Conditional Average Treatment Effect Distillation for Hypothetical State-Intervention Targeting Under Budget Constraints

Vasundhara Acharya; Bülent Yener

cs.LGOct 19, 2026

Conditional Average Treatment Effect Distillation for Hypothetical State-Intervention Targeting Under Budget Constraints

Authors: Vasundhara Acharya, Bülent Yener

Organizations: Rensselaer Polytechnic Institute, Troy, USA

Abstract

Personalized public-health targeting must capture heterogeneous intervention benefits while remaining interpretable under resource constraints. Flexible causal-effect models can be hard to operationalize, while coarse subgroup policies may lose individualized effect information. We study budget-constrained hypothetical state interventions in observational data without a logged historical decision policy. For obesity, elevated glucose, and lifetime smoking-history states, causal-structure-informed pretreatment covariates are prespecified, with capacity capped at 70% of those observed in the adverse state. We evaluate causal distillation where Causal Forest and Bayesian Causal Forest teachers estimate conditional average treatment effects (CATEs), which decision tree and random forest students approximate. We compare individualized policies with unsupervised K-means subgroup policies and supervised causal-tree policies with leaf counts matched to K-means cluster counts. Observational policies are assessed on held-out data using a doubly robust estimator, bootstrap uncertainty, and decision-focused teacher-student fidelity measures. Across PIMA obesity and glucose and NHANES smoking-history experiments, each distilled student was within 0.011 absolute utility of its teacher, with allocation Jaccard similarities of 0.738-0.937. K-means and causal-tree policies were also competitive, with the highest point-estimate policy varying by application. Prespecified paired comparisons were nonsignificant after Holm adjustment, while near-budget-boundary agreement was less stable than global fidelity for some teacher-student pairs. Findings support causal distillation for translating flexible effect models into simpler budgeted targeting rules, although observational estimates depend on causal-identification assumptions and require longitudinal or prospective validation.

Figures & tables

\widehat{\tau}^{(m)}_{i},\qquad i\in\mathcal{D}_{\mathrm{eval}}.

Algorithm 1 Teacher–Student Causal Policy Learning and Held-out Evaluation

Policy experiment	Analytic $N$	Discovery $N$	Evaluation $N$	Eval_eligible $N$	Max_budget $B_{q}$
PIMA BMI	757	454	303	190	133
PIMA glucose	752	451	301	76	53
NHANES smoking history	7,217	4,330	2,887	1,273	891

Table 1: Analytic cohorts and intervention capacity across observational policy experiments.

Model	Hyperparameters used
Causal Forest teacher	$n_{\mathrm{estimators}}=300$ , $\mathrm{max\_depth}=\mathrm{None}$ , $\mathrm{min\_samples\_split}=10$ , $\mathrm{min\_samples\_leaf}=5$
Decision Tree student	$\mathrm{max\_depth}=6$ , $\mathrm{min\_samples\_leaf}=10$ , $\mathrm{max\_leaf\_nodes}=12$
Random Forest student	$n_{\mathrm{estimators}}=6$ , $\mathrm{max\_depth}=6$ , $\mathrm{min\_samples\_leaf}=4$ , $\mathrm{max\_leaf\_nodes}=\mathrm{None}$
BCF teacher	Binary probit outcome model; prognostic forest: $200$ trees, $\alpha=0.95$ , $\beta=2$ , $\mathrm{min\_samples\_leaf}=5$ , $\mathrm{max\_depth}=8$ ; treatment-effect forest: $50$ trees, $\alpha=0.25$ , $\beta=3$ , $\mathrm{min\_samples\_leaf}=5$ , $\mathrm{max\_depth}=5$ ; $1000$ Markov chain Monte Carlo (MCMC) draws
K-means subgroup policy	Number of clusters $K=3$
Teacher–student training	OOF teacher-target generation: $K_{\mathrm{OOF}}=5$ folds

Table 2: PIMA BMI policy: Model hyperparameters used in the teacher–student and subgroup-policy analyses. The same decision-tree and random-forest configurations were subsequently retained for the Bayesian causal forest teacher rather than retuned separately, so that changes in student fidelity and policy performance primarily reflect the change in teacher signal rather than changes in student-model capacity.

Model	Hyperparameters / Training Settings
Causal Forest Teacher	$n_{\text{estimators}}=400$ , $\mathrm{max\_depth}=\mathrm{None}$ , $\mathrm{min\_samples\_split}=10$ , $\mathrm{min\_samples\_leaf}=5$
BCF Teacher	Binary probit outcome model; prognostic forest: $200$ trees, $\alpha=0.95$ , $\beta=2$ , $\mathrm{min\_samples\_leaf}=5$ , $\mathrm{max\_depth}=8$ ; treatment-effect forest: $50$ trees, $\alpha=0.25$ , $\beta=3$ , $\mathrm{min\_samples\_leaf}=5$ , $\mathrm{max\_depth}=5$ ; $1000$ MCMC draws
Decision Tree	$\mathrm{max\_depth}=4$ , $\mathrm{min\_samples\_leaf}=10$ , $\mathrm{max\_leaf\_nodes}=12$
Random Forest	$n_{\text{estimators}}=5$ , $\mathrm{max\_depth}=6$ , $\mathrm{min\_samples\_leaf}=3$ , $\mathrm{max\_leaf\_nodes}=\mathrm{None}$
K-means subgroup policy	$\mathrm{number\_of\_clusters}=4$
Causal Tree subgroup policy	Causal splitting criterion: $\mathrm{causal\_mse}$ ; $\mathrm{max\_leaf\_nodes}=4$ ; honest discovery split: $50\%/50\%$ for structure learning and leaf-effect estimation; $5$ -fold cross-fitted AIPW leaf-effect scores;

Table 3: PIMA Glucose policy: Model hyperparameters used in the teacher–student and subgroup-policy analyses.

