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Prediction-Powered Data Fusion for Treatment Effect Estimation

Yonghan Jung, Shu Yang

Year: 2026Venue: arXiv preprintArea: stat.MLType: PreprintEmbeddings: 129

Intelligence

Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 95%

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Summary

This paper proposes a framework for fusing small, unbiased Randomized Controlled Trials (RCTs) with large, potentially confounded Observational Studies (OBS) to improve the precision of treatment effect estimation. The authors introduce AIPW-Fusion (AIPWF) for Average Treatment Effect (ATE) estimation and two learners, DR-Fusion (DRF) and R-Fusion, for Conditional Average Treatment Effect (CATE) estimation. These methods preserve the unbiasedness of RCT-based estimates while borrowing statistical power from OBS data without requiring specific assumptions about the OBS confounding structure or modeling the confounding function directly.

Entities (11)

OBS · data-source · 99%RCT · data-source · 99%AIPW-Fusion · estimator · 98%DR-Fusion · estimator · 98%R-Fusion · estimator · 98%ATE · metric · 95%CATE · metric · 95%Shu Yang · person · 95%Yonghan Jung · person · 95%ACIC 2016 · dataset · 90%IHDP · dataset · 90%

Relation Signals (13)

AIPW-Fusion → estimates → ATE

confidence 98% · we build an ATE estimator, AIPW-Fusion

DR-Fusion → estimates → CATE

confidence 98% · two CATE learners, DR-Fusion and R-Fusion

R-Fusion → estimates → CATE

confidence 98% · two CATE learners, DR-Fusion and R-Fusion

Shu Yang → authored → Prediction-Powered Data Fusion for Treatment Effect Estimation

confidence 95% · and Shu Yang2

Yonghan Jung → authored → Prediction-Powered Data Fusion for Treatment Effect Estimation

confidence 95% · Prediction-Powered Data Fusion for Treatment Effect Estimation Yonghan Jung1

DR-Fusion → fuses → RCT

confidence 95% · DRF and RF borrow through the same two channels as AIPWF

DR-Fusion → fuses → OBS

confidence 95% · DRF and RF borrow through the same two channels as AIPWF

R-Fusion → fuses → RCT

confidence 95% · DRF and RF borrow through the same two channels as AIPWF

R-Fusion → fuses → OBS

confidence 95% · DRF and RF borrow through the same two channels as AIPWF

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Abstract

Abstract:Randomized controlled trials (RCTs) identify treatment effects without confounding but are often small, whereas observational studies (OBS) are large but may be confounded. Many estimators combining a small RCT with a large OBS have been developed for the average treatment effect (ATE) and the conditional ATE (CATE). However, existing ATE estimators either make assumptions on the OBS or do not borrow enough power from them. The CATE has been studied less than the ATE. Existing CATE methods either assume the OBS are unconfounded, rely on a model of the confounding function, or accept bias in exchange for lower variance. We therefore propose a framework that, without special assumptions on the OBS, fuses the OBS and the RCT by preserving the unbiasedness of RCT-based estimation while borrowing power from the large OBS to boost precision. Applying this principle, we build an ATE estimator, AIPW-Fusion, with closed-form weights and confidence intervals, and two CATE learners, DR-Fusion and R-Fusion. Experiments corroborate our findings.

