery

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