Model	Hyperparameters used
Causal Forest teacher	$n_{\mathrm{estimators}}=200$ , $\mathrm{max\_depth}=10$ , $\mathrm{min\_samples\_split}=20$ , $\mathrm{min\_samples\_leaf}=20$ , $\mathrm{max\_samples}=0.45$
Decision Tree student	$\mathrm{max\_depth}=4$ , $\mathrm{min\_samples\_leaf}=25$ , $\mathrm{max\_leaf\_nodes}=10$
Random Forest student	$n_{\mathrm{estimators}}=8$ , $\mathrm{max\_depth}=5$ , $\mathrm{min\_samples\_leaf}=50$ , $\mathrm{max\_leaf\_nodes}=12$
BCF teacher	Binary probit outcome model; prognostic forest: $200$ trees, $\alpha=0.95$ , $\beta=2$ , $\mathrm{min\_samples\_leaf}=5$ , $\mathrm{max\_depth}=8$ ; treatment-effect forest: $50$ trees, $\alpha=0.25$ , $\beta=3$ , $\mathrm{min\_samples\_leaf}=5$ , $\mathrm{max\_depth}=5$ ; $10$ GFR initialization sweeps and $1000$ MCMC draws
K-means subgroup policy	Number of clusters $K=6$ ; $n_{\mathrm{init}}=50$ ; K-means++ initialization; clustering performed on the $7$ -component discovery-fitted MCA representation
Teacher–student training	OOF teacher-target generation: $K_{\mathrm{OOF}}=5$ folds

Table 4: NHANES smoking-history policy: Model hyperparameters used in the teacher–student and subgroup-policy analyses.

Model	Hyperparameters used
Causal Forest teacher	$n_{\mathrm{estimators}}=400$ , $\mathrm{max\_depth}=\mathrm{None}$ , $\mathrm{min\_samples\_leaf}=5$ , $\mathrm{max\_samples}=0.45$ , $\mathrm{max\_features}=\mathrm{None}$
Decision Tree student	$\mathrm{max\_depth}=3$ , $\mathrm{min\_samples\_leaf}=30$ , $\mathrm{max\_leaf\_nodes}=8$
Random Forest student	$n_{\mathrm{estimators}}=6$ , $\mathrm{max\_depth}=6$ , $\mathrm{min\_samples\_leaf}=4$
BCF teacher	Continuous outcome model; prognostic forest: $200$ trees, $\alpha=0.95$ , $\beta=2$ , $\mathrm{min\_samples\_leaf}=5$ , $\mathrm{max\_depth}=8$ ; treatment-effect forest: $50$ trees, $\alpha=0.25$ , $\beta=3$ , $\mathrm{min\_samples\_leaf}=5$ , $\mathrm{max\_depth}=5$ ; adaptive coding enabled; $10$ GFR warm-start draws and $1000$ MCMC draws
K-Prototypes subgroup policy	Number of clusters $K=3$
Teacher–student training	OOF teacher-target generation: $K_{\mathrm{OOF}}=5$ folds

Table 5: IHDP experiment: Model hyperparameters and training settings used in the teacher–student and subgroup-policy analyses.

Dataset/policy	Treatment	Adjustment variables	Candidate effect modifiers
PIMA–BMI	Obesity status	Age, Pregnancies, DiabetesPedigreeFunction(DPF)	Age, Pregnancies, DPF
PIMA–Glucose	Elevated glucose	Age, BMI, DPF	Age, BMI, DPF
NHANES–Smoking	Smoking history	Gender, Income, Age, Race	Gender, Income, Age, Race, Epilepsy, Education

Table 6: Prespecified covariate roles used in the policy experiments. For PIMA and NHANES, these roles were informed by the fixed causal structures developed in our previous work [ 2 ] . For IHDP, we did not estimate a causal graph using the proposed causal-discovery procedure. All available pretreatment covariates were used for confounding adjustment and treatment-effect modeling. The overlap between the adjustment-variable and effect-modifier sets is intentional, because a pretreatment covariate may contribute to confounding control while also modifying the conditional treatment effect.

Policy family	Policy	Distillation source	Policy risk	Policy utility	95% CI for utility
Reference	Shift all eligible	—	0.1629	0.837	[0.7613,0.8988]
Reference	No shift	—	0.3267	0.6733	[0.6205,0.7294]
Causal forest	Teacher	Causal forest	0.2142	0.7858	[0.7201, 0.8511]
Causal forest	Decision-tree student	Causal forest	0.2146	0.7854	[0.7197, 0.8460]
Causal forest	Random-forest student	Causal forest	0.2244	0.7756	[0.7096, 0.8433]
BCF	Teacher	BCF	0.2099	0.7901	[0.7249, 0.8533]

Table 7: PIMA BMI policy: Held-out performances. Among the budget-constrained learned policies, the Causal Tree subgroup policy achieved the highest point-estimate utility, followed closely by the BCF teacher , causal forest teacher, and their distilled students. K-means achieved a lower point-estimate utility. The 95% confidence intervals overlapped substantially across the learned policies, so the observed ordering should be interpreted descriptively rather than as evidence of statistically significant superiority.