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Prediction-Powered Data Fusion for Treatment Effect Estimation Yonghan Jung1 and Shu Yang2 1 University of Illinois Urbana-Champaign yonghan@illinois.edu 2 North Carolina State University syang24@ncsu.edu October 8, 2026 Abstract Randomized controlled trials (RCTs) identify treatment effects without confounding but are often small, whereas observational studies (OBS) are large but may be confounded. Many estimators combining a small RCT with a large OBS have been developed for the average treatment effect (ATE) and the conditional ATE (CATE). However, existing ATE estimators either make assumptions on the OBS or do not borrow enough power from them. The CATE has been studied less than the ATE. Existing CATE methods either assume the OBS are unconfounded, rely on a model of the confounding function, or accept bias in exchange for lower variance. We therefore propose a framework that, without special assumptions on the OBS, fuses the OBS and the RCT by preserving the unbiasedness of RCT-based estimation while borrowing power from the large OBS to boost precision. Applying this principle, we build an ATE estimator, AIPW-Fusion, with closed-form weights and confidence intervals, and two CATE learners, DR-Fusion and R-Fusion. Experiments corroborate our findings. Contents 1 Introduction 1.1 Related Work 2 Background and Setting 3 Fusion for the ATE 3.1 Estimator, Variance, and Oracles 3.2 Estimating the Density Ratio 3.3 Algorithm and Confidence Intervals 4 Fusion for the CATE 4.1 CATE Risks 4.2 Fitting and Selecting Coefficients 4.3 Linear Sieves: Risk and Calibration 5 Experiments 5.1 Designs 5.2 Estimators and Metrics 5.3 Main Results 5.4 Bias and Observational Sample Size 5.5 Ablations and Further Designs 6 Conclusion References A Proofs B Additional Simulation Results 1 Introduction Randomized controlled trials (RCTs) are the gold standard for identifying treatment effects but are often small, whereas observational studies (OBS) are large but may be confounded. This paper fuses the two to estimate the trial-population conditional and average treatment effects. Transportability and data-fusion methods (Bareinboim and Pearl, 2016; Colnet et al., 2024; Lee et al., 2022; Shi et al., 2023; Jung et al., 2023b; Jung et al., 2023a; Lee et al., 2023; Wu and Yang, 2023; Jung et al., 2024) identify and estimate effects that no single source identifies, such as effects in a new target population or of joint treatments. Most of them do not address cases where the RCT already identifies the target but is too small to estimate it precisely. Our work is closer to robust borrowing methods. For the ATE, they improve a trial estimate by combining it with OBS estimates or OBS-based predictions (Yang and Ding, 2020; Rosenman et al., 2023; Gao et al., 2025; Gagnon-Bartsch et al., 2023; De Bartolomeis et al., 2025), or by prediction-powered inference (Angelopoulos et al., 2023; Angelopoulos et al., 2026; Demirel et al., 2024, PPI;). Compared with ATE fusion, only a few fusion methods have been developed for CATE estimation, which use the trial to correct the bias of OBS-based estimates or the OBS to reduce the variance of trial-based ones (Kallus et al., 2018; Cheng and Cai, 2021; Wu and Yang, 2022; Li et al., 2022; Yang et al., 2023; Yang et al., 2025; Hatt et al., 2022; Gu et al., 2023; Asiaee et al., 2025). Section 1.1 compares them with ours. In this paper, we enhance treatment effect estimators from a small RCT with large, potentially heavily confounded OBS data while keeping them unbiased. Specifically, our contributions are: (i) An ATE estimator, AIPW-Fusion (AIPWF), that stays unbiased under any OBS confounding and combines two approaches, leveraging the OBS regressions plugged into the RCT’s AIPW score and a PPI correction, with a valid interval under an estimated ratio (Sec. 3). (i) Two CATE learners, DR-Fusion (DRF) and R-Fusion (RF), that keep the trial’s CATE target under any OBS confounding, borrow without modeling the confounding function, and whose data-driven borrowing provably does almost no worse than the trial alone (Sec. 4). (i) Experiments on synthetic data and the IHDP and ACIC 2016 benchmarks, where our estimators outperform trial-only estimation, PPI, and naive pooling (Sec. 5). Code is available at https://github.com/CausalDataScience/DataFusionPPI. Section 2 sets up the problem. Sections 3 and 4 develop the ATE estimator and the CATE learners. Section 5 reports the experiments. The appendix contains the proofs and additional results. 1.1 Related Work Transportability and data fusion. Unlike the transportability and data-fusion methods cited in Section 1, we target an effect that the trial already identifies and use the OBS only to estimate it more precisely. ATE fusion. Table 1 compares the ATE fusion methods. Yang and Ding (2020) [YD20] combine a large main sample with unmeasured confounders and a smaller validation sample with all confounders, using the main sample as a control variate that improves the validation-sample estimator. Rosenman et al. (2023) [R23] shrink the experimental estimates toward the observational ones by Stein-type estimators, with lower risk than the experimental estimator under finite-sample conditions. Gao et al. (2025) [G25] borrow external controls selectively, penalizing their bias to keep a comparable subset, and attain the efficiency bound when the controls are comparable. These three methods borrow OBS outcomes, and Table 1 marks them as not unbiased under every OBS confounding. Lee et al. (2023) [L23] weight the trial to balance its covariates with an observational target population, and augment the weights with outcome models for double robustness; they use only the OBS covariates. The remaining three methods keep the trial estimate unbiased under any OBS confounding. Gagnon-Bartsch et al. (2023) [GB23] use predictions from a model trained on auxiliary observational data as a covariate in the trial’s estimator, which stays unbiased because the propensity is known. De Bartolomeis et al. (2025) [DB25] combine several AIPW estimators of the trial, each with a different outcome regression from external predictors. Demirel et al. (2024) [D24] apply PPI (Angelopoulos et al., 2023) to RCT–OBS fusion to generalize the ATE to a target population; their gain shrinks as confounding grows. AIPWF combines the two channels that [GB23, DB25] and [D24] use separately: it blends the OBS outcome regressions into the trial’s AIPW score and adds a PPI correction averaged over the OBS covariates, choosing both coefficients jointly in closed form, with a valid interval under an estimated density ratio (Sec. 3). Table 1: ATE fusion methods. A dash means that the property does not apply. The second column is evaluated for a trial and an OBS with different assignment mechanisms. [DB25] borrows predictions from foundation models trained on external data rather than from an OBS. [D24] averages its predictions over a target covariate sample and, in its appendix, also uses the augmented regression in a doubly robust score. Method Borrows OBS outcomes Unbiased under any OBS confounding OBS regressions in the trial’s AIPW score OBS covariates used to reduce the trial variance [YD20] Yang and Ding (2020) ✓ × × × [L23] Lee et al. (2023) × – × × [R23] Rosenman et al. (2023) ✓ × × × [G25] Gao et al. (2025) ✓ × × × [GB23] Gagnon-Bartsch et al. (2023) ✓ ✓ ✓ × [DB25] De Bartolomeis et al. (2025) ✓ ✓ ✓ × [D24] Demirel et al. (2024) ✓ ✓ × ✓ AIPWF (ours) ✓ ✓ ✓ ✓ CATE fusion. Table 2 compares the CATE fusion methods by what they assume about the OBS. Li et al. (2022) [L22] learn the CATE directly from multiple sources with a doubly robust pseudo-outcome and information-aware weights, and keep the trial’s target only if the OBS are unconfounded. Four methods model the OBS bias. Kallus et al. (2018) [K18] learn a CATE model on the OBS and correct its hidden confounding with a parametric correction fitted on the trial. Wu and Yang (2022) [WY22] jointly learn the CATE and the confounding function with an orthogonal R-loss, at least as efficient as the trial-only estimator. Yang et al. (2025) [Y25] identify the CATE and the confounding function jointly under structural models and derive semiparametric efficient integrative estimators. Hatt et al. (2022) [H22] learn a shared representation from the OBS and trial-specific structure from the RCT, with the OBS bias in a restricted (linear) class. These four methods keep the trial’s target only if the OBS bias follows their model. Three methods trade bias for variance. Cheng and Cai (2021) [C21] combine trial and OBS CATE estimators linearly, with weights that minimize an estimated mean squared error and move toward the trial estimator when bias is detected. Yang et al. (2023) [Y23] test whether the real-world data are compatible with the trial and integrate them only if so, under a structural model with known effect modifiers. Gu et al. (2023) [G23] build shrinkage trees that weight trial and OBS estimates within each leaf. These three methods keep the trial’s target only up to a bias that they accept for lower variance under weak confounding. Asiaee et al. (2025) [A25] calibrate OBS outcome predictions to the trial and use them to augment a trial-based CATE estimator, which keeps the trial’s target under any OBS confounding. DRF and RF borrow through the same two channels as AIPWF inside the DR- and R-learner losses, and select the coefficients by held-out validation (Sec. 4). Like [A25], they keep the trial’s target under any OBS confounding; unlike [A25], they need no model of the discrepancy between the trial and OBS-based CATEs. Table 2: CATE fusion methods. In the last column, × means that the trial’s target is kept only if the OBS are unconfounded [L22], only if the OBS bias follows the method’s model [K18, WY22, Y25, H22], or only up to a bias traded for variance under weak confounding [C21, Y23, G23]. Method Borrows when the OBS is confounded No model of the OBS bias Flexible CATE class Orthogonal loss or score Keeps the trial’s target under any OBS confounding [K18] Kallus et al. (2018) ✓ × ✓ × × [C21] Cheng and Cai (2021) × ✓ ✓ ✓ × [WY22] Wu and Yang (2022) ✓ × ✓ ✓ × [L22] Li et al. (2022) × ✓ ✓ ✓ × [Y23] Yang et al. (2023) × ✓ × ✓ × [Y25] Yang et al. (2025) ✓ × × ✓ × [H22] Hatt et al. (2022) ✓ × ✓ × × [G23] Gu et al. (2023) ✓ ✓ ✓ × × [A25] Asiaee et al. (2025) ✓ × ✓ ✓ ✓ DRF/RF (ours) ✓ ✓ ✓ ✓ ✓ 2 Background and Setting XXAAYY (a) RCT XXAAYYUU (b) OBS Figure 1: Causal diagrams; U is unmeasured. Fig. 1 describes the data-generating processes for RCT and OBS. Each unit has covariates X∈X , a binary treatment A∈0,1A∈\0,1\, and an outcome Y; PRP_R and POP_O denote the RCT and OBS distributions of V=(X,A,Y)V=(X,A,Y). In the RCT, treatment A depends only on the covariates through a known propensity e⁡(x)e(x). In the OBS, an unmeasured confounder U affects both treatment A and outcome Y. With potential outcomes Y⁡(1)Y(1) and Y⁡(0)Y(0), the targets are the RCT-population conditional and average treatment effects τ0(x) _0(x) and θ0 _0. Tables 3 and 4 collect the notation. Assumption 1 (Trial identification). (i) Y=Y⁡(A)Y=Y(A) a.s.; (i) Y⁡(0),Y⁡(1)⟂A|X\Y(0),Y(1)\ A X under PRP_R; (i) e is known and bounded away from 00 and 11. Each source S∈R,OS∈\R,O\ is split into three i.i.d. samples from PSP_S: nuisance, tuning, and evaluation samples DSnuisD_S^nuis, DStuneD_S^tune, and DSevalD_S^eval (Table 3). The three roles can be rotated across the samples and the resulting estimates averaged (cross-fitting), as in Sec. 5; our exact results are for a single split. Table 3: Notation for the ATE (Secs. 2 and 3). τ0(x)≜R[Y⁡(1)−Y⁡(0)∣X=x] _0(x) _R[Y(1)\!-\!Y(0)\! \!X\!=\!x] CATE (target) θ0≜R[τ0(X)] _0 _R[ _0(X)] ATE (target) e⁡(x)≜PR(A=1∣X=x)e(x) P_R(A=1 X=x) RCT propensity r0≜dPRX/dPOXr_0 dP_R^X/dP_O^X; r r density ratio; estimate DSnuisD_S^nuis, DStuneD_S^tune, DSevalD_S^eval samples by role n, N RCT, OBS eval. sizes ℙR,nP_R,n, ℙO,NP_O,N eval.-sample means r^RZ r_RZ, r^CLS r_CLS base ratios (Sec. 3.2) f(r~)≜O[r~f]−R[f] imb_f( r) _O[ rf]-E_R[f] imbalance along f (Prop. 1) μS(x,a)≜S[Y∣X=x,A=a] _S(x,a) _S[Y\! \!X\!=\!x,A\!=\!a] outcome regression μ^S μ_S fit of μS _S on DSnuisD_S^nuis g^≜μ^O(⋅,1)−μ^O(⋅,0) g μ_O(·,1)- μ_O(·,0) OBS effect prediction λ OBS weight in score ω PPI-correction weight μ^λ≜(1−λ)μ^R+λμ^O μ_λ (1-λ) μ_R+λ μ_O blended regressor φ⁡(V,q) (V;q) AIPW score (Eq. (1)) Zλ≜φ⁡(V,μ^λ)Z^λ (V; μ_λ); Z0Z^0 fused score; RCT score Δ≜Z1−Z0 Z^1-Z^0 score difference Assumption 2 (Sample separation). The six samples DRnuisD_R^nuis, DRtuneD_R^tune, DRevalD_R^eval, DOnuisD_O^nuis, DOtuneD_O^tune, and DOevalD_O^eval are mutually independent. All displayed moments are finite; stronger moment conditions are stated where needed. We consider two settings for the covariate distributions PRXP_R^X and POXP_O^X: the RCT and OBS share the same domain, or they come from heterogeneous domains. In both, the OBS may be confounded. Whenever PRXP_R^X is absolutely continuous with respect to POXP_O^X, written PRX≪POXP_R^X P_O^X (every covariate region with positive probability in the RCT also has positive probability in the OBS), the density ratio r0r_0 exists. Assumption 3 (Covariate domains). One of the following holds. (i) (Same domain.) PRX=POXP_R^X=P_O^X, so r0≡1r_0≡ 1. (i) (Heterogeneous domains.) PRX≪POXP_R^X P_O^X, so r0r_0 exists; it need not be known. Lemma 1 (RCT-valid AIPW score). For any fixed measurable q:×0,1→ℝq:X×\0,1\ , the AIPW score, with e evaluated at X, φ⁡(V,q)≜q⁡(X,1)−q⁡(X,0)+AeY−q⁡(X,1)−1−A1−eY−q⁡(X,0) (V;q) q(X,1)-q(X,0)+ Ae\Y-q(X,1)\- 1-A1-e\Y-q(X,0)\ (1) satisfies R[φ⁡(V,q)∣X]=τ0(X)E_R[ (V;q) X]= _0(X) under Assumption 1 (Appendix A). Hence the RCT-only AIPW estimator ℙR,nZ0P_R,nZ^0 is conditionally unbiased for θ0 _0 given DRnuisD_R^nuis. 3 Fusion for the ATE In this section, we build the ATE estimator, called AIPWF, that uses a small RCT sample and borrows power from possibly confounded OBS. 3.1 Estimator, Variance, and Oracles Our estimator starts from Lemma 1: ℙR,nZ0P_R,nZ^0 is the RCT-only AIPW estimator, and ℙR,nZλP_R,nZ^λ is also unbiased for any λ∈ℝλ ; that is, we can use the blended regressor μ^λ μ_λ (Table 3) rather than the RCT-only regressor μ^R μ_R to borrow power from the OBS (Gagnon-Bartsch et al., 2023; De Bartolomeis et al., 2025). Augmenting ℙR,nZλP_R,nZ^λ with the PPI correction (Angelopoulos et al., 2023; Angelopoulos et al., 2026), we define the AIPWF estimator: Definition 1 (AIPWF estimator). For fixed r:→[0,∞)r:X→[0,∞) and (λ,ω)(λ,ω), the AIPWF estimator of θ0 _0 is θ^r(λ,ω)≜ℙR,nZλ+ω⁡ℙO,N(rg^)−ℙR,ng^, θ_r(λ,ω) _R,nZ^λ+ω\P_O,N(r g)-P_R,n g\, (2) with r and g g evaluated at X. Equivalently, θ^r(λ,ω) θ_r(λ,ω) is a sum of two independent sample means, an RCT term and an OBS term: θ^r(λ,ω)=ℙR,n(Zλ−ωg^)+ℙO,N(ωrg^). θ_r(λ,ω)=P_R,n(Z^λ-ω g)+P_O,N(ω r g). (3) At (λ,ω)=(0,0)(λ,ω)=(0,0) this is the RCT-only AIPW estimator ℙR,nZ0P_R,nZ^0 (Robins et al., 1994). The coefficient λ sets the blended regressor μ^λ μ_λ in the AIPW score, and ω scales a transported control variate built from g g (Fig. 2(a)). In this subsection, the density ratio is exact, r=r0r=r_0; Sec. 3.2 estimates it. Theorem 1 (Mean and variance). For every fixed (λ,ω)(λ,ω) and given the fitted regressions μ^R μ_R and μ^O μ_O, ⁡[θ^r0(λ,ω)]=θ0, [ θ_r_0(λ,ω)]= _0, (4) where E and VarVar are over DRevalD_R^eval and DOevalD_O^eval. Let A A ≜VarR(Δ), _R( ), C C ≜CovR(Z0,Δ), _R(Z^0, ), B B ≜VarR(g^)+nNVarO(r0g^), _R( g)+ nNVar_O(r_0 g), D D ≜CovR(Z0,g^). _R(Z^0, g). If these moments are finite, then nVarθ^r0(λ,ω)=VarR(Z0)+(Aλ2+2Cλ)+(Bω2−2Dω).nVar\ θ_r_0(λ,ω)\=Var_R(Z^0)+(Aλ^2+2Cλ)+(Bω^2-2Dω). (5) If A,B>0A,B>0, its unique minimizer is λ⋆=−C/Aandω⋆=D/B. λ =-C/A ω =D/B. (6) in the AIPW scoreμ^R μ_Rμ^O μ_Oblend by λ scoreZλZ^λ on RCTg gOBS mean −-RCT meanof g g++θ θ×ω×\, correction (a) Where λ and ω act (schematic) λω−D2/B-D^2/B−C2/A-C^2/ARCT onlyPPI++λ onlyAIPWF(λ⋆,ω⋆)(λ ,ω ) (b) Quadratic in (λ,ω)(λ,ω) (schematic) r^RZ r_RZr^CLS r_CLSr^base r_basecalibrationr^BAL r_BALr^base r_baser^BAL r_BAL±δ^k± δ_k00f1f_1f2f_2⋯·sfKf_K (c) Ratio calibration (schematic) Figure 2: ATE fusion with AIPWF (Def. 1). (a) Where λ and ω enter θ θ. (b) Contours of Eq. (5). (c) Ratio calibration (Def. 2): bars are the imbalances ℙOnuis[r^fk]−ℙRnuis[fk]P_O^nuis[ rf_k]-P_R^nuis[f_k]. Corollary 1.1 (Population oracles). Under the conditions of Thm. 1 with A,B>0A,B>0, let V~(λ,ω)≜nVarθ^r0(λ,ω) V(λ,ω) nVar\ θ_r_0(λ,ω)\, with RCT-only, PPI++, and AIPWF oracle values V~RCT≜V~(0,0) V_RCT V(0,0), V~PPI+⁣+≜V~(0,ω⋆) V_PPI^++ V(0,ω ), and V~AIPWF≜V~(λ⋆,ω⋆) V_AIPWF V(λ ,ω ). Then V~AIPWF=V~RCT−C2/A−D2/B≤V~PPI+⁣+=V~RCT−D2/B≤V~RCT. V_AIPWF= V_RCT-C^2/A-D^2/B≤ V_PPI^++= V_RCT-D^2/B≤ V_RCT. In Fig. 2(b), PPI++ moves from the RCT-only origin along the ω axis, and AIPWF moves along both axes. Remark 1 shows when each axis helps. Remark 1 (When each channel helps). Since D=CovR(τ0(X),g^(X))D=Cov_R( _0(X), g(X)), the Cauchy–Schwarz inequality gives D2≤VarRτ0(X)VarR(g^)≤VarRτ0(X)BD^2 _R\ _0(X)\Var_R( g) _R\ _0(X)\\,B. Hence the reduction D2/BD^2/B that ω brings in Cor. 1.1 is at most VarRτ0(X)Var_R\ _0(X)\, with or without λ, because the variance is separable in (λ,ω)(λ,ω). Since R[Zλ∣X]=τ0(X)E_R[Z^λ X]= _0(X) for every λ (Lemma 1), VarR(Zλ)=RVarR(Zλ∣X)+VarRτ0(X)Var_R(Z^λ)=E_R\Var_R(Z^λ X)\+Var_R\ _0(X)\, so this cap is the effect-heterogeneity part of the score variance. It is the fraction VarRτ0(X)/VarR(Z0)Var_R\ _0(X)\/Var_R(Z^0) of V~RCT=VarR(Z0) V_RCT=Var_R(Z^0), and the larger fraction VarRτ0(X)/VarR(Zλ⋆)Var_R\ _0(X)\/Var_R(Z^λ ) of the variance VarR(Zλ⋆)=V~RCT−C2/AVar_R(Z^λ )= V_RCT-C^2/A that remains after λ. Conversely, VarR(Zλ⋆)≥VarRτ0(X)Var_R(Z^λ ) _R\ _0(X)\ gives C2/A≤RVarR(Z0∣X)C^2/A _R\Var_R(Z^0 X)\: λ acts only on the within-covariate part of the score variance, and its reduction is zero when μ^R μ_R equals the trial regression (Appendix A). Thus ω can help much only when effect heterogeneity is a large part of the score variance, and λ helps only when μ^R μ_R differs from the trial regression, for example when it is fitted on a small trial. 3.2 Estimating the Density Ratio So far we have assumed r=r0r=r_0. This subsection gives the density-ratio estimator used in AIPWF. AIPWF is unbiased under the correct ratio (Thm. 1), but a misspecified ratio introduces bias. Proposition 1 (Drift under a candidate ratio). For a candidate ratio r~ r, let f(r~)≜O[r~f]−R[f]=O[(r~−r0)f] imb_f( r) _O[ rf]-E_R[f]=E_O[( r-r_0)f] (7) be its imbalance along a feature f. Then, for fixed (λ,ω)(λ,ω) and given the fitted regressions μ^R μ_R and μ^O μ_O, ⁡[θ^r~(λ,ω)]−θ0=ωg^(r~).E[ θ_ r(λ,ω)]- _0=ω\, imb_ g( r). (8) Under Assumption 3(i) we set r^≡1 r≡ 1. Under Assumption 3(i) the ratio must be estimated, and two characterizations of r0r_0 give the estimators. Proposition 2 (Characterizations of r0r_0). Under Assumption 3(i): (i) (Reweighting.) O[r0h]=R[h]E_O[r_0h]=E_R[h] for every h∈L1(PRX)h∈ L^1(P_R^X), and no other weight does so for all bounded h. (i) (Riesz risk.) If r0∈L2(POX)r_0∈ L^2(P_O^X), then r0=arg⁡mina∈L2(POX)O[a2]−2R[a]r_0= _a∈ L^2(P_O^X)\E_O[a^2]-2E_R[a]\. Item (i) follows from item (i), since O[a2]−2R[a]=O[(a−r0)2]−O[r02]E_O[a^2]-2E_R[a]=E_O[(a-r_0)^2]-E_O[r_0^2]. Two base ratios are standard. The Riesz estimator r^RZ r_RZ minimizes the empirical Riesz risk ℙOnuis(a2)−2ℙRnuis(a)P_O^nuis(a^2)-2P_R^nuis(a) of Prop. 2(i) over a linear class (Chernozhukov et al., 2022), with a ridge penalty. The classifier estimator is r^CLS≜(1−π^)/π^s^/(1−s^) r_CLS \(1- π)/ π\\, s/(1- s), where s^(x) s(x) is a fitted probability of trial membership given covariates x, trained on the pooled nuisance covariates, and π π is the trial share. By Prop. 1, an estimated ratio biases AIPWF only through one moment, g^(r^)=O[r^g^]−R[g^] imb_ g( r)=E_O[ r g]-E_R[ g]. A base ratio, such as r^RZ r_RZ or r^CLS r_CLS, approximates r0r_0 as a whole and leaves this moment nonzero in general. We therefore use calibration to achieve O[r^g^]−R[g^]=0E_O[ r g]-E_R[ g]=0. Definition 2 (Calibrated balancing ratio). Given a base ratio r^base>0 r_base>0 and features f⁡(X)=(f1(X),…,fK(X))f(X)=(f_1(X),…,f_K(X)), the calibrated ratio is r^BAL(x)≜r^base(x)eξ^⊤f(x)ℙOnuisr^base(X)eξ^⊤f(X), r_BAL(x) r_base(x)\,e ξ f(x)P_O^nuis\ r_base(X)e ξ f(X)\, where ξ ξ solves the balance equations, for all k≤Kk≤ K, ℙOnuisr^BAL(X)fk(X)=ℙRnuisfk(X).P_O^nuis\ r_BAL(X)f_k(X)\=P_R^nuis\f_k(X)\. (9) Among ratios that average to one on the OBS and satisfy Eq. (9), r^BAL r_BAL is the closest to r^base r_base in Kullback–Leibler divergence (Hainmueller, 2012). For AIPWF, f=g^f= g suffices and is balanced exactly. Fig. 2(c) illustrates it: calibration drives each feature’s imbalance to zero, or within its tolerance δ^k δ_k. Computing the calibrated ratio. Eq. (9) is the stationarity condition of the convex objective log⁡ℙOnuisr^baseeξ⊤f−ξ⊤ℙRnuis(f) _O^nuis\ r_basee^ξ f\-ξ P_R^nuis(f), whose minimizer gives the Kullback–Leibler projection of r^base r_base onto the ratios that satisfy Eq. (9) and average to one. For one feature, the stationarity condition says that the calibrated OBS mean of f equals ℙRnuis(f)P_R^nuis(f); its derivative in ξ is the calibrated variance, which is nonnegative, so the root is unique when ℙRnuis(f)P_R^nuis(f) lies strictly inside the range of f on DOnuisD_O^nuis. With many features (Secs. 4.3 and 5), we add the penalty ∑kδ^k|ξk| _k δ_k| _k| (approximate balance). The features f have no intercept, δ^k δ_k is the standard error of the kkth balance residual, estimated by δ^k2=Var^Onuis(r^basefk)/Nnuis+Var^Rnuis(fk)/nnuis δ_k^2= Var_O^nuis( r_basef_k)/N_nuis+ Var_R^nuis(f_k)/n_nuis with sample variances on DSnuisD_S^nuis of sizes nnuisn_nuis and NnuisN_nuis, and ξ ξ minimizes the objective below. The penalized dual objective log⁡ℙOnuisr^baseeξ⊤f−ξ⊤ℙRnuis(f)+∑kδ^k|ξk| _O^nuis\ r_basee^ξ f\-ξ P_R^nuis(f)+ _k δ_k| _k| (10) is convex in ξ. At a minimizer ξ ξ, the subgradient condition is ℙOnuis(r^BALfk)−ℙRnuis(fk)+δ^kςk=0P_O^nuis( r_BALf_k)-P_R^nuis(f_k)+ δ_k _k=0 with ςk∈∂|ξ^k| _k∈∂| ξ_k|. Hence the kkth residual is at most δ^k δ_k in absolute value, with equality when ξ^k≠0 ξ_k≠ 0. Normalization gives ℙOnuis(r^BAL)=1P_O^nuis( r_BAL)=1. 3.3 Algorithm and Confidence Intervals This subsection gives an algorithm for the AIPWF estimate and its confidence interval. We restrict (λ,ω)(λ,ω) to [0,1]2[0,1]^2, which contains the RCT-only estimator at (0,0)(0,0) and caps the PPI correction at its full weight ω=1ω=1; λ∈[0,1]λ∈[0,1] keeps μ^λ μ_λ a convex blend of μ^R μ_R and μ^O μ_O. Since Thm. 1 needs A,B>0A,B>0 for the oracle coefficients −C/A-C/A and D/BD/B, we also prespecify thresholds ϵA,ϵB>0 _A, _B>0 and require A^>ϵA A> _A and B^>ϵB B> _B. With Proj[0,1]Proj_[0,1] the projection onto [0,1][0,1], we select λ λ ≜Proj[0,1](−C^/A^) 1A^>ϵA, _[0,1](- C/ A)\,1\ A> _A\, ω ω ≜Proj[0,1](D^/(B^+B^cal)) 1B^>ϵB. _[0,1]( D/( B+ B_cal))\,1\ B> _B\. (11) Here B^cal≜n⁡Var^Rnuis(g^)/nnuis+Var^Onuis(r^g^)/Nnuis B_cal n\ Var_R^nuis( g)/n_nuis+ Var_O^nuis( r g)/N_nuis\, or 00 if r^≡1 r≡ 1, is the calibration noise of Thm. 2(i). Algorithm 1 uses the three samples of Assumption 2 in turn: it fits the nuisances, with r r calibrated by Def. 2 with g g among the features; selects (λ^,ω^)( λ, ω) by Eq. (11) from tuning-sample analogues of A,B,C,DA,B,C,D in Thm. 1; and evaluates Eq. (2), adding V^cal V_cal (Thm. 2) to V V. Algorithm 1 AIPWF Input: DSnuis,DStune,DSevalD_S^nuis,D_S^tune,D_S^eval for S∈R,OS∈\R,O\; ϵA,ϵB>0 _A, _B>0. Output: θ^≜θ^r^(λ^,ω^) θ θ_ r( λ, ω), V V, and V^cal V_cal. 3: 3: Fit μ^R,μ^O,g^,r μ_R, μ_O, g, r and form B^cal B_cal on DSnuisD_S^nuis (r^≡1 r≡ 1 under Assumption 3(i)) 4: 4: Form A^,B^,C^,D A, B, C, D on DStuneD_S^tune; select (λ^,ω^)( λ, ω) by Eq. (11) 5: 5: On DSevalD_S^eval, compute θ θ by Eq. (2) with r=r^r= r 6: 6: Return θ^=θ^r^(λ^,ω^) θ= θ_ r( λ, ω) and V^≜Var^R,eval(Zλ^−ω^g^)/n+Var^O,eval(ω^r^g^)/N V Var_R,eval(Z λ- ω g)/n+ Var_O,eval( ω r g)/N, and V^cal V_cal (Thm. 2) Remark 2 (Adaptive reuse). Algorithm 1 selects (λ^,ω^)( λ, ω) on the tuning samples and evaluates θ θ on separate samples, because selecting and evaluating on the same samples can bias the estimate. If X is degenerate, Z0=0Z^0=0, and Δi _i are i.i.d. Rademacher, every fixed λ centers λℙR,nΔ _R,n , but λ^=sign⁡(ℙR,nΔ) λ=sign(P_R,n ) yields |ℙR,nΔ||P_R,n | with strictly positive mean. An analogous construction applies to ω under a common marginal. Bias comes only from reuse. Thm. 2(i) uses the following high-level conditions. Assumption 4 (Conditions for Thm. 2(i)). n,N→∞n,N→∞ with n/Nn/N, nnuis/n_nuis/n, and Nnuis/N_nuis/N converging; ω^→pω¯ ω _p ω; (ℙOnuis−O)(r^g^)=(ℙOnuis−O)(r¯g¯)+op(Nnuis−1/2)(P_O^nuis-E_O)( r g)=(P_O^nuis-E_O)( r g)+o_p(N_nuis^-1/2) for fixed r¯,g¯ r, g with g^→g¯ g→ g in L2(PRX)L^2(P_R^X); for some η>0η>0, (2+η)(2+η)-th moments of Zλ^−ω^g^Z λ- ω g, ω^r^g ω r g, g¯ g, and r¯g¯ r g bounded uniformly in n; n⁡(Vn+Vcal)n(V_n+V_cal) converging to a positive limit with Vcal≜ω¯2VarR(g¯)/nnuis+VarO(r¯g¯)/NnuisV_cal ω^2\Var_R( g)/n_nuis+Var_O( r g)/N_nuis\; and (V^+V^cal)/(Vn+Vcal)→p1( V+ V_cal)/(V_n+V_cal) _p1. Theorem 2 (Confidence interval of the ATE estimator). Let θ^=θ^r^(λ^,ω^) θ= θ_ r( λ, ω), let g imb_ g be as in Prop. 1, and let Vn≜VarR(Zλ^−ω^g^)/n+ω^2VarO(r^g^)/N V_n _R(Z λ- ω g)/n+ ω^2Var_O( r g)/N (12) be the variance of θ θ given the nuisance and tuning samples, i.e., the sum of the variances of the two terms of Eq. (3). (i) (Mean.) Given the nuisance and tuning samples, ⁡[θ^]=θ0+ω^g^(r^)and⁡[V^]=Vn; [ θ]= _0+ ω\, imb_ g( r) [ V]=V_n; (13) in particular, θ θ is unbiased when g^(r^)=0 imb_ g( r)=0 (e.g., r^=r0 r=r_0, including r^≡r0≡1 r≡ r_0≡ 1 under Assumption 3(i), or r r perfectly calibrated, O[r^g^]=R[g^]E_O[ r g]=E_R[ g]). (i) (Coverage.) Let r^=r^BAL r= r_BAL with g g among its features, and let V^cal≜ω^2B^cal/n V_cal ω^2 B_cal/n with B^cal B_cal from Eq. (11). Under Assumption 4, limn→∞P⁡(θ0∈[θ^±z1−α/2(V^+V^cal)1/2])=1−α. _n→∞P ( _0∈[ θ± z_1-α/2( V+ V_cal)^1/2] )=1-α. (14) Item (i) shows that r r moves the mean only through the drift ω^g^(r^) ω\, imb_ g( r), while V V stays unbiased for VnV_n. Calibration does not make the drift zero. Eq. (9) holds on the nuisance samples, so g^(r^)=(ℙRnuis−R)g^−(ℙOnuis−O)(r^g^) imb_ g( r)=(P_R^nuis-E_R) g-(P_O^nuis-E_O)( r g), a difference of nuisance-sample averages. It has mean approximately zero and standard deviation of order nnuis−1/2n_nuis^-1/2, the order of the standard error. Thus, with r^BAL r_BAL, item (i) gives a nonzero bias ω^g^(r^) ω\, imb_ g( r) given the nuisance samples. Averaged over the nuisance samples, this bias is zero to first order, so θ θ is unbiased to first order. Item (i) therefore treats it as noise and adds its variance V^cal V_cal. It follows from the Lyapunov central limit theorem (van der Vaart, 1998, Prop. 2.27) for the four independent nuisance and evaluation samples. 4 Fusion for the CATE In this section, we develop two CATE learners that borrow from the OBS in the same two ways as AIPWF: DRF, built on the AIPW pseudo-outcome of the DR-learner (Kennedy, 2023), and RF, built on the R-learner loss (Nie and Wager, 2021). Throughout this section, Assumptions 1 and 2 hold, either Assumption 3(i) (r0≡1r_0≡ 1) or Assumption 3(i) holds, and λ∈[0,1]λ∈[0,1], so the blended regressor μ^λ μ_λ is a convex combination of μ^R μ_R and μ^O μ_O and R[Zλ∣X]=τ0(X)E_R[Z^λ X]= _0(X) (Lemma 1). Table 4 collects the CATE notation. Table 4: Notation for the CATE (Sec. 4); the notation of Table 3 carries over. ℛ0(t)≜R[t⁡(X)−τ0(X)2]R_0(t) _R[\t(X)\!