Teacher	Student	Fidelity ( $R^{2}$ )	Decision Jaccard	Spearman	Kendall $\tau_{b}$	Boundary Kendall $\tau_{b}$
Causal forest	Decision tree	0.7089	0.7500	0.8315	0.6805	0.1188
Causal forest	Random forest	0.7855	0.7733	0.8930	0.7245	$-0.1534$
BCF	Decision tree	0.7306	0.8472	0.8650	0.7138	$-0.0259$
BCF	Random forest	0.801	0.8095	0.8977	0.7340	$-0.1056$

Table 8: PIMA BMI policy: Teacher–student decision overlap and ranking fidelity at the prespecified eligible-population inclusion rate

q=0.70

. Global agreement was generally high, with decision Jaccard similarities ranging from

0.7500

0.8472

and eligible-population Spearman correlations ranging from

0.8315

0.8977

. The BCF-distilled decision tree achieved the highest decision overlap, whereas the BCF-distilled random forest achieved the strongest global rank correlation. In contrast, Kendall

\tau_{b}

near the allocation boundary was weak or negative for three of the four students, indicating that local ordering around the treatment cutoff was less stable than the overall ranking.

Fig 1: PIMA BMI policy: The causal-tree comparator achieved the highest utility among budget-constrained policies, closely followed by the BCF teacher, causal forest teacher, and distilled decision-tree students. All learned policies outperformed the no-shift reference, while shift-all achieved the highest overall utility

Fig 2: PIMA BMI policy: Structural complexity of teacher and distilled student models. The distilled decision-tree students provide the most compact representations, while the causal-forest teacher is substantially more complex

Fig 3: PIMA BMI policy: Paired differences in held-out policy utility at the prespecified 70% eligible-individual intervention budget.

Student	Readable rule	Eligible $N$	Student benefit	CF teacher benefit	Policy status	Illustrative interpretation
DT	Pregnancies $\leq 3.5$ and DPF $>0.738$	21	44.9 pp	45.7 pp	Fully prioritized	This is an exact DT path. An eligible individual satisfying both thresholds is assigned directly to this leaf, which predicts high positive benefit and is fully included within the 70% intervention capacity.
DT	Age $>28.5$ and $0.190<$ DPF $\leq 0.426$	45	23.0 pp	23.3 pp	Capacity boundary	The exact rule identifies a positive-benefit leaf at which the available capacity is exhausted. The rule therefore identifies the boundary subgroup, while final allocation follows the prespecified fractional boundary policy.
DT	Age $\leq 42.5$ and DPF $\leq 0.190$	19	19.2 pp	17.2 pp	Positive benefit, outside capacity	The rule predicts positive benefit, but this leaf lies below the current capacity cutoff. The individual is therefore not prioritized at the prespecified budget despite positive estimated benefit.
RF surrogate	Pregnancies $\leq 3.5$ and DPF $>0.745$	20	44.4 pp	45.8 pp	Fully prioritized RF region	This surrogate region summarizes a portion of the full RF in which all eligible held-out individuals were selected. It provides a concise, clinician-readable description of the RF behavior, although the actual policy decision remains that of the full RF ensemble.
RF surrogate	$0.190<$ DPF $\leq 0.426$	80	25.1 pp	25.2 pp	Partially prioritized RF region	The surrogate identifies a broad positive-benefit region, but the full RF makes different policy decisions within it. Thus, the rule explains the general covariate pattern associated with prioritization without replacing the full RF decision.
RF surrogate	Age $\leq 42.5$ and DPF $\leq 0.190$	19	18.9 pp	17.2 pp	Positive region, outside capacity	The surrogate identifies a region with positive RF benefit, but the full RF does not prioritize these profiles at the current 70% capacity. The rule therefore provides a readable explanation for a lower-priority region of the RF policy.

Table 9: PIMA BMI policy: Representative clinician-facing examples from the distilled decision-tree and random-forest students. The shallow RF surrogate showed good fidelity to the full random-forest predictions (

R^{2}=0.892

Student	Readable rule	Eligible $N$	Student benefit	BCF teacher benefit	Policy status	Illustrative interpretation
DT	$0.5<$ Pregnancies $\leq 5.5$ and $0.589<$ DPF $\leq 1.280$	23	33.1 pp	34.3 pp	Fully prioritized	This is an exact DT path. An eligible individual satisfying these thresholds is assigned directly to this terminal leaf, which predicts positive benefit and is fully included within the prespecified 70% intervention capacity.
DT	Age $>40.5$ , Pregnancies $>5.5$ , and DPF $>0.304$	15	22.5 pp	22.8 pp	Capacity boundary	The exact DT rule identifies a positive-benefit leaf at which the intervention capacity is exhausted. At the prespecified budget, 60% of eligible individuals in this leaf receive allocation through the boundary mechanism.
DT	Age $>24.5$ and DPF $\leq 0.182$	21	19.8 pp	19.0 pp	Positive benefit, outside capacity	The rule predicts positive benefit, but the leaf falls below the current capacity cutoff.
RF surrogate	Pregnancies $\leq 4.5$ and DPF $>0.735$	24	36.3 pp	36.9 pp	Fully prioritized RF region	The surrogate identifies a readable region in which all eligible evaluation profiles were selected by the full RF.
RF surrogate	Age $\leq 40.5$ and $0.318<$ DPF $\leq 0.735$	67	28.7 pp	29.5 pp	Partially prioritized RF region	The surrogate identifies a positive-benefit region in which the full RF prioritized approximately 98.5% of eligible individuals. .
RF surrogate	Age $>24.5$ and DPF $\leq 0.182$	21	20.4 pp	19.0 pp	Positive region, outside capacity	The surrogate identifies a region with positive mean RF benefit, but none of the eligible profiles in this region are prioritized under the current 70% intervention capacity.

Table 10: PIMA BMI policy: Representative clinician-facing examples from the BCF-distilled decision-tree and random-forest students. The shallow RF surrogate achieved an

R^{2}

0.792

with respect to the full BCF-distilled RF predictions and is used only for interpretation, not for policy allocation.

Policy family	Policy	Distillation source	Policy risk	Policy utility	95% CI for utility
Reference	Shift all eligible	—	0.2375	0.7625	[0.7010, 0.8204]
Reference	No shift	—	0.3522	0.6478	[0.5847, 0.6977]
Causal forest	Teacher	Causal forest	0.2730	0.7270	[0.6607, 0.7814]
Causal forest	Decision-tree student	Causal forest	0.2693	0.7307	[0.6720, 0.7841]
Causal forest	Random-forest student	Causal forest	0.2712	0.7288	[0.6649, 0.7820]
BCF	Teacher	BCF	0.2735	0.7265	[0.6656, 0.7800]

Table 11: PIMA glucose policy: Held-out performance of the causal-forest and Bayesian-causal-forest teachers, their distilled decision-tree and random-forest students, and the K-means and causal-tree subgroup policies. Among the learned policies, the causal-tree subgroup policy achieved the highest point-estimate utility, followed closely by K-means and the causal-forest-distilled decision-tree student. The remaining teacher and student policies also produced similar point estimates. The 95% confidence intervals overlapped across all learned policies.

Fig 4: PIMA glucose policy: The causal-tree comparator achieved the highest utility among the budget-constrained learned policies, followed closely by K-means and the causal-forest-distilled decision-tree student.

Teacher	Student	Fidelity ( $R^{2}$ )	Decision Jaccard	Spearman	Kendall $\tau_{b}$	Boundary Kendall $\tau_{b}$
Causal forest	Decision tree	0.6108	0.7377	0.7313	0.5672	0.402911
Causal forest	Random forest	0.7078	0.8596	0.8007	0.6118	0.642857
BCF	Decision tree	0.7398	0.8596	0.7234	0.5889	0.267261
BCF	Random forest	0.7634	0.7667	0.6778	0.5435	-0.254588

Table 12: PIMA glucose policy: Teacher–student decision overlap and ranking fidelity at the prespecified eligible-population inclusion rate. Decision overlap was generally high, with policy Jaccard similarities ranging from

0.7377

0.8596

. The CF-distilled random forest and BCF-distilled decision tree achieved the highest decision overlap, while the CF-distilled random forest showed the strongest global rank agreement with its teacher. The BCF-distilled random forest achieved the highest teacher–student prediction fidelity (

R^{2}=0.7634

), although its policy overlap and rank correlations were lower than those of the BCF-distilled decision tree.

Fig 5: PIMA glucose policy: Teacher–student structural complexity. Both distilled student models exhibit substantially fewer terminal leaves compared to the causal-forest teacher. The decision-tree student is also more compact than the BCF treatment-effect forest, according to the reported per-draw measure, while the BCF-distilled random forest has more leaves.

Fig 6: PIMA glucose policy: Paired differences in held-out policy utility at the prespecified eligible-individual intervention budget. Positive values favor policy $A$ and negative values favor policy $B$ . Holm-adjusted $p$ -values are shown at the right to account for multiplicity across the eight prespecified comparisons. All confidence intervals include zero, and none of the comparisons are statistically significant after Holm adjustment.

Student	Readable rule	Eligible $N$	Student benefit	CF teacher benefit	Policy status	Illustrative interpretation
DT	$33.700<\mathrm{BMI}\leq 41.800$ and DPF $\leq 0.440$	13	37.1 pp	38.8 pp	Fully prioritized	This is an exact DT path. An eligible individual satisfying these thresholds is assigned directly to this terminal leaf, which predicts positive benefit and is fully included within the prespecified 70% intervention capacity.
DT	Age $\leq 48.000$ , BMI $>30.750$ , and DPF $>0.440$	21	27.5 pp	28.7 pp	Capacity boundary	This exact DT leaf lies at the intervention-capacity boundary. Individuals within the leaf receive the same DT benefit score, and approximately 42.9% of eligible profiles in this leaf are prioritized through the boundary allocation mechanism.
DT	$25.500<\mathrm{Age}\leq 46.500$ and $26.750<\mathrm{BMI}\leq 30.750$	9	24.0 pp	20.8 pp	Positive benefit, outside capacity	The exact DT rule predicts positive benefit, but the leaf falls below the current capacity cutoff and none of its eligible evaluation profiles are prioritized at the prespecified budget.
RF surrogate	$33.750<\mathrm{BMI}\leq 43.050$ and DPF $\leq 0.440$	12	37.0 pp	39.1 pp	Fully prioritized RF region	The surrogate identifies a readable region in which all eligible evaluation profiles were selected by the full RF. Because this is a surrogate region, the rule summarizes rather than exactly reproduces the ensemble.
RF surrogate	$24.500<\mathrm{Age}\leq 46.500$ and BMI $\leq 33.750$	19	28.0 pp	27.1 pp	Partially prioritized RF region	The surrogate identifies a positive-benefit region in which the full RF prioritizes approximately 47.4% of eligible profiles, demonstrating within-region variation in individual RF allocation decisions.
RF surrogate	Age $\leq 24.500$ and $30.150<\mathrm{BMI}\leq 33.750$	3	24.8 pp	23.8 pp	Positive region, outside capacity	The surrogate identifies a region with positive mean RF-estimated benefit, but none of the eligible profiles in this region are prioritized under the current 70% intervention capacity.