-\! _0(X)\^2] CATE risk (Sec. 4.1) ∈DRF,RF mdl∈\DRF,RF\ learner mλ≜eμ^λ(⋅,1)+(1−e)μ^λ(⋅,0)m_λ e μ_λ(·,1)+(1-e) μ_λ(·,0) marginal regression κ _ mdl unit weight (Def. 3) ZλZ^λ_ mdl pseudo-outcome (Def. 3) ℙStuneP_S^tune average over DStuneD_S^tune ℛ^(t,λ,ω,r) R_ mdl(t;λ,ω,r) proxy risk (Eq. (17)) τ^(⋅,λ,ω) τ_ mdl(·;λ,ω) fitted learner (Sec. 4.2) =(λm,ωm)m=1MG=\( _m, _m)\_m=1^M candidate grid (Def. 4) ℛ^val(τ^m,ωm) R_val( τ_m; _m) validation score (Def. 4) ϱ(λ,ω) _ mdl(λ,ω) transport error (Thm. 4) ζ:→ℝpζ:X ^p sieve basis (Def. 5) ζ≜ζ⊤βT_ζ \ζ β\ sieve class (Def. 5) β^(λ,ω) β_ mdl(λ,ω) coefficients (Def. 5) ρ≥0ρ≥ 0 ridge penalty (Def. 5) H^(ω) H_ mdl(ω) Gram estimate (Prop. 4) ΓR≜R[ζζ⊤] _R _R[ζ ] sieve Gram matrix βp _p best sieve coefficients τp⋆≜ζ⊤βpτ _p ζ _p best sieve approximation nR≜|DRtune|n_R |D_R^tune|, nO≜|DOtune|n_O |D_O^tune| tuning sample sizes β¯(ω) β(ω) mean coefficients (Prop. 5) H¯ρ H_ρ mean Hessian (Prop. 5) ⁡(ω) bias(ω) squared bias (Prop. 5) (λ,ω)J_ mdl(λ,ω) variance (Thm. 5) 4.1 CATE Risks For a prespecified class ⊆L2(PRX)T L^2(P_R^X), the CATE risk of t∈t is ℛ0(t)≜R[t⁡(X)−τ0(X)2]. _0(t) _R[\t(X)- _0(X)\^2]. (15) The risk ℛ0R_0 requires the unknown τ0 _0. We therefore define a computable proxy: Definition 3 (DRF and RF proxy risks). For learner ∈DRF,RF mdl∈\DRF,RF\, let κDRF _DRF ≜1, 1, ZDRFλ Z_DRF^λ ≜Zλ, Z^λ, κRF _RF ≜(A−e)2e⁡(1−e), (A-e)^2e(1-e), ZRFλ Z_RF^λ ≜Y−mλA−e. Y-m_λA-e. The proxy risk of learner mdl is ℛ(t,λ)≜R[κZλ−t⁡(X)2].R_ mdl(t;λ) _R[ _ mdl\Z_ mdl^λ-t(X)\^2]. (16) With ℙStuneP_S^tune the average over DStuneD_S^tune, ω∈ℝω , and a candidate weight r, its empirical proxy risk is ℛ^(t,λ,ω,r)≜ℙRtune[κZλ−t⁡(X)2]+ωℙOtune[r(g^−t)2]−ωℙRtune[κ(g^−t)2]. R_ mdl(t;λ,ω,r) _R^tune[ _ mdl\Z_ mdl^λ-t(X)\^2]+ω\,P_O^tune[r( g-t)^2]-ω\,P_R^tune[ _ mdl( g-t)^2]. (17) DRF regresses the AIPW score ZDRFλ=ZλZ^λ_DRF=Z^λ, as in the DR-learner (Kennedy, 2023). RF is the R-learner (Nie and Wager, 2021) with each squared residual divided by e⁡(1−e)e(1-e), since Y−mλ−(A−e)t2=e⁡(1−e)κRFZRFλ−t2\Y-m_λ-(A-e)t\^2=e(1-e)\, _RF\Z_RF^λ-t\^2. When r=r0r=r_0, in expectation, each empirical risk equals ℛ0R_0 up to an additive constant that does not depend on t, so the two have the same minimizer. Theorem 3 (Exact CATE targets). Fix ω∈ℝω , t∈t , and r. Let Q(λ)≜ℛ(τ0,λ)Q_ mdl(λ) _ mdl( _0;λ) be the loss at the true CATE. Then for ∈DRF,RF mdl∈\DRF,RF\, ⁡[ℛ^(t,λ,ω,r)]=Q(λ)+ℛ0(t)+ωO[(r−r0)(g^−t)2].E[ R_ mdl(t;λ,ω,r)]=Q_ mdl(λ)+R_0(t)+ _O[(r-r_0)( g-t)^2]. Here Q(λ)Q_ mdl(λ) does not depend on t, and the last term is zero if r=r0r=r_0. In particular, if r=r0r=r_0, arg⁡mint∈⁡[ℛ^(t,λ,ω,r0)]=arg⁡mint∈ℛ0(t) _t E[ R_ mdl(t;λ,ω,r_0)]= _t R_0(t) (18) for all (λ,ω)(λ,ω). At r=r0r=r_0, borrowing from the OBS never shifts the target, and (λ,ω)(λ,ω) change only the variance of the fitted learner (Secs. 4.2 and 4.3). At r≠r0r≠ r_0, the last term can shift the minimizer when ω>0ω>0. The shift does not depend on λ, and for linear sieves calibration reduces it to nuisance-sampling error (Sec. 4.3). The proposed empirical risk ℛ R_ mdl is, in expectation, an orthogonal risk (Foster and Syrgkanis, 2023) with respect to the RCT regression μ^R μ_R. Proposition 3 (Orthogonality of the learning equation). For t,s∈t,s and any fixed r, not necessarily r0r_0, let Ψ(t)[s;μ^R,μ^O,r]≜−12Dt[ℛ^(t,λ,ω,r)][s] _ mdl(t)[s; μ_R, μ_O,r] - 12D_tE[ R_ mdl(t;λ,ω,r)][s] (19) be the learning equation of learner mdl at t in the direction s, where DtD_t is the directional derivative with respect to t; the learned CATE solves Ψ(t)[s]=0 _ mdl(t)[s]=0 for all s∈s . Then, for every h:×0,1→ℝh:X×\0,1\ and every hrh_r with O[hr]=0E_O[h_r]=0, writing Δh≜h⁡(⋅,1)−h⁡(⋅,0) h h(·,1)-h(·,0), the derivative of Ψ(t)[s] _ mdl(t)[s] at ε=0 =0 is (i) 00 along μ^R+εh μ_R+ h; (i) ωO[(r−r0)Δhs] _O[(r-r_0)\, h\,s] along μ^O+εh μ_O+ h; (i) ωO[hrs(g^−t)] _O[h_r\,s\,( g-t)] along r+εhrr+ h_r. Thus Ψ _ mdl is orthogonal to μ^R μ_R for every r, to μ^O μ_O only at r=r0r=r_0, and to r only if g^=t g=t. This is the orthogonality of the AIPW score and the R loss to their outcome regressions under a known propensity (Chernozhukov et al., 2018; Kennedy, 2023; Nie and Wager, 2021). At t=τ0t= _0, g^=t g=t requires no unmeasured confounding in the OBS. In practice, we set r=r^r= r, fitted on the nuisance samples by the classifier estimator r^CLS r_CLS (Sec. 3.2) or the calibrated ratio r^BAL r_BAL (Def. 2). in the pseudo-outcomeμ^R μ_Rμ^O μ_Oblend by λ loss ofZλ−tZ^λ_ mdl-tg gOBS loss −-RCT lossof g^−t g-t++τ τ_ mdl×ω×\, _tPPI correction of the loss (a) DRF and RF (schematic) nuisancetuningevaluationfit μ^R,μ^O,g^,r μ_R, μ_O, g, rτ^m← τ_m Eq. (17)ℛ^val(τ^m,ωm) R_val( τ_m; _m)Eq. (21)keep thebest mmfor each mmgrid =(λm,ωm)m=1MG=\( _m, _m)\_m=1^M,fixed in advance (b) Algorithm 2 (schematic) ⋆ onlyλω⋆ candidateselectedwithin 2εeval2 _evalof the bestshading:CATE risk (c) Validation guarantee (schematic) Figure 3: CATE fusion with DRF and RF. (a) Where λ and ω enter the loss (Def. 3). (b) Algorithm 2. (c) Thm. 4: shading is the CATE risk; the selected candidate is within 2εeval(α)2 _eval(α) of the best (dashed) when r^=r0 r=r_0. 4.2 Fitting and Selecting Coefficients We now select the coefficients. We define a grid =(λm,ωm)m=1M⊆[0,1]2G=\( _m, _m)\_m=1^M [0,1]^2, compute for each m the minimizer of the empirical proxy risk in Eq. (17), and choose the (λm,ωm)( _m, _m) whose minimizer has the smallest estimated CATE risk (Def. 4). We write τ^(⋅,λ,ω)∈arg⁡mint∈ℛ^(t,λ,ω,r^) τ_ mdl(·;λ,ω)∈ _t R_ mdl(t;λ,ω, r) (20) for this minimizer, the fitted fusion learner, with ties broken by a fixed rule (e.g., the smallest R[t(X)2]E_R[t(X)^2]). For grid point m, write τ^m≜τ^(⋅,λm,ωm) τ_m τ_ mdl(·; _m, _m). We pick the τ^m τ_m with the smallest validation score, defined below. Definition 4 (Validation score on a finite grid). Let ⊆[0,1]2G [0,1]^2 be a prespecified finite grid containing (0,0)(0,0). For m=1,…,Mm=1,…,M, ℛ^val(τ^m,ωm)≜ℙR,n[Z0−τ^m(X)2]+ωmℙO,N[r^(g^−τ^m)2]−ωmℙR,n[(g^−τ^m)2], R_val( τ_m; _m) _R,n[\Z^0- τ_m(X)\^2]+ _m\,P_O,N[ r( g- τ_m)^2]- _m\,P_R,n[( g- τ_m)^2], (21) and (λ^,ω^)≜(λm^,ωm^)( λ_ mdl, ω_ mdl) ( _ m, _ m) with m^∈arg⁡minmℛ^val(τ^m,ωm) m∈ _m R_val( τ_m; _m). The score depends on λm _m through τ^m τ_m. That is, ℛ^val(τ^m,ωm) R_val( τ_m; _m) is the DRF empirical proxy risk of Eq. (17) with λ=0λ=0 and r=r^r= r, evaluated at t=τ^mt= τ_m on the evaluation samples. Since ℛ^val(τ^m,ωm) R_val( τ_m; _m) is the DRF proxy risk ℛ^DRF(τ^m,0,ωm,r^) R_DRF( τ_m;0, _m, r) evaluated at a candidate fitted on other samples, Thm. 3 applies with λ=0λ=0. When r^=r0 r=r_0, conditional on the fitted objects, ⁡[ℛ^val(τ^m,ωm)]=QDRF(0)+ℛ0(τ^m). [ R_val( τ_m; _m)]=Q_DRF(0)+R_0( τ_m). (22) The constant QDRF(0)Q_DRF(0) is the same for every m, so the score ranks the candidates by CATE risk. Since ℛ^val R_val uses finite samples and r r, its minimizer may miss the best candidate; Thm. 4 bounds this loss. Theorem 4 (Validation guarantee). Suppose |Z0||Z^0|, |g^|| g|, and all τ^m τ_m are bounded by B, and 0≤r^≤r¯0≤ r≤ r. Let ϱ(λ,ω)≜ωO[(r^−r0)g^−τ^(⋅,λ,ω)2] _ mdl(λ,ω) _O[( r-r_0)\ g- τ_ mdl(·;λ,ω)\^2] (23) be the transport error of r r in Eq. (21), which is zero when r^=r0 r=r_0. Conditional on the nuisance and tuning samples, with probability at least 1−α1-α, ℛ0τ^(⋅,λ^,ω^)≤min(λ,ω)∈⁡ℛ0τ^(⋅,λ,ω)+2εeval(α)+2max|ϱ|,R_0\ τ_ mdl(·; λ_ mdl, ω_ mdl)\≤ _(λ,ω) R_0\ τ_ mdl(·;λ,ω)\+2 _eval(α)+2 _G| _ mdl|, (24) where M≜||M |G| and εeval(α)≜2B22log⁡(4M/α)(2n−1/2+r¯N−1/2). _eval(α) 2B^2 2 (4M/α)\,(2n^-1/2+ rN^-1/2). (25) That is, the risk of the selected learner exceeds the smallest risk on the grid by at most a sampling term of order n−1/2+N−1/2n^-1/2+N^-1/2 and a term from the error in r r. Since (0,0)∈(0,0) , when r^=r0 r=r_0 the selected learner is at most 2εeval(α)2 _eval(α) worse than the RCT-only learner. 4.3 Linear Sieves: Risk and Calibration We now analyze DRF and RF when T is a linear sieve (Newey, 1997; Chen, 2007). Definition 5 (Linear-sieve fusion learner). Let ζ:→ℝpζ:X ^p be fixed before tuning (e.g., splines, random features, or a neural network11 1 A frozen neural representation trained on the nuisance samples: the network is fit on DSnuisD_S^nuis, and its weights are held fixed during tuning.), and ζ≜ζ⊤β:β∈ℝpT_ζ \ζ β:β ^p\. For ρ≥0ρ≥ 0, τ^(x,λ,ω)≜ζ(x)⊤β^(λ,ω) τ_ mdl(x;λ,ω) ζ(x) β_ mdl(λ,ω) with β^(λ,ω)∈arg⁡minβℛ^(ζ⊤β,λ,ω,r^)+ρ‖β‖22. β_ mdl(λ,ω)∈ _β\ R_ mdl(ζ β;λ,ω, r)+ρ\|β\|_2^2\. (26) Let ΓR≜R[ζζ⊤] _R _R[ζ ], assumed invertible, and βp≜ΓR−1R[ζτ0] _p _R^-1E_R[ζ _0]. Then τp⋆≜ζ⊤βp=arg⁡mint∈ζℛ0(t)τ _p ζ _p= _t _ζR_0(t) is the best approximation of τ0 _0 in the sieve class ζT_ζ of Def. 5. The estimator β β_ mdl has a closed form. Algorithm 2 DRF and RF Input: DSnuis,DStune,DSevalD_S^nuis,D_S^tune,D_S^eval for S∈R,OS∈\R,O\; class T; grid ∋(0,0)G (0,0); a density-ratio estimator. Output: (λ^,ω^)( λ_ mdl, ω_ mdl) and τ^(⋅,λ^,ω^) τ_ mdl(·; λ_ mdl, ω_ mdl). 1 Fit μ^R,μ^O,g μ_R, μ_O, g on DSnuisD_S^nuis, and r r by Def. 2 2 For each m, fit τ^m τ_m on DStuneD_S^tune by minimizing Eq. (17) 3 For each m, compute ℛ^val(τ^m,ωm) R_val( τ_m; _m) by Eq. (21) on DSevalD_S^eval 4 Select (λ^,ω^)( λ_ mdl, ω_ mdl) by minimizing ℛ^val R_val over G, with deterministic tie-breaking Proposition 4 (Normal equation). Let H^(ω)≜(1−ω)ℙRtune(κζζ⊤)+ωℙOtune(r^ζζ⊤) H_ mdl(ω) (1-ω)P_R^tune( _ mdlζ\,ζ )+ _O^tune( rζ\,ζ ). If H^(ω)+ρI H_ mdl(ω)+ρ I is invertible, β^(λ,ω) β_ mdl(λ,ω) in Def. 5 is the unique solution of the normal equation, which sets the gradient of its objective with respect to β to zero: H^(ω)+ρIβ=ℙRtuneκζ(Zλ−ωg^)+ωℙOtune(r^ζg^).\ H_ mdl(ω)+ρ I\β=P_R^tune\ _ mdlζ(Z^λ_ mdl-ω g)\+ω\,P_O^tune( rζ g). (27) Next, we bound how far β β_ mdl is from βp _p on average. Proposition 5 (Bias of β β_ mdl). Under Assumptions 1–3, fix (λ,ω,ρ)(λ,ω,ρ) and condition on the fitted nuisances, r r, and ζ, so that E is over the tuning samples and oPo_P is over the nuisance samples. Let H¯ρ≜ΓR+ωζζ⊤(r^)+ρI H_ρ _R+ω\, imb_ζ ( r)+ρ I be invertible, and let β¯(ω) β(ω) solve Eq. (27) with each average ℙStuneP_S^tune replaced by its mean SE_S. Let ψR, _R, mdl ≜κζZλ−ωg^−(1−ω)ζ⊤β¯, _ mdlζ\Z^λ_ mdl-ω g-(1-ω)ζ β\, ψO _O ≜ωr^ζ(g^−ζ⊤β¯) ω\, r\,ζ( g-ζ β) be the contributions of one RCT and one OBS tuning unit to those equations, and V≜VarR(ψR,)/nR+VarO(ψO)/nOV_ mdl _R( _R, mdl)/n_R+Var_O( _O)/n_O. Suppose the remainder ≜β^−β¯−H¯ρ−1(ℙRtuneψR,+ℙOtuneψO−ρβ¯) rem β_ mdl- β- H_ρ^-1(P_R^tune _R, mdl+P_O^tune _O-ρ β) satisfies ‖ΓR2=oP(nR−1+nO−1)E\| rem\|^2_ _R=o_P(n_R^-1+n_O^-1), where ‖v‖ΓR2≜v⊤ΓRv\|v\|^2_ _R v _Rv. Then [β^]=β¯(ω)+oP(nR−1/2+nO−1/2)E[ β_ mdl]= β(ω)+o_P(n_R^-1/2+n_O^-1/2), and β¯(ω)−βp=H¯ρ−1ωO[(r^−r0)ζ(g^−τp⋆)]−ρβp. β(ω)- _p= H_ρ^-1 \ω\,E_O[( r\!-\!r_0)ζ( g\!-\!τ _p)]-ρ _p \. Hence the squared bias ⁡(ω)≜‖β¯(ω)−βp‖ΓR2 bias(ω) \| β(ω)- _p\|^2_ _R obeys the bound (ω)1/2≤‖ΓR1/2H¯ρ−1‖op[ρ‖βp‖+ωO(r^−r0)2O‖ζ⁡(g^−τp⋆)‖21/2]. bias(ω)^1/2≤\| _R^1/2 H_ρ^-1\|_op [ρ\| _p\|+ω\E_O( r-r_0)^2\,E_O\|ζ( g-τ _p)\|^2\^1/2 ]. The ω term is a product of the ratio error r^−r0 r-r_0 and the gap g^−τp⋆ g-τ _p of the OBS prediction from the best sieve fit, and vanishes when either factor is zero. Theorem 5 (Risk of a sieve candidate). Under the conditions of Prop. 5, suppose also that []⊤ΓRβ¯(ω)−βp=oP(nR−1+nO−1)E[ rem] _R\ β(ω)- _p\=o_P(n_R^-1+n_O^-1). Then ⁡[ℛ0(τ^)]=ℛ0(τp⋆)+⁡(ω)+(λ,ω)+oP(nR−1+nO−1),E[R_0( τ_ mdl)]=R_0(τ _p)+ bias(ω)+J_ mdl(λ,ω)+o_P(n_R^-1+n_O^-1), (28) where ⁡(ω) bias(ω) is from Prop. 5, and (λ,ω)≜tr⁡(ΓRH¯ρ−1VH¯ρ−1) _ mdl(λ,ω) ( _R H_ρ^-1V_ mdl H_ρ^-1) (29) is the ΓR _R-norm variance of β β_ mdl. Here ℛ0(τp⋆)R_0(τ _p) is the approximation error of the sieve, bias comes from the ratio imbalances and the ridge penalty, and J_ mdl is the estimation error. At r^=r0 r=r_0, both imbalances vanish, so only the ridge part =ρ2‖(ΓR+ρI)−1βp‖ΓR2 bias=ρ^2\|( _R+ρ I)^-1 _p\|^2_ _R remains. Def. 2 calibrates r r along a chosen feature vector f, and the right f depends on the target. The next result compares the ATE and the sieve CATE. Proposition 6 (Calibration for ATE and sieve CATE). Let r r satisfy Eq. (9) on the nuisance samples up to the tolerances δ^k δ_k. Then, for each balanced feature fkf_k, fk(r^)=(ℙRnuis−R)fk−(ℙOnuis−O)(r^fk)+ek,|ek|≤δ^k, imb_f_k( r)=(P_R^nuis-E_R)f_k-(P_O^nuis-E_O)( rf_k)+e_k, |e_k|≤ δ_k, and likewise for their linear combinations. (i) (ATE) If f contains g g, the AIPWF drift ωg^(r^)ω\, imb_ g( r) is this nuisance-sampling error, which Thm. 2(i) absorbs into V^cal V_cal. (i) (Sieve CATE) If f contains g^2 g^2, ζg^ζ g, and ζkζl _k _l for all k≤lk≤ l, the ω term of β¯(ω)−βp β(ω)- _p in Prop. 5 and ϱ _ mdl for every candidate in ζT_ζ are linear in these errors, and hence of order nnuis−1/2+Nnuis−1/2n_nuis^-1/2+N_nuis^-1/2 under Assumption 4. Hence, with a calibrated r r, the ratio part of ⁡(ω) bias(ω) in Thm. 5 is of order nnuis−1+Nnuis−1n_nuis^-1+N_nuis^-1 and no longer depends on how well the base ratio approximates r0r_0, and the transport error in Thm. 4 is of order nnuis−1/2+Nnuis−1/2n_nuis^-1/2+N_nuis^-1/2. The ATE needs one feature, while the sieve CATE needs 1+p+p⁡(p+1)/21+p+p(p+1)/2, because the ratio enters the CATE through a p-vector and a p×p× p matrix rather than a scalar. For a general class T, the ratio enters through O[(r^−r0)s(g^−t)]E_O[( r-r_0)s( g-t)] for every direction s∈s (Prop. 3), which no finite f removes. Exact balance on many features is fragile, so each gap need only be within its standard error δ^k δ_k (Sec. 3.2). 