Table 13: PIMA glucose policy: Representative clinician-facing examples from the causal-forest-distilled decision-tree and random-forest students. The shallow RF surrogate achieved an

R^{2}

0.782

with respect to the full causal-forest-distilled RF predictions and is used only for interpretation, not for policy allocation.

Student	Readable rule	Eligible $N$	Student benefit	BCF teacher benefit	Policy status	Illustrative interpretation
DT	$33.250<\mathrm{BMI}\leq 43.100$ and DPF $\leq 1.011$	30	32.4 pp	32.7 pp	Fully prioritized	This is an exact DT path. An eligible individual satisfying these thresholds is assigned directly to this terminal leaf, which predicts positive benefit and is fully included within the prespecified 70% intervention capacity.
DT	Age $>24.500$ , $28.750<\mathrm{BMI}\leq 33.250$ , and DPF $\leq 1.011$	19	30.0 pp	31.0 pp	Capacity boundary	This exact DT leaf lies at the intervention-capacity boundary. Individuals within the leaf receive the same DT benefit score, and approximately 94.7% of eligible profiles in this leaf are prioritized under the boundary allocation mechanism.
DT	BMI $>43.100$ and DPF $\leq 1.011$	6	28.8 pp	27.6 pp	Positive benefit, outside capacity	The exact DT rule predicts positive benefit, but the leaf falls below the current capacity cutoff and none of its eligible evaluation profiles are prioritized at the prespecified budget.
RF surrogate	BMI $>32.850$ and DPF $\leq 1.086$	38	31.9 pp	31.8 pp	Predominantly prioritized RF region	The surrogate identifies a readable positive-benefit region in which the full RF prioritizes approximately 94.7% of eligible profiles. Because this is a surrogate region, individual RF decisions may still differ within the region.
RF surrogate	$28.750<\mathrm{BMI}\leq 32.850$ and DPF $\leq 1.086$	19	29.7 pp	31.0 pp	Partially prioritized RF region	The surrogate identifies a positive-benefit region in which the full RF prioritizes approximately 68.4% of eligible profiles, illustrating within-region variation in the ensemble’s individual allocation decisions.
RF surrogate	BMI $>28.750$ and DPF $>1.086$	6	21.2 pp	18.2 pp	Positive region, outside capacity	The surrogate identifies a region with positive mean RF-estimated benefit, but none of the eligible profiles in this region are prioritized under the current 70% intervention capacity.

Table 14: PIMA glucose policy: Representative clinician-facing examples from the BCF-distilled decision-tree and random-forest students. The shallow RF surrogate achieved an

R^{2}

0.899

with respect to the full BCF-distilled RF predictions and is used only for interpretation, not for policy allocation.

Policy family	Policy	Distillation source	Policy risk	Policy utility	95% CI for utility
Reference	Shift all eligible	—	0.2182	0.7818	[0.7581, 0.7984]
Reference	No shift	—	0.2490	0.7510	[0.7333, 0.7669]
Causal forest	Teacher	Causal forest	0.2261	0.7739	[0.7530, 0.7889]
Causal forest	Decision-tree student	Causal forest	0.2336	0.7664	[0.7450, 0.7820]
Causal forest	Random-forest student	Causal forest	0.2335	0.7665	[0.7453, 0.7821]
BCF	Teacher	BCF	0.2294	0.7706	[0.7507, 0.7853]

Table 15: NHANES smoking-history policy: Held-out performances. Among the learned policies evaluated under the

q=0.70

maximum eligible-population budget, K-means achieved the highest point-estimate utility (

0.7754

), followed closely by the causal-forest teacher (

0.7739

). The Bayesian causal-forest teacher and its distilled students achieved similar utility (

0.7706

–

0.7716

), while the Causal Tree achieved utility

0.7701

. The shift-all policy is shown only as an unconstrained reference and shifts all eligible individuals. Confidence intervals substantially overlapped across the learned policies.

Fig 7: NHANES smoking-history policy: held-out utility by method. Among the budget-constrained learned policies, K-means achieved the highest point-estimate utility ( $0.7754$ ), followed by the causal-forest teacher ( $0.7739$ ).