5 Experiments 5.1 Designs ATECATE Figure 4: ATE MSE and its ratio to trial-only AIPW (top); CATE risk ratio to trial-only DR-learner (bottom); 95% bootstrap intervals. Columns 1–4: versus n; synthetic with r0=1r_0=1, then synthetic, IHDP, and ACIC 2016 under a covariate shift. Column 5: shifted synthetic at n=3,000n=3,000, versus confounding bias/sdRτ0(X)/sd_R\, _0(X). A latent U confounds the OBS in a synthetic design and two semi-synthetic benchmarks, IHDP (Hill, 2011) and ACIC 2016 (Dorie et al., 2019); the bias has root mean square 1.5sdRτ0(X)1.5\,sd_R\ _0(X)\ in the synthetic design. The benchmarks keep only their real covariates and noiseless surfaces μ0,μ1 _0, _1, and both samples draw rows of this table. The trial randomizes A, and the OBS assigns A through X and U, which also enters Y=μA(X)+U+εY= _A(X)+U+ . The OBS covariate law equals the trial law (r0=1r_0=1) or tilts it by exp⁡ah(X) \a\,h(X)\ for a standardized direction h (the first principal component in the benchmarks), so r0∝exp⁡(−ah)r_0 (-a\,h). Synthetic design. The covariates are X=lU+0.9ϵX∈ℝ10X=lU+0.9\, _X ^10 with U∼N⁡(0,1)U N(0,1) and ϵX∼N⁡(0,I10) _X N(0,I_10), where l loads U on the first two coordinates with weights ±0.5± 0.5, so that Var⁡(U∣X)=0.62Var(U X)=0.62. The outcome is Y=m⁡(X)+2U+Aτ(X)+0.7εY=m(X)+2U+Aτ(X)+0.7 with ε∼N⁡(0,1) N(0,1), where m is a fixed combination of linear, sine, quadratic, and pairwise-product terms and τ is a nonlinear function with Rτ(X)=0.8E_Rτ(X)=0.8 and VarRτ(X)=1.47Var_Rτ(X)=1.47. The trial sets A∼Bernoulli⁡(0.35)A (0.35). The OBS sets A∼Bernoulli⁡[clip⁡expit⁡(η⁡(X)+ν⁡(X)U),0.05,0.95]A [clip\expit(η(X)+ν(X)U),0.05,0.95\] for a fixed index η, with ν⁡(x)=ν01+ηstd(x)ν(x)= _0\1+ _std(x)\ and ηstd _std the index standardized over the trial covariates, so the confounding is strongest for units that are likely to be treated. The bias of the OBS contrast is δ(x)=2O[U∣A=1,X=x]−O[U∣A=0,X=x]δ(x)=2\E_O[U A=1,X=x]-E_O[U A=0,X=x]\, computed by Gauss–Hermite quadrature, and ν0 _0 sets Rδ(X)21/2=1.5sdRτ0(X)\E_Rδ(X)^2\^1/2=1.5\,sd_R\ _0(X)\. The bias sweep changes only ν0 _0 so that this root mean square is 0,0.25,…,1.250,0.25,…,1.25 times sdRτ0(X)sd_R\ _0(X)\. Benchmarks. IHDP (Hill, 2011) has 747 units, 25 covariates, and, in the version of Shalit et al. (2017), 100 realizations of its response surfaces. ACIC 2016 (Dorie et al., 2019) has 4,802 units and 58 covariates, three of them categorical and one-hot encoded, and we use its settings 1–10. We discard the treatments and outcomes recorded in both data sets and keep only the covariates and the noiseless surfaces. Both samples draw rows of the covariate table with replacement. The benchmarks are thus finite-population designs. An OBS sample of 15,000 rows contains each IHDP row about 20 times on average and each ACIC row about 3 times, and a trial of n=3,000n=3,000 contains each IHDP row about 4 times. The CATE risk averages over the rows of the table (Metrics), so it is computed at covariate values that also appear in the OBS sample, and the repeated rows may make the OBS regression more accurate there than it would be at new covariate values. Replication k uses IHDP realization kmod100k 100 and ACIC setting (kmod10)+1(k 10)+1. The published noiseless surfaces μ0,μ1 _0, _1 are scaled so that VarRμA(X)=1Var_R _A(X)=1, and Y=μA(X)+U+εY= _A(X)+U+ with U,ε∼N⁡(0,1)U, N(0,1) independent of X. The trial draws rows uniformly and sets A∼Bernoulli⁡(0.35)A (0.35). The OBS treatment follows the rule of the synthetic design with η linear in three standardized covariates, and ν0 _0 sets |Rδ(X)|=0.50sdR(Y)|E_Rδ(X)|=0.50\,sd_R(Y), where δ(x)=O[U∣A=1,X=x]−O[U∣A=0,X=x]δ(x)=E_O[U A=1,X=x]-E_O[U A=0,X=x]. Covariate shift. A shift changes the law of X only, not that of (U,A,Y)(U,A,Y) given X. In the synthetic design, the OBS covariates have mean ada\,d instead of zero for a fixed direction d, which is the exponential tilt exp⁡(ad⊤Σ−1x) (a\,d ^-1x) with Σ=Cov⁡(X) =Cov(X). In the benchmarks, the OBS draws row i with probability proportional to exp⁡(ahi) (ah_i), where h is the standardized first principal component of the standardized covariates. Since the trial draws rows uniformly, the density ratio at row i is r0(xi)=eah¯/eahir_0(x_i)= e^ah/e^ah_i, where eah¯ e^ah is the average of eahje^ah_j over the rows, and 1/O[r02]=1/(eah¯e−ah¯)1/E_O[r_0^2]=1/( e^ah\, e^-ah). In the synthetic design, with h⁡(x)=d⊤Σ−1xh(x)=d ^-1x and d⊤Σ−1d=1d ^-1d=1, r0(x)=exp⁡a2/2−ah(x)r_0(x)= \a^2/2-a\,h(x)\ and 1/O[r02]=e−a21/E_O[r_0^2]=e^-a^2. The constant a sets the effective sample fraction 1/O[r02]1/E_O[r_0^2] to 0.400.40 (the shift of Fig. 4) or 0.700.70 (the weak shift of Tables 5 and 6). 5.2 Estimators and Metrics We run Algorithms 1 and 2, cross-fitted by rotating the roles of the three trial samples, for n from 300 to 3,000 with 1,000 replications. We compare with trial-only estimation, PPI++ (λ=0λ=0, B^cal≡0 B_cal≡ 0), a naive pool of all rows, Shrinkage (Rosenman et al., 2023, after) and Pretest, which pool the trial-only and naive-pool ATEs, the 2-step CATE learner (Kallus et al., 2018, after), and a ridge variant of the integrative R-learner (IR) of Wu and Yang (2022). Samples and estimators. A trial of size n is split into equal nuisance, tuning, and evaluation samples, and the OBS has N=15,000N=15,000 rows split 60/20/2060/20/20; Fig. 6 varies N. Every split-based method is cross-fitted: the three trial roles rotate over three equal blocks, each method runs once per rotation, and the three ATE estimates or CATE predictions are averaged. The naive pool instead fits one regression on all trial and OBS rows, and Shrinkage and Pretest combine it with the cross-fitted trial-only estimate. The OBS keeps one split because it is large. Here n and N are the total trial and OBS sizes, so the evaluation sizes in each rotation, written n and N in Secs. 3 and 4, are n/3n/3 and 0.2N0.2N. Thm. 2 applies to each rotation. The Wald interval of the averaged ATE uses a first-order standard error of the average: each trial unit contributes its score Zλ^−ω^g^Z λ- ω g from the rotation that evaluates its block and, with an estimated ratio, the calibration term ω^g^(X) ω g(X) from the rotation whose nuisance sample contains it, and the OBS adds the variances of its rotation-averaged evaluation scores and calibration terms ω^r^g ω r g. The same OBS random numbers are used under every covariate law. Both outcome regressions are ridge regressions (penalty 5) of Y on (f,A,Af)(f,A,Af), where f standardizes X, X2X^2, sin⁡X X, and the pairwise products of the first six covariates, and their predictions are clipped to the 0.5% to 99.5% range of each arm on the fitting sample. Z1,…,Z5Z_1,…,Z_5 are the first five principal components of the standardized trial nuisance covariates, scaled to unit variance. When r0=1r_0=1, we set r^≡1 r≡ 1 (Assumption 3(i)). Under a shift, the base ratio is the classifier ratio of Sec. 3.2 with a logistic regression on (Z1,…,Z5)(Z_1,…,Z_5), calibrated by Def. 2: for the ATE it balances g g exactly, and for the CATE it balances g^2 g^2, ζg^ζ g, and ζkζl _k _l (k≤lk≤ l) within their standard errors δ^k δ_k (Prop. 6). AIPWF uses ϵA=ϵB=10−3Var^R(Z0) _A= _B=10^-3 Var_R(Z^0) in Eq. (11), and PPI++ fixes λ=0λ=0 and uses ω^=Proj[0,1](D^/B^) ω=Proj_[0,1]( D/ B), the rule of Angelopoulos et al. (2026) with our projection, without B^cal B_cal. Shrinkage weights the naive pool by seR2/(seR2+gap2)se_R^2/(se_R^2+gap^2), where gapgap is the difference of the naive-pool and trial-only ATEs and seRse_R is the trial-only standard error, and Pretest uses the naive pool when the gap is within 1.96seR1.96\,se_R. Shrinkage is a single-estimate rule in the spirit of Rosenman et al. (2023), not their Stein-type estimator, which shrinks the stratum estimates jointly. DRF and RF use the sieve ζ=(1,Z1,…,Z5,g^)ζ=(1,Z_1,…,Z_5, g), ridge ρ=0.01ρ=0.01, and the grid 0,0.5,12\0,0.5,1\^2, with ties broken toward smaller (λ,ω)(λ,ω); their PPI++ versions use the grid points with λ=0λ=0. The trial-only learners use μ^R μ_R and the sieve (1,Z1,…,Z5)(1,Z_1,…,Z_5) without g g. 2-step fits g^+(1,Z1,…,Z5)⊤β g+(1,Z_1,…,Z_5) β, with β a ridge regression of the trial pseudo-outcome minus g g (Kallus et al., 2018). IR (Wu and Yang, 2022) minimizes one R-loss over the trial and OBS rows, in which the OBS rows add a bias function c⁡(x)c(x) to τ⁡(x)τ(x). It takes τ in (1,Z1,…,Z5)(1,Z_1,…,Z_5) and c in (1,Z1O,…,Z5O,g^)(1,Z^O_1,…,Z^O_5, g), where ZkOZ^O_k are the principal components of the OBS nuisance covariates, and uses an OBS propensity from logistic regression. A ridge penalty replaces their SCAD penalty, and the penalty ratio λτ/λc∈0,0.5,1,1.5 _τ/ _c∈\0,0.5,1,1.5\ and λc∈10−3,10−2,…,102 _c∈\10^-3,10^-2,…,10^2\ are chosen by the R-loss of both sources on the evaluation samples instead of cross-validation. The naive pool fits the same outcome regression to all trial and OBS rows, with no source indicator or weight and without clipping. Metrics. Each cell has 1,000 replications, seeded by the base seed 20261005, the design, the trial size, and the replication index, and no replication failed except 2 of the 10,000 misspecified-shift replications (IHDP, n=600n=600 and 3,0003,000), whose ratio calibration did not converge and which are left out. The ATE error is the estimate minus θ0 _0, and the CATE risk is Rτ^(X)−τ0(X)2E_R\ τ(X)- _0(X)\^2 over 50,000 trial covariate draws (synthetic design) or all rows (benchmarks). The figures and Tables 5 and 6 report ratios of mean errors to the trial-only AIPW estimator or DR-learner; the figures add paired percentile 95% intervals from 2,000 bootstrap resamples of the replications. Table 5: ATE mean squared error relative to trial-only AIPW, by trial size n and covariate law of the OBS: common (r0=1r_0=1), weak shift (effective sample fraction 0.70), and strong shift (0.40, as in Fig. 4). Synthetic IHDP ACIC 2016 n Method r0=1r_0=1 weak strong r0=1r_0=1 weak strong r0=1r_0=1 weak strong 300 PPI++ 0.95 1.01 1.01 1.00 1.04 1.05 1.00 1.02 1.02 Fusion 0.55 0.60 0.60 0.46 0.47 0.48 0.45 0.46 0.47 Shrinkage 1.17 1.18 1.18 1.21 1.21 1.22 1.25 1.26 1.26 Pretest 1.17 1.16 1.20 1.71 1.70 1.76 2.07 2.09 2.16 Naive pool 17.41 16.92 16.24 11.05 10.92 10.69 9.16 8.93 8.56 900 PPI++ 0.97 1.04 1.05 1.00 1.02 1.02 1.00 1.01 1.01 Fusion 0.75 0.79 0.80 0.82 0.84 0.84 0.42 0.42 0.43 Shrinkage 1.03 1.03 1.03 1.05 1.05 1.06 1.10 1.11 1.12 Pretest 1.00 1.00 1.00 1.00 1.00 1.00 1.03 1.04 1.02 Naive pool 72.74 67.82 61.71 51.00 48.97 45.79 24.50 23.25 21.62 3,000 PPI++ 0.99 1.06 1.06 0.97 1.04 1.00 1.00 1.02 1.01 Fusion 0.96 1.00 1.00 0.92 0.98 0.96 0.77 0.79 0.79 Shrinkage 1.01 1.01 1.01 1.02 1.02 1.02 1.03 1.03 1.04 Pretest 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 Naive pool 184.50 160.01 133.47 163.58 148.30 126.35 113.49 101.57 88.22 Table 6: CATE risk relative to the trial-only DR-learner; otherwise as Table 5. RCT (R) is the trial-only R-learner, and IR the integrative R-learner. Synthetic IHDP ACIC 2016 n Method r0=1r_0=1 weak strong r0=1r_0=1 weak strong r0=1r_0=1 weak strong 300 DRF 0.58 0.59 0.59 0.56 0.57 0.57 0.54 0.54 0.56 RF 0.58 0.58 0.59 0.56 0.56 0.57 0.53 0.53 0.56 PPI++ DRF 0.81 0.80 0.81 0.79 0.79 0.79 0.82 0.80 0.83 PPI++ RF 0.74 0.74 0.74 0.70 0.70 0.70 0.78 0.76 0.79 2-step DR 0.96 0.95 0.94 1.06 1.07 1.08 1.06 1.10 1.14 2-step R 0.92 0.91 0.90 0.99 0.99 1.01 1.04 1.08 1.12 IR 1.02 1.02 1.02 0.94 0.96 0.96 0.61 0.61 0.61 RCT (R) 0.96 0.96 0.96 0.92 0.92 0.92 0.98 0.98 0.98 Naive pool 2.35 2.30 2.24 1.52 1.52 1.51 1.18 1.21 1.22 900 DRF 0.64 0.65 0.65 0.60 0.61 0.62 0.58 0.61 0.62 RF 0.64 0.65 0.65 0.60 0.61 0.62 0.58 0.60 0.62 PPI++ DRF 0.69 0.69 0.69 0.63 0.64 0.65 0.74 0.76 0.78 PPI++ RF 0.67 0.67 0.68 0.62 0.63 0.64 0.68 0.70 0.72 2-step DR 0.97 0.95 0.94 1.11 1.12 1.14 1.10 1.16 1.22 2-step R 0.96 0.94 0.93 1.10 1.11 1.13 1.05 1.11 1.17 IR 1.20 1.20 1.20 1.08 1.09 1.10 0.87 0.87 0.88 RCT (R) 0.99 0.99 0.99 0.99 0.99 0.99 0.95 0.95 0.95 Naive pool 3.72 3.54 3.32 2.38 2.33 2.24 1.80 1.80 1.79 3,000 DRF 0.64 0.64 0.64 0.55 0.56 0.57 0.62 0.65 0.67 RF 0.64 0.64 0.64 0.55 0.56 0.57 0.62 0.65 0.67 PPI++ DRF 0.64 0.64 0.64 0.55 0.56 0.57 0.64 0.66 0.68 PPI++ RF 0.64 0.64 0.64 0.55 0.56 0.57 0.63 0.66 0.68 2-step DR 0.97 0.94 0.93 1.11 1.12 1.15 1.15 1.22 1.30 2-step R 0.97 0.94 0.93 1.11 1.12 1.15 1.14 1.22 1.29 IR 1.08 1.08 1.08 1.02 1.03 1.04 1.00 1.00 1.00 RCT (R) 1.00 1.00 1.00 1.00 1.00 1.00 0.99 0.99 0.99 Naive pool 3.28 3.02 2.73 2.13 2.00 1.84 1.95 1.90 1.84 5.3 Main Results Fig. 4 shows each error relative to the trial-only estimator on a log scale, so a curve below 1 improves on the trial alone. For the ATE, we compare AIPWF (Fusion) with trial-only AIPW (Robins et al., 1994), PPI++ (Angelopoulos et al., 2026), Shrinkage (Rosenman et al., 2023, after), Pretest (Sec. 5.2), and the naive pool. Fusion has the smallest MSE in every design, covariate law, and trial size (columns 1–4). It is also robust to the size of the confounding (column 5): its error stays flat as the bias grows. For the CATE, we compare DRF and RF with the trial-only learners (RCT), their PPI++ versions, 2-step, IR, and the naive pool (Sec. 5.2). DRF and RF have the smallest risk in every design, covariate law, and trial size. They stay below the trial-only risk at every bias. Tables 5 and 6 give the numbers. Figure 5: Bias sweep on the synthetic design under the strong shift at n=300n=300, 900900, and 3,0003,000, against the confounding bias in units of sdRτ0(X)sd_R\, _0(X). Top: ATE MSE relative to trial-only AIPW; bottom: CATE risk relative to the trial-only DR-learner. 95% bootstrap intervals. Figure 6: Effect of the observational sample size N (split 60/20/20) under the covariate shift, for the synthetic design and IHDP at n=300n=300 and 900900. Top: ATE MSE relative to RCT only; bottom: CATE risk relative to the trial-only DR-learner. 95% bootstrap intervals. Patterns in Fig. 4. The ATE gap of fusion below 1 is widest at small n and narrows by n=3,000n=3,000, so the OBS help the ATE most when the trial is small; the CATE ratios of DRF and RF stay between 0.53 and 0.67 at every n. PPI++ stays near 1 for the ATE and approaches DRF and RF for the CATE only at n=3,000n=3,000, so the gain of fusion at small n comes from λ. For the ATE, this is the cap in Remark 1: the PPI correction can remove at most the share of effect heterogeneity in the score variance, which is small in these designs: at r0=1r_0=1, VarRτ0(X)/VarR(Z0)Var_R\ _0(X)\/Var_R(Z^0) is 0.02 to 0.10 on the tuning samples. For the CATE, the trial-only learners use the sieve (1,Z1,…,Z5)(1,Z_1,…,Z_5) without g g, whereas DRF, RF, and their PPI++ versions use (1,Z1,…,Z5,g^)(1,Z_1,…,Z_5, g), so the ratios to the trial-only learners combine the fusion loss with the gain from g g as a basis function. The trial loss on the same sieve separates the two (Table 9): at large n the gap comes from g g as a basis function, and at small n mostly from the fusion loss. IR, in contrast, reaches or exceeds the trial-only risk by n=3,000n=3,000 and is below it only on ACIC and on IHDP at n=300n=300. IR models the confounding function in a linear class, so the OBS rows inform τ only as far as the OBS bias is close to that class. Its CATE class (1,Z1,…,Z5)(1,Z_1,…,Z_5) is the sieve of the trial-only learners, which is consistent with its ratio approaching or exceeding 1 as n grows. DRF and RF do not model the confounding function, keep the trial’s target under any OBS confounding (Thm. 3), and use the OBS to reduce variance. The naive pool rises with n, because its confounding bias stays fixed while the trial-only error shrinks. Shrinkage and Pretest never fall below 1 in columns 1–4. ATE. Fusion has the smallest ATE MSE in all 45 cells (three designs, three covariate laws, five trial sizes), with 0.40 to 1.00 of the trial-only MSE (Table 5). In the cells of Fig. 4, its ratio is 0.47 to 0.60 at n=300n=300 and 0.79 to 1.00 at n=3,000n=3,000. PPI++ stays between 0.95 and 1.07, and Shrinkage and Pretest stay between 1.00 and 2.16. The naive pool is worse than the trial alone in every cell, by a factor of 8.6 to 184 in Table 5. Tables 13 and 14 report the bias, spread, and Wald intervals: the coverage of fusion is 0.94 to 0.96, and its intervals are 63% to 99% as wide as the trial-only ones. CATE. DRF and RF have 0.53 to 0.67 of the trial-only risk in all 45 cells, and they differ from each other by at most 0.01 (Table 6). In the cells of Fig. 4, they have 0.56 to 0.59 at n=300n=300. DRF or RF has the smallest risk in all 45 cells. The ratio of IR is 0.61 to 1.02 at n=300n=300 and 1.00 to 1.08 at n=3,000n=3,000, whereas DRF and RF stay between 0.55 and 0.67 at n=3,000n=3,000. PPI++ approaches DRF and RF only by n=3,000n=3,000, and 2-step stays between 0.90 and 1.30 in the cells of Fig. 4. The naive pool is worse than the trial alone in every cell. Table 7: The two channels of fusion. ATE MSE relative to trial-only AIPW of λ only (ω=0ω=0), PPI++ (λ=0λ=0, ω^=Proj[0,1](D^/B^) ω=Proj_[0,1]( D/ B)), fusion with ω ω of Eq. (11) or with ω^=Proj[0,1](D^/B^) ω=Proj_[0,1]( D/ B). Synthetic IHDP ACIC 2016 n Method r0=1r_0=1 weak strong r0=1r_0=1 weak strong r0=1r_0=1 weak strong 300 λ only 0.60 0.60 0.60 0.47 0.46 0.46 0.46 0.46 0.46 PPI++ 0.95 1.01 1.01 1.00 1.04 1.05 1.00 1.02 1.02 Fusion 0.55 0.60 0.60 0.46 0.47 0.48 0.45 0.46 0.47 Fusion, ω^=D^/B ω= D/ B 0.55 0.62 0.62 0.46 0.49 0.50 0.45 0.47 0.48 900 λ only 0.78 0.78 0.78 0.83 0.84 0.83 0.42 0.42 0.43 PPI++ 0.97 1.04 1.05 1.00 1.02 1.02 1.00 1.01 1.01 Fusion 0.75 0.79 0.80 0.82 0.84 0.84 0.42 0.42 0.43 Fusion, ω^=D^/B ω= D/ B 0.75 0.82 0.83 0.82 0.86 0.85 0.42 0.43 0.44 3,000 λ only 0.97 0.97 0.97 0.95 0.96 0.97 0.78 0.77 0.78 PPI++ 0.99 1.06 1.06 0.97 1.04 1.00 1.00 1.02 1.01 Fusion 0.96 1.00 1.00 0.92 0.98 0.96 0.77 0.79 0.79 Fusion, ω^=D^/B ω= D/ B 0.96 1.04 1.04 0.92 1.01 0.97 0.77 0.80 0.80 Table 8: Ratios to the practitioner’s trial-only estimators (3-fold cross-fitting, nuisances fitted on two thirds of the trial, no tuning sample), marked (2/3). The rotated trial-only estimators of Tables 5 and 6, whose nuisances use one third, are marked (1/3). Synthetic IHDP ACIC 2016 n Method r0=1r_0=1 weak strong r0=1r_0=1 weak strong r0=1r_0=1 weak strong 300 Fusion / trial-only (2/3) 0.70 0.76 0.76 0.63 0.64 0.65 0.40 0.41 0.41 Trial-only (1/3) / (2/3) 1.27 1.27 1.27 1.37 1.37 1.37 0.89 0.89 0.89 DRF / DR (2/3) 0.67 0.67 0.68 0.67 0.68 0.69 0.56 0.57 0.59 RF / DR (2/3) 0.66 0.67 0.67 0.66 0.67 0.68 0.56 0.56 0.58 RF / R (2/3) 0.69 0.70 0.70 0.70 0.70 0.71 0.58 0.58 0.61 DR (1/3) / DR (2/3) 1.14 1.14 1.14 1.20 1.20 1.20 1.05 1.05 1.05 900 Fusion / trial-only (2/3) 0.84 0.89 0.90 0.92 0.95 0.94 0.64 0.64 0.66 Trial-only (1/3) / (2/3) 1.12 1.12 1.12 1.12 1.12 1.12 1.52 1.52 1.52 DRF / DR (2/3) 0.63 0.64 0.64 0.60 0.61 0.62 0.65 0.67 0.69 RF / DR (2/3) 0.63 0.64 0.64 0.60 0.61 0.62 0.64 0.66 0.68 RF / R (2/3) 0.64 0.64 0.64 0.60 0.61 0.62 0.67 0.69 0.71 DR (1/3) / DR (2/3) 0.99 0.99 0.99 1.00 1.00 1.00 1.11 1.11 1.11 3,000 Fusion / trial-only (2/3) 0.97 1.00 1.01 0.95 1.01 0.99 0.90 0.92 0.92 Trial-only (1/3) / (2/3) 1.01 1.01 1.01 1.03 1.03 1.03 1.17 1.17 1.17 DRF / DR (2/3) 0.61 0.61 0.61 0.54 0.55 0.57 0.63 0.66 0.68 RF / DR (2/3) 0.61 0.61 0.61 0.54 0.55 0.57 0.63 0.66 0.68 RF / R (2/3) 0.61 0.61 0.61 0.54 0.55 0.57 0.63 0.66 0.68 DR (1/3) / DR (2/3) 0.95 0.95 0.95 0.99 0.99 0.99 1.01 1.01 1.01 Table 9: CATE with the trial loss alone on the sieve with g g: its risk relative to the trial-only DR-learner, and the risk of DRF and RF relative to it. Synthetic IHDP ACIC 2016 n Method r0=1r_0=1 weak strong r0=1r_0=1 weak strong r0=1r_0=1 weak strong 300 Trial loss, sieve with g g (DR) 0.84 0.83 0.83 0.84 0.84 0.85 0.90 0.92 0.93 Trial loss, sieve with g g (R) 0.79 0.78 0.78 0.75 0.76 0.76 0.88 0.90 0.91 DRF / same sieve (DR) 0.69 0.70 0.72 0.67 0.68 0.68 0.60 0.59 0.60 RF / same sieve (R) 0.74 0.74 0.76 0.74 0.74 0.74 0.61 0.59 0.61 900 Trial loss, sieve with g g (DR) 0.68 0.67 0.67 0.63 0.64 0.65 0.76 0.78 0.79 Trial loss, sieve with g g (R) 0.67 0.66 0.65 0.62 0.63 0.63 0.70 0.72 0.73 DRF / same sieve (DR) 0.94 0.96 0.97 0.95 0.96 0.96 0.77 0.78 0.79 RF / same sieve (R) 0.96 0.98 1.00 0.97 0.97 0.98 0.82 0.84 0.84 3,000 Trial loss, sieve with g g (DR) 0.64 0.63 0.62 0.55 0.56 0.57 0.64 0.66 0.67 Trial loss, sieve with g g (R) 0.64 0.63 0.62 0.55 0.56 0.57 0.63 0.65 0.67 DRF / same sieve (DR) 1.00 1.02 1.02 1.00 1.01 1.01 0.98 0.99 1.00 RF / same sieve (R) 1.01 1.02 1.03 1.00 1.01 1.01 0.99 1.00 1.01 Table 10: Misspecified shift (tilt exp⁡a⁡(h2−1)/2 \a(h^2-1)/ 2\, effective sample fraction 0.40): fusion with the calibrated ratio r^BAL r_BAL and with the uncalibrated logistic ratio r^CLS r_CLS and ω^=Proj[0,1](D^/B^) ω=Proj_[0,1]( D/ B). MSE ratio: relative to RCT only. Design n Method Bias SD RMSE Cov. MSE ratio Synthetic 300 RCT only 0.007 0.302 0.302 0.95 1.00 Fusion 0.002 0.232 0.232 0.94 0.59 Fusion, uncalibrated r^CLS r_CLS 0.123 0.233 0.264 0.90 0.76 900 RCT only -0.007 0.139 0.139 0.95 1.00 Fusion -0.005 0.123 0.123 0.96 0.78 Fusion, uncalibrated r^CLS r_CLS 0.105 0.124 0.163 0.87 1.36 3,000 RCT only -0.002 0.073 0.073 0.95 1.00 Fusion -0.002 0.072 0.072 0.95 0.99 Fusion, uncalibrated r^CLS r_CLS 0.080 0.074 0.109 0.76 2.24 IHDP 300 RCT only -0.007 0.255 0.255 0.96 1.00 Fusion -0.002 0.177 0.176 0.95 0.48 Fusion, uncalibrated r^CLS r_CLS -0.032 0.182 0.184 0.94 0.52 900 RCT only -0.003 0.114 0.114 0.96 1.00 Fusion -0.003 0.105 0.105 0.95 0.84 Fusion, uncalibrated r^CLS r_CLS -0.030 0.118 0.121 0.91 1.13 3,000 RCT only 0.000 0.056 0.056 0.96 1.00 Fusion 0.001 0.055 0.055 0.96 0.97 Fusion, uncalibrated r^CLS r_CLS -0.019 0.068 0.071 0.88 1.58 Table 11: Synthetic design with large effect heterogeneity (outcome noise 0.2, VarRτ0(X)=5.88Var_R _0(X)=5.88, confounding bias 0.75sdRτ0(X)0.75\,sd_R _0(X)), under r0=1r_0=1 and the strong shift. MSE ratio: relative to RCT only. Law n Method Bias SD RMSE Cov. MSE ratio r0=1r_0=1 300 RCT only 0.008 0.307 0.307 0.96 1.00 PPI++ 0.007 0.281 0.281 0.96 0.84 λ only 0.013 0.237 0.238 0.95 0.60 Fusion 0.012 0.205 0.206 0.95 0.45 Fusion, ω^=D^/B ω= D/ B 0.012 0.205 0.206 0.95 0.45 900 RCT only 0.000 0.149 0.149 0.95 1.00 PPI++ -0.003 0.136 0.136 0.95 0.83 λ only 0.000 0.138 0.138 0.95 0.86 Fusion -0.003 0.122 0.122 0.95 0.67 Fusion, ω^=D^/B ω= D/ B -0.003 0.122 0.122 0.95 0.67 3,000 RCT only 0.002 0.074 0.074 0.95 1.00 PPI++ 0.003 0.072 0.072 0.95 0.93 λ only 0.002 0.073 0.073 0.96 0.96 Fusion 0.003 0.070 0.070 0.95 0.89 Fusion, ω^=D^/B ω= D/ B 0.003 0.070 0.070 0.95 0.89 strong 300 RCT only 0.008 0.307 0.307 0.96 1.00 PPI++ 0.008 0.317 0.317 0.96 1.07 λ only 0.013 0.238 0.238 0.96 0.60 Fusion 0.014 0.241 0.241 0.95 0.62 Fusion, ω^=D^/B ω= D/ B 0.013 0.250 0.250 0.95 0.66 900 RCT only 0.000 0.149 0.149 0.95 1.00 PPI++ -0.004 0.156 0.156 0.96 1.10 λ only 0.000 0.138 0.138 0.95 0.86 Fusion -0.001 0.140 0.140 0.95 0.89 Fusion, ω^=D^/B ω= D/ B -0.003 0.147 0.147 0.95 0.97 3,000 RCT only 0.002 0.074 0.074 0.95 1.00 PPI++ 0.003 0.085 0.085 0.95 1.30 λ only 0.002 0.073 0.073 0.96 0.96 Fusion 0.003 0.077 0.077 0.95 1.06 Fusion, ω^=D^/B ω= D/ B 0.003 0.083 0.083 0.95 1.26 5.4 Bias and Observational Sample Size Fig. 5 repeats the bias sweep at n=300n=300 and 900900. The ATE ratio of fusion is flat in the bias at every n, and at n=3,000n=3,000 the CATE ratio of IR stays between 1.08 and 1.10 at every bias. The naive pool, Shrinkage, and Pretest beat fusion when the OBS are nearly unconfounded. At n=3,000n=3,000 the naive pool falls behind fusion at a bias between 0 and 0.25 (ATE) or between 0.25 and 0.5 (CATE) standard deviations. This threshold decreases as n grows: for the CATE, it lies between 0.5 and 0.75 standard deviations at n=300n=300 and between 0.25 and 0.5 at n=3,000n=3,000. In Fig. 6, the ATE ratio of fusion changes little with N, and its CATE ratio falls as N grows. Fig. 7 gives the absolute errors. 5.5 Ablations and Further Designs Two channels. Table 7 separates λ and ω. At r0=1r_0=1, B^cal=0 B_cal=0, and fusion is at or below λ only in 14 of 15 cells, for example 0.55 against 0.60 on the synthetic design at n=300n=300. Under a shift, λ only is slightly below fusion in 28 of 30 cells, by at most 0.03, so once the ratio is estimated the PPI correction does not help in these designs. The rule ω^=Proj[0,1](D^/B^) ω=Proj_[0,1]( D/ B), which ignores B^cal B_cal, does worse: Eq. (11) lowers the MSE of fusion by 1% to 4% in all 30 shift cells, and the old rule exceeds the trial-only MSE in three cells at n=3,000n=3,000 (1.01 to 1.04). Practitioner’s baselines. A practitioner without a tuning sample would use 3-fold cross-fitted trial-only AIPW and DR- and R-learners with the nuisances fitted on two thirds of the trial. This AIPW has lower MSE than the rotated trial-only AIPW of Table 5 in 42 of 45 cells (Table 8). Fusion is still below it in 42 of 45 cells; the exceptions are three shift cells at n=3,000n=3,000, at 1.00 to 1.01. DRF and RF are below the practitioner’s DR- and R-learners in all 45 cells, at 0.54 to 0.71 of their risk. Same sieve. Table 9 fits the trial loss alone on the sieve (1,Z1,…,Z5,g^)(1,Z_1,…,Z_5, g), the grid point (λ,ω)=(0,0)(λ,ω)=(0,0) of Algorithm 2. It uses g g, so it is not a trial-only learner. It has 0.75 to 0.93 of the trial-only DR risk at n=300n=300 and 0.55 to 0.67 at n=3,000n=3,000. DRF and RF have 0.59 to 0.76 of its risk at n=300n=300 and 0.98 to 1.03 at n=3,000n=3,000. Thus at small n most of the gain of DRF and RF over the trial-only learners comes from the fusion loss, and at large n it comes from g g as a basis function. Misspecified ratio. Table 10 uses a shift that the base ratio cannot represent. On the synthetic design and IHDP, the OBS tilts the trial law by exp⁡a⁡(h2−1)/2 \a(h^2-1)/ 2\ with a>0a>0 and effective sample fraction 0.40, so the OBS spread more widely along h, while the logistic classifier on (Z1,…,Z5)(Z_1,…,Z_5) is log-linear in h. With the uncalibrated r^CLS r_CLS, for which ω^=Proj[0,1](D^/B^) ω=Proj_[0,1]( D/ B) and the standard error omits V^cal V_cal, fusion is biased (0.08 to 0.12 on the synthetic design and 0.02 to 0.03 on IHDP), its coverage falls to 0.76 to 0.94, and its MSE exceeds the trial-only MSE at n≥900n≥ 900 (1.13 to 2.24). With r^BAL r_BAL, the absolute bias is at most 0.005, the coverage is 0.94 to 0.96, and the MSE is 0.48 to 0.99 of the trial-only MSE. DRF and RF have 0.52 to 0.62 of the trial-only CATE risk here. Large heterogeneity. Table 11 changes the synthetic design so that ω can matter: the outcome noise is 0.2 instead of 0.7, VarRτ0(X)=5.88Var_R _0(X)=5.88 instead of 1.47, and ν0 _0 is unchanged, so the confounding bias is 0.75sdRτ0(X)0.75\,sd_R _0(X). The heterogeneity share VarRτ0(X)/VarR(Z0)Var_R\ _0(X)\/Var_R(Z^0) rises to 0.20 to 0.33. At r0=1r_0=1, PPI++ has 0.83 to 0.93 of the trial-only MSE, and fusion improves on λ only at every n, from 0.60 to 0.45 at n=300n=300 and from 0.96 to 0.89 at n=3,000n=3,000. Under the strong shift, PPI++ is above 1 (1.07 to 1.30), and fusion is above λ only (0.62 against 0.60 at n=300n=300, 1.06 against 0.96 at n=3,000n=3,000). Eq. (11) reduces this loss relative to the rule ω^=Proj[0,1](D^/B^) ω=Proj_[0,1]( D/ B) (0.66 and 1.26) but does not remove it. The coverage of fusion is 0.95 to 0.96 throughout. 