Teacher	Student	Fidelity ( $R^{2}$ )	Decision Jaccard	Spearman	Kendall $\tau_{b}$	Boundary Kendall $\tau_{b}$
Causal forest	Decision tree	0.6272	0.7749	0.7663	0.6029	-0.3686
Causal forest	Random forest	0.6923	0.7856	0.7813	0.6151	-0.2872
BCF	Decision tree	0.8503	0.9120	0.9316	0.8118	0.5985
BCF	Random forest	0.8512	0.9370	0.9468	0.8227	0.6301

Table 16: NHANES smoking-history policy: Teacher–student decision overlap and ranking fidelity at the prespecified eligible-population inclusion rate

q=0.70

. Teacher–student decision overlap was high across all distilled policies, with Jaccard similarities ranging from

0.7749

0.9370

. The BCF-distilled random forest achieved the highest decision overlap, the strongest global rank agreement with its teacher, and the highest boundary-region rank agreement. The BCF-distilled decision tree also showed strong agreement with its teacher. Boundary-region rank agreement was positive for the BCF-distilled students but negative for the CF-distilled students.

Fig 8: NHANES smoking-history policy: teacher–student structural complexity. Structural complexity is summarized using the total number of terminal leaves on a logarithmic scale. Variation in realized leaf count of BCF reflects the adaptively learned posterior tree structures.

Fig 9: NHANES smoking-history policy: paired utility comparisons. Paired held-out utility differences are shown for the prespecified policy comparisons, with 95% paired-bootstrap confidence intervals and Holm-adjusted $p$ -values. Differences are defined as $\Delta U=U(A)-U(B)$ , such that positive values favor the first policy named in each comparison. None of the comparisons remained statistically significant after Holm adjustment at the 0.05 level.

Student	Readable rule	Eligible $N$	Student benefit (pp)	CF teacher benefit (pp)	Policy status	Illustrative interpretation
Decision tree	Age $\leq 60$ years, race/ethnicity other than non-Hispanic White, and high-school education or less	245	8.33	7.87	Fully prioritized	This subgroup had comparatively high predicted benefit and was fully prioritized under the $q=0.70$ intervention budget.
Decision tree	Age $>60$ years, moderate-to-high income (PIR $\geq 2$ ), and race/ethnicity other than non-Hispanic White	88	1.89	1.36	Capacity boundary	This subgroup had positive predicted benefit but lay near the intervention capacity boundary, so only part of the subgroup was selected.
Decision tree	Age $>60$ years, low income (PIR $<2$ ), and high-school education	65	1.53	0.68	Outside capacity	Predicted benefit remained positive, but the subgroup was not prioritized within the available intervention capacity.
Random forest Surrogate	Age $\leq 60$ years, race/ethnicity other than non-Hispanic White, and high-school education or less	245	8.20	7.87	Fully prioritized	This surrogate region represents individuals with comparatively high predicted benefit who were consistently prioritized by the RF policy.
Random forest Surrogate	Age $\leq 60$ years, race/ethnicity other than non-Hispanic White, and more than high-school education	229	2.26	2.00	Intermediate / mixed	This surrogate region contained individuals with positive benefit but mixed allocation decisions around the policy boundary.
Random forest Surrogate	Age $>60$ years, moderate-to-high income (PIR $\geq 2$ ), and race/ethnicity other than non-Hispanic White	88	1.97	1.36	Outside capacity	Although the region showed positive predicted benefit, it fell outside the allocation capacity of the RF policy.

Table 17: NHANES smoking-history policy: Representative clinician-facing examples from the causal-forest-distilled decision-tree and random-forest students. The shallow RF surrogate achieved an

R^{2}

0.797

with respect to the full causal-forest-distilled RF predictions and is used only for interpretation, not for policy allocation.

Student	Readable rule	Eligible $N$	Student benefit (pp)	BCF teacher benefit (pp)	Policy status	Illustrative interpretation
Decision tree	Age $<40$ years, at least high-school education, and race/ethnicity other than Mexican American	274	6.07	6.04	Fully prioritized	This subgroup had comparatively high predicted benefit and was fully prioritized under the $q=0.70$ intervention budget.
Decision tree	Age 40–60 years, at least high-school education, and male	170	3.58	3.84	Capacity boundary	This subgroup had positive predicted benefit but lay near the intervention capacity boundary, so only part of the subgroup was selected.
Decision tree	Age $>60$ years, at least high-school education, and female	157	3.02	2.30	Outside capacity	Predicted benefit remained positive, but the subgroup was not prioritized within the available intervention capacity.
Random forest	Age $<40$ years and at least high-school education	309	5.94	5.82	Fully prioritized	This surrogate region represents individuals with comparatively high predicted benefit who were consistently prioritized by the RF policy.
Random forest	Age 40–60 years and at least high-school education	373	4.38	4.43	Intermediate / mixed	This surrogate region contained individuals with positive benefit but mixed allocation decisions near the policy boundary.
Random forest	Age $>60$ years, at least high-school education, and male	182	1.52	1.11	Outside capacity	Although the region showed positive predicted benefit, it fell outside the allocation capacity of the RF policy.

Table 18: NHANES smoking-history policy: Representative clinician-facing examples from the BCF-distilled decision-tree and random-forest students. The shallow RF surrogate achieved an

R^{2}

0.897

with respect to the full BCF-distilled RF predictions and is used only for interpretation, not for policy allocation.

Fig 10: BMI policy: sensitivity to the outcome nuisance-model specification. Change in held-out policy utility after replacing the logistic-regression outcome nuisance model in the doubly robust evaluator with a HistGradientBoosting classifier. The dashed vertical line at zero indicates no change in estimated utility. Negative values indicate lower utility estimates under the HistGradientBoosting outcome nuisance model.Changes are shown in percentage points for visual readability.