6 Conclusion For the ATE, AIPWF has closed-form optimal weights (Thm. 1, Cor. 1.1) and valid Wald intervals with a calibrated ratio (Thm. 2). For the CATE, DRF and RF keep the trial’s CATE risk as their target (Thm. 3), validation selects the weights with a guarantee relative to the RCT-only learner (Thm. 4), and for linear sieves, calibration on finitely many moments controls the ratio error in the risk (Thm. 5, Prop. 6). Fusion has the lowest error in every design and trial size. References Angelopoulos et al. (2023) A. N. Angelopoulos, S. Bates, C. Fannjiang, M. I. Jordan, and T. Zrnic. Prediction-powered inference. Science, 382(6671):669–674, 2023. Angelopoulos et al. (2026) A. N. Angelopoulos, J. C. Duchi, and T. Zrnic. PPI++: Efficient prediction-powered inference. The Annals of Applied Statistics, 20(3):2235–2252, 2026. Asiaee et al. (2025) A. Asiaee, C. Di Gravio, C. Beck, Y. Mei, S. Pal, and J. D. Huling. Improving precision of RCT-based CATE estimation using data borrowing with double calibration. arXiv preprint arXiv:2306.17478, 2025. Bareinboim and Pearl (2016) E. Bareinboim and J. Pearl. Causal inference and the data-fusion problem. Proceedings of the National Academy of Sciences, 113(27):7345–7352, 2016. Chen (2007) X. Chen. Large sample sieve estimation of semi-nonparametric models. In J. J. Heckman and E. E. Leamer, editors, Handbook of Econometrics, volume 6B, pages 5549–5632. Elsevier, 2007. Cheng and Cai (2021) D. Cheng and T. Cai. Adaptive combination of randomized and observational data. arXiv preprint arXiv:2111.15012, 2021. Chernozhukov et al. (2018) V. Chernozhukov, D. Chetverikov, M. Demirer, E. Duflo, C. Hansen, W. Newey, and J. Robins. Double/debiased machine learning for treatment and structural parameters. The Econometrics Journal, 21(1):C1–C68, 2018. Chernozhukov et al. (2022) V. Chernozhukov, W. K. Newey, and R. Singh. Automatic debiased machine learning of causal and structural effects. Econometrica, 90(3):967–1027, 2022. Colnet et al. (2024) B. Colnet, I. Mayer, G. Chen, A. Dieng, R. Li, G. Varoquaux, J.-P. Vert, J. Josse, and S. Yang. Causal inference methods for combining randomized trials and observational studies: a review. Statistical Science, 39(1):165–191, 2024. De Bartolomeis et al. (2025) P. De Bartolomeis, J. Abad, G. Wang, K. Donhauser, R. M. Duch, F. Yang, and I. J. Dahabreh. Efficient randomized experiments using foundation models. In Advances in Neural Information Processing Systems, volume 38, pages 159470–159509, 2025. Demirel et al. (2024) I. Demirel, A. Alaa, A. Philippakis, and D. Sontag. Prediction-powered generalization of causal inferences. In Proceedings of the 41st International Conference on Machine Learning, volume 235 of Proceedings of Machine Learning Research, pages 10385–10408, 2024. Dorie et al. (2019) V. Dorie, J. Hill, U. Shalit, M. Scott, and D. Cervone. Automated versus do-it-yourself methods for causal inference: Lessons learned from a data analysis competition. Statistical Science, 34(1):43–68, 2019. Foster and Syrgkanis (2023) D. J. Foster and V. Syrgkanis. Orthogonal statistical learning. The Annals of Statistics, 51(3):879–908, 2023. Gagnon-Bartsch et al. (2023) J. A. Gagnon-Bartsch, A. C. Sales, E. Wu, A. F. Botelho, J. A. Erickson, L. W. Miratrix, and N. T. Heffernan. Precise unbiased estimation in randomized experiments using auxiliary observational data. Journal of Causal Inference, 11(1):20220011, 2023. Gao et al. (2025) C. Gao, S. Yang, M. Shan, W. Ye, I. Lipkovich, and D. Faries. Improving randomized controlled trial analysis via data-adaptive borrowing. Biometrika, 112(2):asae069, 2025. Gu et al. (2023) J. Gu, C. Tang, H. Yan, Q. Cui, L. Li, and J. Zhou. FAST: A fused and accurate shrinkage tree for heterogeneous treatment effects estimation. In Advances in Neural Information Processing Systems, volume 36, pages 504–522, 2023. Hainmueller (2012) J. Hainmueller. Entropy balancing for causal effects: A multivariate reweighting method to produce balanced samples in observational studies. Political Analysis, 20(1):25–46, 2012. Hatt et al. (2022) T. Hatt, J. Berrevoets, A. Curth, S. Feuerriegel, and M. van der Schaar. Combining observational and randomized data for estimating heterogeneous treatment effects. arXiv preprint arXiv:2202.12891, 2022. Hill (2011) J. L. Hill. Bayesian nonparametric modeling for causal inference. Journal of Computational and Graphical Statistics, 20(1):217–240, 2011. Jung et al. (2023a) Y. Jung, I. Díaz, J. Tian, and E. Bareinboim. Estimating causal effects identifiable from a combination of observations and experiments. In Advances in Neural Information Processing Systems, volume 36, pages 46446–46490, 2023a. Jung et al. (2023b) Y. Jung, J. Tian, and E. Bareinboim. Estimating joint treatment effects by combining multiple experiments. In Proceedings of the 40th International Conference on Machine Learning, volume 202 of Proceedings of Machine Learning Research, pages 15451–15527, 2023b. Jung et al. (2024) Y. Jung, J. Tian, and E. Bareinboim. Unified covariate adjustment for causal inference. In Advances in Neural Information Processing Systems, volume 37, pages 6448–6499, 2024. Kallus et al. (2018) N. Kallus, A. M. Puli, and U. Shalit. Removing hidden confounding by experimental grounding. In Advances in Neural Information Processing Systems, volume 31, 2018. Kennedy (2023) E. H. Kennedy. Towards optimal doubly robust estimation of heterogeneous causal effects. Electronic Journal of Statistics, 17(2):3008–3049, 2023. Lee et al. (2022) D. Lee, S. Yang, and X. Wang. Doubly robust estimators for generalizing treatment effects on survival outcomes from randomized controlled trials to a target population. Journal of Causal Inference, 10(1):415–440, 2022. Lee et al. (2023) D. Lee, S. Yang, L. Dong, X. Wang, D. Zeng, and J. Cai. Improving trial generalizability using observational studies. Biometrics, 79(2):1213–1225, 2023. Li et al. (2022) X. Li, Y. Li, Q. Cui, L. Li, and J. Zhou. Robust direct learning for causal data fusion. In Proceedings of the 14th Asian Conference on Machine Learning, volume 189 of Proceedings of Machine Learning Research, pages 611–626, 2022. Newey (1997) W. K. Newey. Convergence rates and asymptotic normality for series estimators. Journal of Econometrics, 79(1):147–168, 1997. Nie and Wager (2021) X. Nie and S. Wager. Quasi-oracle estimation of heterogeneous treatment effects. Biometrika, 108(2):299–319, 2021. Robins et al. (1994) J. M. Robins, A. Rotnitzky, and L. P. Zhao. Estimation of regression coefficients when some regressors are not always observed. Journal of the American Statistical Association, 89(427):846–866, 1994. Rosenman et al. (2023) E. Rosenman, G. Basse, A. Owen, and M. Baiocchi. Combining observational and experimental datasets using shrinkage estimators. Biometrics, 79(4):2961–2973, 2023. Shalit et al. (2017) U. Shalit, F. D. Johansson, and D. Sontag. Estimating individual treatment effect: generalization bounds and algorithms. In Proceedings of the 34th International Conference on Machine Learning, volume 70 of Proceedings of Machine Learning Research, pages 3076–3085, 2017. Shi et al. (2023) X. Shi, Z. Pan, and W. Miao. Data integration in causal inference. WIREs Computational Statistics, 15(1):e1581, 2023. van der Vaart (1998) A. W. van der Vaart. Asymptotic Statistics. Cambridge University Press, 1998. Wu and Yang (2022) L. Wu and S. Yang. Integrative R-learner of heterogeneous treatment effects combining experimental and observational studies. In Proceedings of the First Conference on Causal Learning and Reasoning, volume 177 of Proceedings of Machine Learning Research, pages 904–926, 2022. Wu and Yang (2023) L. Wu and S. Yang. Transfer learning of individualized treatment rules from experimental to real-world data. Journal of Computational and Graphical Statistics, 32(3):1036–1045, 2023. Yang and Ding (2020) S. Yang and P. Ding. Combining multiple observational data sources to estimate causal effects. Journal of the American Statistical Association, 115(531):1540–1554, 2020. Yang et al. (2023) S. Yang, C. Gao, D. Zeng, and X. Wang. Elastic integrative analysis of randomised trial and real-world data for treatment heterogeneity estimation. Journal of the Royal Statistical Society Series B, 85(3):575–596, 2023. Yang et al. (2025) S. Yang, S. Liu, D. Zeng, and X. Wang. Data fusion methods for the heterogeneity of treatment effect and confounding function. Bernoulli, 31(4):2987–3012, 2025. Supplement of Prediction-Powered Data Fusion for Treatment Effect Estimation Appendix A Proofs Table 12: Section-specific notation. Notation used throughout the paper is in Tables 3 and 4. Symbol Meaning Where Sec. 3: ATE A,B,C,DA,B,C,D variance coefficients Thm. 1 λ⋆,ω⋆λ ,\ ω oracle coefficients Thm. 1 V~(λ,ω) V(λ,ω) scaled variance nVarθ^r0(λ,ω)nVar\ θ_r_0(λ,ω)\ Cor. 1.1 ϵA,ϵB _A,\ _B coefficient thresholds Eq. (11) A^,B^,C^,D A, B, C, D tuning-sample estimates of A,…,DA,…,D Sec. 3.3 ψR≜Zλ^−ω^g _R Z λ- ω g, ψO≜ω^r^g _O ω r g, V V RCT and OBS terms of θ θ; variance estimate Algorithm 1 VnV_n variance of θ θ Thm. 2 r^RZ,r^CLS,r^BAL r_RZ,\ r_CLS,\ r_BAL Riesz, classifier, and balancing ratio estimates Sec. 3.2 f,δ^kf,\ δ_k balance features and tolerances Def. 2 Sec. 4: CATE ℛ0(t)R_0(t) CATE risk Sec. 4.1 ℛ,ℛ^R_ mdl,\ R_ mdl RCT loss and fusion loss of learner ∈DRF,RF mdl∈\DRF,RF\ Sec. 4.1, Def. 3 κ,Zλ,mλ _ mdl,\ Z_ mdl^λ,\ m_λ weight, pseudo-outcome, marginal regression Def. 3 Q(λ)Q_ mdl(λ) noise floor Thm. 3 Ψ _ mdl population learning equation Prop. 3 ,ℛ^valG,\ R_val candidate grid and validation score Def. 4 εeval(α),ϱ _eval(α),\ _ mdl evaluation error and transport error Thm. 4 ζ,ζ,β^ζ,\ T_ζ,\ β_ mdl sieve features, class, and coefficients Def. 5 τp⋆,τ _p,\ J_ mdl projection of τ0 _0 and first-order variance Thm. 5 Orthogonality. Sec. 4 states which fitted nuisances enter the first-order condition of a loss. The following two definitions fix the language. Definition 6 (Gateaux derivative (van der Vaart, 1998, Chapter 20)). Let F be a functional of a function-valued argument η, and let h be a direction such that η+εhη+ h stays in the domain of F for all sufficiently small |ε|| |. Whenever the limit exists, the Gateaux derivative of F at η in the direction h is DηF(η)[h] D_ηF(η)[h] ≜dεF(η+εh)|ε=0 . dd F(η+ h) |_ =0 (30) Definition 7 (Learning equation and orthogonality (Chernozhukov et al., 2018; Foster and Syrgkanis, 2023)). Let L⁡(t,η)L(t;η) be a population loss with a target argument t in a linear class and a nuisance argument η, and let η0 _0 denote the true nuisance. The learning equation of L is Ψ(t,η)[s] (t;η)[s] ≜−12DtL(t,η)[s] - 12D_tL(t;η)[s] (31) for target directions s in the class. The loss is orthogonal to η if DηΨ(t⋆,η0)[h,s]=0D_η (t ; _0)[h,s]=0 for every admissible nuisance direction h and every target direction s. When L is convex in t, a function t⋆t minimizes L⁡(⋅,η0)L(·\,; _0) over the class exactly when Ψ⁡(t⋆,η0)[s]=0 (t ; _0)[s]=0 for every s in the class, which is why Ψ is called the learning equation. Lemma 1. Condition on X. Randomization and consistency give R[(A/e⁡(X))Y−q⁡(X,1)∣X]=R[Y⁡(1)−q⁡(X,1)∣X]E_R[(A/e(X))\Y-q(X,1)\ X]=E_R[Y(1)-q(X,1) X] and the analogous identity for A=0A=0. Substituting into Eq. (1) cancels q and leaves τ0(X) _0(X). The same argument with q=μ^λq= μ_λ yields R[Zλ∣X]=τ0(X)E_R[Z^λ X]= _0(X) and R[Δ∣X]=0E_R[ X]=0. Thm. 1. RCT-validity gives R[Zλ]=θ0E_R[Z^λ]= _0. Exact transport gives O[r0g^]=R[g^]E_O[r_0 g]=E_R[ g]. These two identities prove unbiasedness of θ^r0 θ_r_0. Replacing r0r_0 by r~ r leaves the drift Eq. (8). Centering gives θ^r0−θ0=(ℙR,n−PR)(Zλ−ωg^)+(ℙO,N−PO)(ωr0g^) θ_r_0- _0=(P_R,n-P_R)(Z^λ-ω g)+(P_O,N-P_O)(ω r_0 g). Independence of the evaluation blocks and the usual variance of an empirical mean give Var⁡θ^r0(λ,ω)=1nVarR(Zλ−ωg^)+ω2NVarO(r0g^)Var\ θ_r_0(λ,ω)\= 1nVar_R(Z^λ-ω g)+ ω^2NVar_O(r_0 g); substituting Zλ=Z0+λΔZ^λ=Z^0+λ and CovR(Δ,g^)=0Cov_R( , g)=0 gives Eq. (5). Thm. 1 (minimizer) and Cor. 1.1. Expand VarR(Z0+λΔ−ωg^)Var_R(Z^0+λ -ω g) and use CovR(Δ,g^)=0Cov_R( , g)=0. The resulting quadratic is separated. The unique critical point when A,B>0A,B>0 is λ⋆=−C/Aλ =-C/A, ω⋆=D/Bω =D/B. Completing the square yields the stated minimum. The identity D=CovR(τ0(X),g^(X))D=Cov_R( _0(X), g(X)) follows from R[Z0∣X]=τ0(X)E_R[Z^0 X]= _0(X). Also, C=R[CovR(Z0,Δ∣X)]+CovRR[Z0∣X],R[Δ∣X]C=E_R[Cov_R(Z^0, X)]+Cov_R\E_R[Z^0 X],E_R[ X]\ and the second term is zero. If μ^R μ_R equals the trial regression, Z0−τ0(X)=A/e⁡(X)ε(1)−(1−A)/(1−e⁡(X))ε(0)Z^0- _0(X)=\A/e(X)\ (1)-\(1-A)/(1-e(X))\ (0) with ε⁡(a)≜Y−μR(X,a) (a) Y- _R(X,a) and R[ε(a)∣A=a,X]=0E_R[ (a) A=a,X]=0, while Δ is a function of (A,X)(A,X), so C=0C=0. Prop. 1. Under Assumption 2, the evaluation samples are independent of μ^R μ_R and μ^O μ_O, so ZλZ^λ and g g are fixed functions given the fits, and r~ r is fixed. Taking the mean of each sample average in Eq. (2) gives ⁡[θ^r~(λ,ω)]=R[Zλ]+ω⁡O[r~g^]−R[g^]E[ θ_ r(λ,ω)]=E_R[Z^λ]+ω\E_O[ r g]-E_R[ g]\. Lemma 1 with q=μ^λq= μ_λ gives R[Zλ]=R[τ0(X)]=θ0E_R[Z^λ]=E_R[ _0(X)]= _0, and the braces equal g^(r~) imb_ g( r) by definition, which proves Eq. (8). The second form of the imbalance, O[r~f]−R[f]=O[(r~−r0)f]E_O[ rf]-E_R[f]=E_O[( r-r_0)f], is Prop. 2(i) with h=fh=f. Prop. 2. (i) Under Assumption 3(i), r0=dPRX/dPOXr_0=dP_R^X/dP_O^X exists, and the Radon–Nikodym theorem gives O[r0B]=PRX(B)=R[B]E_O[r_01_B]=P_R^X(B)=E_R[1_B] for every measurable B⊆B . Linearity extends this identity to simple functions, and monotone convergence extends it to every nonnegative measurable h. For h∈L1(PRX)h∈ L^1(P_R^X), applying it to h+h^+ and h−h^- shows that both parts are finite, so r0h∈L1(POX)r_0h∈ L^1(P_O^X) and O[r0h]=R[h]E_O[r_0h]=E_R[h]. For uniqueness, let w∈L1(POX)w∈ L^1(P_O^X) satisfy O[wh]=R[h]E_O[wh]=E_R[h] for every bounded h. With B≜w>r0B \w>r_0\ and h=Bh=1_B, the hypothesis and the identity just shown give O[(w−r0)B]=PRX(B)−PRX(B)=0E_O[(w-r_0)1_B]=P_R^X(B)-P_R^X(B)=0. The integrand is nonnegative and is positive on B, so POX(B)=0P_O^X(B)=0. The same argument with B=w<r0B=\w<r_0\ gives w=r0w=r_0 POXP_O^X-a.s. (i) Let a∈L2(POX)a∈ L^2(P_O^X). By part (i) applied to |a||a| and the Cauchy–Schwarz inequality, R|a|=O[r0|a|]≤O[r02]O[a2]1/2<∞E_R|a|=E_O[r_0|a|]≤\E_O[r_0^2]\,E_O[a^2]\^1/2<∞, so part (i) applies to h=ah=a and gives R[a]=O[r0a]E_R[a]=E_O[r_0a]. Hence O[a2]−2R[a]=O[(a−r0)2]−O[r02]E_O[a^2]-2E_R[a]=E_O[(a-r_0)^2]-E_O[r_0^2]. The last term does not depend on a, and the first is zero only when a=r0a=r_0 POXP_O^X-a.s., so r0r_0 is the unique minimizer over L2(POX)L^2(P_O^X). Thm. 2. Condition on the fitted and selected objects. Unbiased sample variances on the two independent evaluation blocks prove ⁡[V^]=Var⁡(θ^r)E[ V]=Var( θ_r) for any fixed r. The studentized limit is a two-sample triangular-array CLT under the (2+η)(2+η)-moment (Lyapunov) condition of Assumption 4, plus Slutsky using V^/Vn→p1 V/V_n _p1. Variance unbiasedness is not a CLT. Estimated ratio. Write r^g^=r0g^+(r^−r0)g r g=r_0 g+( r-r_0) g on the OBS tuning sample to see that r r changes B B only. On evaluation, θ^r^−θ^r0=ω^ℙO,N(r^−r0)g θ_ r- θ_r_0= ω\,P_O,N\( r-r_0) g\. The conditional mean is θ0+ω^O[r^g^]−R[g^]=θ0+ω^g^(r^) _0+ ω\E_O[ r g]-E_R[ g]\= _0+ ω\, imb_ g( r), and R[g^]=O[r0g^]E_R[ g]=E_O[r_0 g] gives g^(r^)=O[(r^−r0)g^] imb_ g( r)=E_O[( r-r_0) g]. Calibration noise (Thm. 2(i)). Under Assumption 4, calibration with g g among the features gives ℙOnuis(r^g^)=ℙRnuis(g^)P_O^nuis( r g)=P_R^nuis( g), so g^(r^)=(ℙRnuis−R)g^−(ℙOnuis−O)(r^g^) imb_ g( r)=(P_R^nuis-E_R) g-(P_O^nuis-E_O)( r g). Since g g is fitted on DOnuisD_O^nuis, the first term is, given DOnuisD_O^nuis, an average of nnuisn_nuis independent centered terms, and g^→g¯ g→ g replaces it by (ℙRnuis−R)g¯(P_R^nuis-E_R) g up to op(nnuis−1/2)o_p(n_nuis^-1/2). The second term is linearized by assumption. Because g^(r^)=Op(n−1/2) imb_ g( r)=O_p(n^-1/2), (ω^−ω¯)g^(r^)=op(n−1/2)( ω- ω) imb_ g( r)=o_p(n^-1/2). Hence θ^−θ0 θ- _0 equals, up to op(n−1/2)o_p(n^-1/2), a sum of four centered averages over the independent samples DRevalD_R^eval, DOevalD_O^eval, DRnuisD_R^nuis, and DOnuisD_O^nuis, with total variance Vn+VcalV_n+V_cal to first order. The Lyapunov central limit theorem and Slutsky’s lemma give the coverage. Under Assumption 3(i) with r^≡1 r≡ 1, g^(r^)=0 imb_ g( r)=0 exactly and V^cal=0 V_cal=0. Thm. 3. The cross term in R[(Zλ−τ0)+(τ0−t)2]E_R[\(Z^λ- _0)+( _0-t)\^2] vanishes by R[Zλ−τ0∣X]=0E_R[Z^λ- _0 X]=0, so ℛDRF=QDRF(λ)+ℛ0(t)R_DRF=Q_DRF(λ)+R_0(t). For the R loss, write Y−mλ−(A−e)t=U+(A−e)(τ0−t)Y-m_λ-(A-e)t=U+(A-e)( _0-t) with U≜Y−mλ−(A−e)τ0U Y-m_λ-(A-e) _0; randomization gives R[U⁡(A−e)∣X]=0E_R[U(A-e) X]=0. For the ω terms, R[κ∣X]=1E_R[ _ mdl X]=1 and O[r0h]=R[h]E_O[r_0h]=E_R[h] give ⁡ℙOtune[r(g^−t)2]−ℙRtune[κ(g^−t)2]=O[(r−r0)(g^−t)2]E\P_O^tune[r( g-t)^2]-P_R^tune[ _ mdl( g-t)^2]\=E_O[(r-r_0)( g-t)^2], which is zero at r=r0r=r_0. Prop. 3. Fix the fitted regressions and r. Under Assumption 2, the tuning averages in Eq. (17) are independent of the fits, so ⁡[ℛ^(t,λ,ω,r)]=R[κZλ−t2]+ωO[r(g^−t)2]−ωR[κ(g^−t)2]E[ R_ mdl(t;λ,ω,r)]=E_R[ _ mdl\Z^λ_ mdl-t\^2]+ _O[r( g-t)^2]- _R[ _ mdl( g-t)^2]. Along t+εst+ s this is a quadratic in ε , and Definition 6 gives Ψ(t)[s]= _ mdl(t)[s]= R[κZλ−ts]+ωO[r⁡(g^−t)s]−ωR[κ(g^−t)s]. _R[ _ mdl\Z^λ_ mdl-t\s]+ _O[r( g-t)s]- _R[ _ mdl( g-t)s]. Each perturbation below changes Ψ(t)[s] _ mdl(t)[s] affinely in ε , so its derivative at ε=0 =0 is the coefficient of ε . Since e⁡(X)=PR(A=1∣X)e(X)=P_R(A=1 X), we have R[A−e∣X]=0E_R[A-e X]=0 and R[(A−e)2∣X]=e⁡(1−e)E_R[(A-e)^2 X]=e(1-e), so R[κ∣X]=1E_R[ _ mdl X]=1 for both learners. (i) Along μ^R+εh μ_R+ h, only ZλZ^λ_ mdl changes, through μ^λ+ε⁡(1−λ)h μ_λ+ (1-λ)h. For DRF, the score in Eq. (1) is affine in q, and the coefficient of ε in ZλZ^λ is (1−λ)[h⁡(X,1)1−A/e−h⁡(X,0)1−(1−A)/(1−e)](1-λ)[h(X,1)\1-A/e\-h(X,0)\1-(1-A)/(1-e)\], whose conditional mean given X is zero. For RF, κRFZRFλ=(A−e)(Y−mλ)/e⁡(1−e) _RFZ^λ_RF=(A-e)(Y-m_λ)/\e(1-e)\, and mλm_λ changes by ε⁡(1−λ)eh(X,1)+(1−e)h(X,0) (1-λ)\e\,h(X,1)+(1-e)h(X,0)\, so the coefficient of ε is −(1−λ)(A−e)eh(X,1)+(1−e)h(X,0)/e(1−e)-(1-λ)(A-e)\e\,h(X,1)+(1-e)h(X,0)\/\e(1-e)\, whose conditional mean given X is also zero. Since s is a function of X, the derivative is 00 for every r. (i) Along μ^O+εh μ_O+ h, μ^λ μ_λ changes by ελh λ h and g g by εΔh h. The first change contributes zero by the argument of part (i) with λ in place of 1−λ1-λ. The second contributes ωO[rΔhs]−ωR[κΔhs]=ω⁡O[rΔhs]−R[Δhs] _O[r\, h\,s]- _R[ _ mdl h\,s]=ω\E_O[r\, h\,s]-E_R[ h\,s]\, because R[κ∣X]=1E_R[ _ mdl X]=1. Prop. 2(i) with h replaced by Δhs h\,s gives R[Δhs]=O[r0Δhs]E_R[ h\,s]=E_O[r_0 h\,s], so the derivative is ωO[(r−r0)Δhs] _O[(r-r_0) h\,s]. (i) Along r+εhrr+ h_r, only the OBS term changes, and the derivative is ωO[hrs(g^−t)] _O[h_r\,s( g-t)]. Hence the derivative in (i) is zero for every r, the derivative in (i) is zero for all h and s when r=r0r=r_0, and the derivative in (i) is zero for all hrh_r when g^=t g=t POXP_O^X-a.s. For the converse statements, let ω≠0ω≠ 0, and let T contain the constant function 11 and a function that is not POXP_O^X-a.s. constant. If r≠r0r≠ r_0 on a set of positive POXP_O^X-probability, then h⁡(x,1)=sign⁡r⁡(x)−r0(x)h(x,1)=sign\r(x)-r_0(x)\, h⁡(x,0)=0h(x,0)=0, and s=1s=1 give ωO|r−r0|≠0 _O|r-r_0|≠ 0 in (i). In (i), take hr=B−POX(B)h_r=1_B-P_O^X(B) for measurable B⊆B . A zero derivative for every such hrh_r gives O[W−OWB]=0E_O[\W-E_OW\1_B]=0 for every B, with W≜s⁡(g^−t)W s( g-t), so W is POXP_O^X-a.s. constant. With s=1s=1, g^−t=c g-t=c for a constant c, and with the non-constant s, csc\,s is constant only if c=0c=0. Thus orthogonality to r forces g^=t g=t POXP_O^X-a.s. Thm. 4. Condition on fitted objects. The validation score has mean QDRF(0)+ℛ0(τ^)+ωO[(r^−r0)(g^−τ^)2]Q_DRF(0)+R_0( τ_ mdl)+ _O[( r-r_0)( g- τ_ mdl)^2]. Hoeffding on the M bounded RCT and OBS averages, with a union bound, yields the uniform deviation εeval(α) _eval(α). For ω∈[0,1]ω∈[0,1], each RCT summand Z0−τ^2−ωG\Z^0- τ_ mdl\^2-ω G lies in an interval of length 8B28B^2 and each OBS summand ωrGω rG in an interval of length 4r¯B24 rB^2. A two-sided Hoeffding bound with failure probability α/(2M)α/(2M) for each of the 2M2M averages gives the RCT term 2B22log⁡(4M/α) 2n−1/22B^2 2 (4M/α)\,2n^-1/2 and the OBS term 2B22log⁡(4M/α)r¯N−1/22B^2 2 (4M/α)\, rN^-1/2. The selected pair then lies within 2εeval(α)2 _eval(α) plus twice the largest transport error of the best grid risk, which is at most the (0,0)(0,0) risk. Details for Sec. 4.3. Best sieve approximation. Since R[ζ⁡(τ0−ζ⊤βp)]=0E_R[ζ( _0-ζ _p)]=0, ℛ0(ζ⊤β)=R[ζ⊤(β−βp)+(τp⋆−τ0)2]=‖β−βp‖ΓR2+ℛ0(τp⋆)R_0(ζ β)=E_R[\ζ (β- _p)+(τ _p- _0)\^2]=\|β- _p\|^2_ _R+R_0(τ _p), which is minimized at β=βpβ= _p. Let ΓR≜R[ζζ⊤]≻0 _R _R[ζ\,ζ ] 0 and βp≜ΓR−1R[ζτ(X)] _p _R^-1E_R[ζτ(X)], so τp⋆=ζ⊤βpτ _p=ζ _p. Since ℛ^(ζ⊤β,λ,ω,r^) R_ mdl(ζ β;λ,ω, r) is quadratic in β with Hessian 2H^(ω)⪰02 H_ mdl(ω) 0, every minimizer solves the normal equations H^(ω)+ρIβ=ℙRtuneκζ(Zλ−ωg^)+ωℙOtune(r^ζg^)\ H_ mdl(ω)+ρ I\β=P_R^tune\ _ mdlζ(Z_ mdl^λ-ω g)\+ _O^tune( rζ g), whose right side is affine in λ; this proves Prop. 4. Prop. 5. Because R[κ∣X]=1E_R[ _ mdl X]=1 and R[κZλ∣X]=τ0(X)E_R[ _ mdlZ^λ_ mdl X]= _0(X), Assumption 2 gives ⁡[H^(ω)]+ρI=H¯ρE[ H_ mdl(ω)]+ρ I= H_ρ, and the right side of Eq. (27) has mean ΓRβp+ωζg^(r^) _R _p+ω\, imb_ζ g( r). Hence β¯=H¯ρ−1ΓRβp+ωζg^(r^) β= H_ρ^-1\ _R _p+ω\, imb_ζ g( r)\, and subtracting H^(ω)+ρIβ¯\ H_ mdl(ω)+ρ I\ β from both sides of Eq. (27) gives H^(ω)+ρI(β^−β¯)=ψ¯\ H_ mdl(ω)+ρ I\( β_ mdl- β)= ψ_ mdl, where ψ¯≜ℙRtuneψR,+ℙOtuneψO−ρβ¯ ψ_ mdl _R^tune _R, mdl+P_O^tune _O-ρ β has mean zero and variance V_ mdl. Thus β^−β¯=H¯ρ−1ψ¯+ β_ mdl- β= H_ρ^-1 ψ_ mdl+ rem, so ⁡[β^]=β¯+⁡[]E[ β_ mdl]= β+E[ rem] with ∥[]∥ΓR≤(∥ΓR2)1/2=oP(nR−1/2+nO−1/2)\|E[ rem]\|_ _R≤(E\| rem\|^2_ _R)^1/2=o_P(n_R^-1/2+n_O^-1/2). Subtracting H¯ρβp H_ρ _p from ΓRβp+ωζg^(r^) _R _p+ω\, imb_ζ g( r) gives β¯−βp=H¯ρ−1[ω⁡ζg^(r^)−ζζ⊤(r^)βp−ρβp] β- _p= H_ρ^-1[ω\ imb_ζ g( r)- imb_ζ ( r) _p\-ρ _p]. Since f(r^)=O[(r^−r0)f] imb_f( r)=E_O[( r-r_0)f] under Assumption 3 and ζ⊤βp=τp⋆ζ _p=τ _p, the braces equal O[(r^−r0)ζ(g^−τp⋆)]E_O[( r-r_0)ζ( g-τ _p)], which gives the display for β¯(ω)−βp β(ω)- _p. For the bound, (ω)1/2=‖ΓR1/2(β¯−βp)‖≤‖ΓR1/2H¯ρ−1‖op‖ωO[(r^−r0)ζ(g^−τp⋆)]−ρβp‖ bias(ω)^1/2=\| _R^1/2( β- _p)\|≤\| _R^1/2 H_ρ^-1\|_op\|ω\,E_O[( r-r_0)ζ( g-τ _p)]-ρ _p\|, and Jensen’s and the Cauchy–Schwarz inequalities give ‖O[(r^−r0)ζ(g^−τp⋆)]‖≤O(r^−r0)2O‖ζ⁡(g^−τp⋆)‖21/2\|E_O[( r-r_0)ζ( g-τ _p)]\|≤\E_O( r-r_0)^2\,E_O\|ζ( g-τ _p)\|^2\^1/2, which proves the proposition. Thm. 5. By the identity ℛ0(ζ⊤β)=ℛ0(τp⋆)+‖β−βp‖ΓR2R_0(ζ β)=R_0(τ _p)+\|β- _p\|^2_ _R shown above, ℛ0(τ^)=ℛ0(τp⋆)+‖β^−βp‖ΓR2R_0( τ_ mdl)=R_0(τ _p)+\| β_ mdl- _p\|^2_ _R. Write β^−βp=(β^−β¯)+(β¯−βp) β_ mdl- _p=( β_ mdl- β)+( β- _p). Since H¯ρ−1ψ¯ H_ρ^-1 ψ_ mdl has mean zero and covariance H¯ρ−1VH¯ρ−1 H_ρ^-1V_ mdl H_ρ^-1, the Cauchy–Schwarz inequality gives ‖β^−β¯‖ΓR2=+oP(nR−1+nO−1)E\| β_ mdl- β\|^2_ _R=J_ mdl+o_P(n_R^-1+n_O^-1). The cross term is 2[]⊤ΓR(β¯−βp)=oP(nR−1+nO−1)2E[ rem] _R( β- _p)=o_P(n_R^-1+n_O^-1) by assumption, which proves the theorem. The two conditions on rem are assumptions of Prop. 5 and Thm. 5; they hold under the usual fixed-p expansion of β β_ mdl, which we do not verify here. Prop. 6. By the subgradient condition in Sec. 3.2, ℙOnuis(r^fk)−ℙRnuis(fk)=ekP_O^nuis( rf_k)-P_R^nuis(f_k)=e_k with |ek|≤δ^k|e_k|≤ δ_k. Subtracting this from fk(r^)=O[r^fk]−R[fk] imb_f_k( r)=E_O[ rf_k]-E_R[f_k] gives the display, and linearity of g↦g(r^)g imb_g( r) extends it to linear combinations of the features. Part (i) takes g=g^g= g; Thm. 2(i) treats the resulting drift as noise with variance V^cal V_cal. For part (i), the ω term of Prop. 5 is ζg^(r^)−ζζ⊤(r^)βp imb_ζ g( r)- imb_ζ\,ζ ( r) _p, and for a candidate τ^=ζ⊤β τ=ζ β in ζT_ζ, (g^−τ^)2=g^2−2β⊤ζg^+β⊤ζζ⊤β( g- τ)^2= g^2-2β ζ g+β ζ\,ζ β, so ϱ=ω(g^−τ^)2(r^) _ mdl=ω\, imb_( g- τ)^2( r). Both are fixed linear combinations of the balanced imbalances given the tuning samples. Each nuisance-sample average in the display is OP(nnuis−1/2+Nnuis−1/2)O_P(n_nuis^-1/2+N_nuis^-1/2) under the conditions of Thm. 2(i), and so is δ^k δ_k. Appendix B Additional Simulation Results This appendix gives the bias, standard deviation, and Wald intervals of the ATE estimates (Tables 13 and 14) and the absolute errors behind Fig. 4 (Fig. 7). Table 13: ATE over 1,000 replications on the synthetic design: bias, standard deviation, and root mean squared error, mean estimated standard error (SE), coverage, and mean width of the 95% Wald interval. Shrinkage, Pretest, and the naive pool have no interval. Design n Method Bias SD RMSE SE Cov. Width Synthetic, r0=1r_0=1 300 RCT only 0.007 0.302 0.302 0.298 0.95 1.169 PPI++ 0.007 0.294 0.294 0.294 0.95 1.152 Fusion 0.003 0.225 0.225 0.219 0.94 0.857 Shrinkage 0.077 0.318 0.327 – – – Pretest 0.014 0.326 0.326 – – – Naive pool 1.255 0.117 1.260 – – – 900 RCT only -0.007 0.139 0.139 0.143 0.95 0.562 PPI++ -0.009 0.137 0.137 0.140 0.95 0.550 Fusion -0.009 0.120 0.121 0.126 0.96 0.493 Shrinkage 0.011 0.141 0.141 – – – Pretest -0.007 0.139 0.139 – – – Naive pool 1.185 0.067 1.187 – – – 3000 RCT only -0.002 0.073 0.073 0.072 0.95 0.283 PPI++ -0.002 0.073 0.073 0.071 0.95 0.279 Fusion -0.002 0.072 0.072 0.070 0.95 0.273 Shrinkage 0.003 0.073 0.073 – – – Pretest -0.002 0.073 0.073 – – – Naive pool 0.991 0.039 0.992 – – – Synthetic, r0≠1r_0≠1 300 RCT only 0.007 0.302 0.302 0.298 0.95 1.169 PPI++ 0.002 0.304 0.304 0.303 0.95 1.186 Fusion 0.002 0.234 0.234 0.226 0.94 0.886 Shrinkage 0.080 0.319 0.328 – – – Pretest 0.016 0.331 0.331 – – – Naive pool 1.211 0.117 1.217 – – – 900 RCT only -0.007 0.139 0.139 0.143 0.95 0.562 PPI++ -0.007 0.143 0.143 0.146 0.96 0.574 Fusion -0.005 0.124 0.124 0.130 0.95 0.510 Shrinkage 0.012 0.141 0.142 – – – Pretest -0.007 0.139 0.139 – – – Naive pool 1.091 0.068 1.093 – – – 3000 RCT only -0.002 0.073 0.073 0.072 0.95 0.283 PPI++ -0.003 0.075 0.075 0.074 0.95 0.291 Fusion -0.002 0.073 0.073 0.071 0.95 0.280 Shrinkage 0.004 0.073 0.073 – – – Pretest -0.002 0.073 0.073 – – – Naive pool 0.843 0.043 0.844 – – – Table 14: As Table 13, for IHDP and ACIC 2016. Design n Method Bias SD RMSE SE Cov. Width IHDP 300 RCT only -0.007 0.255 0.255 0.250 0.96 0.981 PPI++ -0.008 0.261 0.261 0.253 0.96 0.991 Fusion -0.002 0.177 0.177 0.182 0.95 0.713 Shrinkage 0.066 0.274 0.281 – – – Pretest 0.034 0.337 0.339 – – – Naive pool 0.831 0.073 0.835 – – – 900 RCT only -0.003 0.114 0.114 0.116 0.96 0.456 PPI++ -0.003 0.116 0.116 0.118 0.96 0.463 Fusion -0.002 0.105 0.105 0.104 0.95 0.409 Shrinkage 0.015 0.117 0.118 – – – Pretest -0.003 0.114 0.114 – – – Naive pool 0.772 0.047 0.773 – – – 3000 RCT only 0.000 0.056 0.056 0.058 0.96 0.228 PPI++ 0.000 0.057 0.057 0.059 0.96 0.232 Fusion 0.000 0.055 0.055 0.057 0.96 0.224 Shrinkage 0.006 0.057 0.057 – – – Pretest 0.000 0.056 0.056 – – – Naive pool 0.633 0.038 0.635 – – – ACIC 300 RCT only 0.003 0.278 0.278 0.287 0.96 1.124 PPI++ -0.001 0.281 0.281 0.289 0.95 1.132 Fusion -0.009 0.190 0.190 0.187 0.94 0.732 Shrinkage 0.095 0.297 0.312 – – – Pretest 0.081 0.401 0.408 – – – Naive pool 0.811 0.063 0.813 – – – 900 RCT only -0.005 0.163 0.163 0.160 0.94 0.629 PPI++ -0.006 0.164 0.164 0.161 0.94 0.632 Fusion -0.003 0.107 0.107 0.107 0.95 0.421 Shrinkage 0.028 0.170 0.173 – – – Pretest -0.005 0.165 0.165 – – – Naive pool 0.759 0.044 0.760 – – – 3000 RCT only 0.003 0.067 0.067 0.066 0.94 0.259 PPI++ 0.004 0.068 0.068 0.067 0.95 0.261 Fusion 0.003 0.060 0.060 0.059 0.95 0.230 Shrinkage 0.010 0.068 0.069 – – – Pretest 0.003 0.067 0.067 – – – Naive pool 0.633 0.031 0.634 – – – Figure 7: Absolute errors for the designs of Fig. 4, columns 1–4. Top: ATE MSE; bottom: CATE risk. 95% bootstrap intervals.

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