Fig 11: Glucose policy: sensitivity to the outcome nuisance-model specification. The dashed vertical line at zero indicates no change in estimated utility. Negative values indicate lower utility estimates under the HistGradientBoosting outcome nuisance model.Changes are shown in percentage points for visual readability.

Fig 12: Smoking policy: sensitivity to the outcome nuisance-model specification. The dashed vertical line at zero indicates no change in estimated utility. Negative values indicate lower utility estimates under the HistGradientBoosting outcome nuisance model.Changes are shown in percentage points for visual readability.

Fig 13: BMI policy: sensitivity of held-out policy utility to the discovery–evaluation split. Points show the mean held-out policy utility across the independent discovery–evaluation splits, and horizontal error bars show one standard deviation across splits. The BCF teacher achieved the highest mean utility point estimate followed closely by the CF teacher.

Fig 14: Glucose policy: sensitivity of held-out policy utility to the discovery–evaluation split. K-means attained the highest mean utility point estimate followed closely by the Causal Tree. Overall, the mean utilities were relatively similar across policies, with overlapping split-to-split variability. Error bars summarize variability across discovery–evaluation splits and are not confidence intervals.

Fig 15: Smoking policy: sensitivity of held-out policy utility to the discovery–evaluation split. The BCF teacher attained the highest mean utility point estimate, followed closely by the BCF DT student. Overall, the policies showed closely comparable mean performance with substantial overlap in split-to-split variability. Error bars summarize variability across discovery–evaluation splits and are not confidence intervals.

Fig 16: BMI policy: sensitivity to score-policy threshold extension. Horizontal bars denote 95% bootstrap confidence intervals, and the dashed vertical line indicates no change in utility. Changes were small across all models

Fig 17: Glucose policy: sensitivity to score-policy threshold extension. Horizontal bars denote 95% bootstrap confidence intervals, and the dashed vertical line indicates no change in utility. Changes were small across all models

Fig 18: Smoking policy: sensitivity to score-policy threshold extension. Horizontal bars denote 95% bootstrap confidence intervals, and the dashed vertical line indicates no change in utility. Changes were small across all models

Method	Method family	$\sqrt{\mathrm{PEHE}}$	True policy value	Mean policy regret
Causal Forest teacher	Causal Forest	$3.2190\pm 5.4669$	$14.0385\pm 15.8533$	$0.3282$
CF Decision Tree student	CF distillation	$3.7957\pm 6.5949$	$13.8532\pm 15.5185$	$0.5135$
CF Random Forest student	CF distillation	$3.5699\pm 6.1307$	$13.9602\pm 15.7249$	$0.4065$
BCF teacher	BCF	$\mathbf{1.8410\pm 3.4889}$	$\mathbf{14.2940\pm 16.3010}$	$\mathbf{0.0727}$
BCF Decision Tree student	BCF distillation	$3.2212\pm 5.6655$	$13.9773\pm 15.7715$	$0.3894$
BCF Random Forest student	BCF distillation	$2.4207\pm 4.4021$	$14.1724\pm 16.0666$	$0.1943$

Table 19: IHDP semi-synthetic analysis: Performance averaged across 10 IHDP realizations, reported as mean

\pm

standard deviation where available. True policy value denotes the average ground-truth outcome under the semi-synthetic IHDP outcome-generating process if treatment were assigned according to the policy, with higher values indicating better performance. Policy regret is the difference between the oracle policy value and the value achieved by the evaluated policy, with lower values indicating performance closer to the oracle and zero denoting the oracle itself. The selective BCF policy achieved a higher true policy value than the unrestricted treat-all reference because budgeted targeting can avoid allocating treatment to individuals with low or negative treatment benefit. The absolute magnitude of policy value does not have a direct clinical interpretation and is used for comparative evaluation within the benchmark. The oracle positive-benefit policy under the 70% budget cap achieved a mean value of

14.3667\pm 16.4195

. PEHE is not applicable to the reference policies. The comparatively large standard deviations reflect known variation in the scale of the original IHDP outcome simulations across realizations [ 25 ] .

		Agreement with teacher			Agreement with ground truth
Method	Teacher	Spearman	Kendall	Top- $q$ Jaccard	Spearman	Kendall	Oracle top- $q$ Jaccard
Causal Forest teacher	–	–	–	–	$0.7607\pm 0.1310$	$0.5760\pm 0.1190$	$0.7681\pm 0.0526$
CF Decision Tree student	Causal Forest	$0.8172\pm 0.1189$	$0.6775\pm 0.1332$	$0.8112\pm 0.0788$	$0.6694\pm 0.1882$	$0.5283\pm 0.1598$	$0.7361\pm 0.0754$
CF Random Forest student	Causal Forest	$0.9001\pm 0.0719$	$0.7432\pm 0.1018$	$0.8621\pm 0.0590$	$0.7308\pm 0.1353$	$0.5477\pm 0.1171$	$0.7555\pm 0.0361$
BCF teacher	–	–	–	–	$\mathbf{0.8572\pm 0.1573}$	$\mathbf{0.7016\pm 0.1643}$	$\mathbf{0.8601\pm 0.0548}$
BCF Decision Tree student	BCF	$0.7864\pm 0.0710$	$0.6359\pm 0.0824$	$0.8052\pm 0.0625$	$0.7252\pm 0.1312$	$0.5792\pm 0.1307$	$0.7742\pm 0.0402$

Table 20: IHDP ranking fidelity and policy overlap: Results are averaged across 10 IHDP realizations and reported as mean

\pm

standard deviation. Spearman and Kendall coefficients measure agreement in the complete treatment-benefit ranking, whereas top-

q

Jaccard measures overlap between the populations selected under the 70% treatment budget. Teacher-agreement measures are reported only for distilled students.

Policy Learning

Policy learning develops decision rules that choose actions or interventions to maximize outcomes from data, feedback, or interaction.

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Jul 29, 2026

From Unsupervised Subgroups to Hypothetical State-Intervention Policies: An Evaluation of Selected Subgrouping Methods in Observational Health Data

Conventional subgroup analyses can yield unstable and difficult-to-interpret conclusions, especially in observational biomedical data where each individual is observed under only one exposure state, true individual treatment effects are unavailable, and causal structure is uncertain. We investigate whether subgroups constructed from pretreatment characteristics, without using exposure, outcome, or estimated treatment-effect information, can serve as interpretable units for budget-constrained policy prioritization. We propose a framework combining causal-discovery-informed covariate selection, discovery-evaluation sample splitting, inductive unsupervised clustering, uncertainty-aware subgroup selection, and held-out doubly robust policy evaluation. We compare K-means, hard, membership-weighted, and stochastic Fuzzy C-means, Bayesian Gaussian mixture models, and a supervised causal-forest-derived CATE-tree comparator. Policies are evaluated under a 70% budget for hypothetical obesity-to-non-obesity and elevated-to-lower-glucose state shifts in the PIMA Indians Diabetes dataset and for a lifetime-smoking-history contrast in NHANES. The highest estimated ungated utilities were 0.799 for the BMI policy using Bayesian GMM, 0.735 for the glucose policy using hard or membership-weighted FCM, and 0.775 for the smoking-history policy using K-means. All paired 95% confidence intervals for policy-risk differences included zero, and no comparison remained statistically significant after Holm adjustment. Bayesian pooling generally preserved ungated allocations, whereas Empirical Bernstein gating was more conservative. Policies with similar estimated utility could nevertheless prioritize different individuals. The findings should be interpreted as assumption-dependent decision-support evidence for hypothetical state contrasts rather than proof of intervention benefit.

Vasundhara Acharya, Bulent Yener

Rensselaer Polytechnic Institute, Troy, USA

Clinical Decision-Making

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Aug 1, 2026

Causal Inference with Unstructured Treatments

Causal inference usually concerns a scalar treatment, yet in many problems the treatment is unstructured: a text, an image, or a sequence of clinical decisions. Consider an instructor writing a course description to attract more students: the treatment is the course description, and the outcome is enrollment. The standard target, the average treatment effect of fixing the treatment to one exact value versus another, runs into two problems. It cannot be estimated, because almost no exact description recurs across courses, leaving no comparable group from which to measure its effect; and it would be of little use even if it could, since no one wants every course to carry the same description. What the instructor actually wants to know is which features of a description raise enrollment, and which of those features can be acted on across many courses. To this end, we propose a causal query for unstructured treatments: the maximally influential feature (MIF), the feature of the treatment that most strongly influences the outcome. We formalize the MIF as a binary feature of the treatment, defined by a feature-scoring function, constrained so that both of its values stay well populated, and chosen to maximize the causal effect it induces. Turning the feature on shifts the distribution of treatments toward those that display it, turning it off shifts away, and the MIF effect contrasts the two average potential outcomes. We study identification conditions for the MIF, develop algorithms to estimate it, and make it actionable through a nudging algorithm that revises a treatment along the MIF into an outcome-improving version. We illustrate the MIF algorithm across applications in text, image, and dynamic treatment sequences.

Kevin Christian Wibisono, Yixin Wang

Department of Statistics University of Michigan, Ann Arbor

Causal Effect Estimation Causal Inference Causal Intervention

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Sep 27, 2026

Wasserstein Causal Forests for Distribution-Valued Outcomes

This paper proposes Wasserstein Causal Forests (WCF) for settings in which each unit's outcome is itself a probability distribution. This study also defines finite-grid transformed average and conditional average treatment effects, including a reference-distance contrast that asks whether treatment moves unit-level distributions toward a prespecified benchmark. Simulations cover null effects, location and shape changes, limited overlap, equal-mean but different laws, heterogeneous effects, multimodality, and structural zeros. WCF is most accurate on the conditional-law metric in most reported designs and sharply improves reference-effect estimation in the principal location-and-shape settings, but it is less accurate than the forest baselines for multimodal settings. WCF is applied to the famous Project STAR \citep{word1990state}, revealing that small classes alter more than the mean: they raise within-grade mathematics achievement by

0.161

standard deviations on average (SE

0.028

); but the gain is not a uniform location shift, it is larger in the upper part of the classroom score distribution (

+0.179

at the ninetieth percentile versus

+0.091

at the tenth) and, in descriptive stratum estimates, largest in the schools serving the most economically disadvantaged students (highest free-lunch quartile,

+0.262

, versus

+0.092

+0.164

elsewhere), while overall dispersion is essentially unchanged.

Hugo Gobato Souto

University of São Paulo

Causal Effect Estimation Conditional Average Treatment Effect Estimation Conditional Distribution Estimation

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Abstract

Policy Learning

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From Unsupervised Subgroups to Hypothetical State-Intervention Policies: An Evaluation of Selected Subgrouping Methods in Observational Health Data

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