Data Fusion for Partial Identification of Causal Effects
Authors: Quinn Lanners
Cynthia Rudin
Alexander Volfovsky
Harsh Parikh
Venue: NeurIPS 2025 poster ()
Keywords: causal inference
partial identification
data fusion
double machine learning
double robustness
causality
Primary area: probabilistic_methods
TL;DR: We develop a framework for partial identification of causal effects under violations of key assumptions when combining heterogeneous data sources.

Abstract: Data fusion techniques integrate information from heterogeneous data sources to improve learning, generalization, and decision-making across data sciences. In causal inference, these methods leverage rich observational data to improve causal effect estimation, while maintaining the trustworthiness of randomized controlled trials. Existing approaches often relax the strong "no unobserved confounding" assumption by instead assuming exchangeability of counterfactual outcomes across data sources. However, when both assumptions simultaneously fail—a common scenario in practice—current methods cannot identify or estimate causal effects. We address this limitation by proposing a novel partial identification framework that enables researchers to answer key questions such as: *Is the causal effect positive/negative?* and *How severe must assumption violations be to overturn this conclusion?* Our approach introduces interpretable sensitivity parameters that quantify assumption violations and derives corresponding causal effect bounds. We develop doubly robust estimators for these bounds and operationalize breakdown frontier analysis to understand how causal conclusions change as assumption violations increase. We apply our framework to the Project STAR study, which investigates the effect of classroom size on students’ third-grade standardized test performance. Our analysis reveals that the Project STAR results are robust to simultaneous violations of key assumptions, both on average and across various subgroups of interest. This strengthens confidence in the study's conclusions despite potential unmeasured biases in the data.

Data Fusion for Partial Identification of Causal Effects
Quinn Lanners Duke University qml@duke edu
Cynthia Rudin Duke University cynthia@cs. duke.
Alexander Volfovsky
Harsh Parikh
Duke University av1360duke edu
Yale University harsh. wwrsh.parikh@yale..cm edu
Abstract
Data fusion techniques integrate information from heterogeneous data sources to improve learning, generalization, and decision-makinn across data sciences. In causal inference, these methods leverage rich observational data to improve causal effect estimation, while maintaining the trustworthiness of randomized controlled trials. Existing approaches often relax the strong "no unobserved confounding" assumption by instead assuming exchangeability of counterfactual outcomes across data sources. However, when both assumptions simultaneously fail-a common scenario in practice-current methods cannot identify or estimate causal effects. We address this limitation by proposing novel partial identification framework that enables researchers to answer key questions such as: the causal effect posiriieenneattive? and How severe must assumption violations be to overturn this conclusion? Our approach introduces interpretable sensitivity parameters that quantify assumption violations and derives corresponding causal effect bounds. We develop doubly robust estimators for these bounds and operationalize breakdown frontier analysis to understand how causal conclusions change assumption violations increase. We apply our framework to the Project STAR study, which investigates the effect of classroom size students' third-grade standardized performance. Our analysis reveals that the Project STAR results are robust to simultaneous violations of key assumptions, both average and across various subgroups of interest. This strengthens confidence in the study's conclusions despite potential unmeasured biases in the data.
Introduction
Modern evidence-based Decision-making increasingly relies on combining information from various sources a practice known as data fusion. From integrating satellite imagery across multiple spatial resolutions (Tang et al., 20166] Yan al. (202)) to merging genetic markers with electronic health records (Hall et al. 2016 Conroy et al. [023] Zawistowski et al. 2023), data fusion enables more robust, generalizable, and efficient analysis (Meng et al., (020). In causal inference, data fusion has emerged popular paradigm, recently recognized as of the top ten research directions for advancing the field (Mitra et al.) 20222] Bareinboim and Pearl 2016). Data fusion approaches in causal inference have focused on generalizing transporting evidence from experimental studies (Degtiar and Rose 2023 Pearl 2015 Dahabreh et al. 2019 Lu al. 2019), precisely estimating heterogeneous causal effects (Brantner et al. 2023 Yang al. al.] 2023). improving efficiency (Rosen-
man et al. 2023 Lin et al.) 2025), and mitigating estimation bias (Kallus et al. 2018 Colnet et al. 2024b), among other things.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Consider the example of the Project STAR study, which investigated the impact of class size on students' academic performance. The dataset comprises both randomized controlled trial (RCT)- where students were randomly assigned to different classroom sizes and an observational cohort
where students self-selected into classrooms (Mosteller [995] Achilles et al. 2008). While the experimental data likely ensures internal validity, it may suffer from limited external validity or generalizability (von Hippel and Wagner 2018; Justman| 2018). Conversely, the observational data better reflects real-world settings but may suffer from unobserved confounding (Athey al. 2020
Parikh et al. 2023). Merging the two sources can yield more precise and externally valid treatment effect estimates under milder, partially testable assumptions (Parikh et al., 2023] Wu and Yang 2022). Specifically, the average treatment effect (ATE) becomes identifiable if either the RCT generalizes well to the target population or the observational data satisfies unmeasured confounding (NUC)
Lin et al. 2025 Yang et al. 2023).
However, if both assumptions fail - i.e., the RCT lacks external validity and the observational data is confounded then the treatment effect no longer point-identifiable, even in the limit of infinite data. In such settings, classical estimators break down. Nevertheless, researchers may still answer important questions like: the treatment effect positive? or How severe must assumption violations be to overturn this conclusion? These are questions of partial identification, where the goal is to estimate plausible range or bounds on the treatment effect rather than a single point estimate (Corfiilld et al. 1959 Manski| 2003).
While a rich literature on partial identification exists, many approaches rely on strong distributional or parametric assumptions and often ignore opportunities to tighten bounds by leveraging multiple
datasets (Rosenbaum and Rubin 1983 [Blackwll]] 2014| Ding and FanderWeele 2016 Bonvini and Kennedy [022] Nguyen et 2017 2018 Nie et al. 20lne
Colnet et al.) 2022] Dahabreh et al. [023]
Huang 2024). This creates critical gap in sensitivity analysis frameworks that are both flexible and informative when combining experimental and observational data.
Contributions. We propose a general framework for partially identifying treatment effects by integrat- ing complementary strengths of experimental and observational studies. Our key contributions are:
1. We introduce interpretable sensitivity parameters, 7 and p. that quantify the extent of external validity violations and unmeasured confounding, respectively.
2. We develop double machine learning estimator based on the efficient influence function (EIF) for estimating treatment effect bounds function of (7, P), without relying strong distributional or parametric assumptions.
3. We operationalize efficient breakdown frontier analysis, which characterizes regions in the (7, p) space where the treatment effect remains conclusively positive (or negative) - allowing for assessment of the robustness of causal conclusions under simultaneous assumption violations.
Our framework enables comprehensive sensitivity analyses in data fusion settings, helping researchers transparently explore the consequences of assumption violations. By leveraging the internal validity of RCTs and the representativeness of observational studies, our approach yields tighter, more interpretable bounds on treatment effects and offers a principled way to assess robustness.
2 Preliminaries
We consider the setting where have sample D,, [1,....nn n. of units across experimental cohort (D.) and an observational study (D.)) drawn identically and independently from P. For each unit E Dn = D. DD. S. Li e D.] is a binary experimental cohort indicator, T, E {0,0 1} is the binary treatment indicator, Y; is the observed outcome, and X; is the vector of pretreatment covariates. We assume the outcome space is bounded and positive, but note that, without loss of generality, our approach can be applied to all bounded outcome scenarios by simply shifting the outcome domain. (0) and (1))) denote the two potential outcomes for unit i.
Typically, one is interested in using these datasets to estimate the following two standard estimands, namely the average treatment effect (ATE): treatment effect (CATE): r(x) = Ep (1)) - X
Ep EEEYYOO (1)) - (O)), and the conditional average
We assume the following standard conditions hold:

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Assumption A1. [SUTVA and Consistency). There is no interference between units, there is single version of each treatment, and for all units, Y; =T) + (1 (())
Assumption A2. [TTrealment Positivity). For s {0, 1} and all 3c > 0 such that
c < P(T = x x, s) 1-
Assumption A3. [Study Positivity]. For all x, 3e such that c < P(S X = x) 1 Assumption A4. [Internal Validity of the Experiment). (Y( Y()) T x
However, we acknowledge the possibility of unobserved confounders that concurrently influence S, T, and Due to such unobserved confounding. the following exchangeability assumptions, which are standard in the literature, may fail to hold:
Assumption A5. /No Unobserved Confounding (NUC) in the Observational Data]. (Y (Y()), (1)) T X x
Assumption A6. [Study Exchangeabilityy- (Y) (1)) 1 X =
In this paper, we explicitly consider scenarios in which A5 and A6 assumptions are simultaneously violated, thereby challenging the point identifiability of and (x))
Discussion of Assumptions. Al imposes the standard SUTVA and consistency conditions, ensuring that potential outcomes well-defined for each treatment level and that the observed outcome corresponds to the treatment received. A2 is the standard treatment positivity assumption, ensuring overlap between treated and control groups. A4 and A5 structurally equivalent, differing only in the sample subset (experimental vs. observational units). Internal validity in RCTs is generally accepted due randomization, whereas NUC is stronger, treatment may depend on unobserved confounders.
Combining experimental and observational samples requires the additional A3 and A6 assumptions. statee that each unit must have a nonzero probability of being an experimental unit and necessary to ensure overlap between the two study cohorts. A6iis is the study exchangeability assumption, which states that, conditional on covariates, potential outcomes are exchangcable across studies. Like NUC, it can be strong assumption, study participation may depend on unobservables.
2.1 Quantifying Assumption Violations
We introduce two additional terms, P and 7. that separately quantify violations of A5 and A6 respectively. The value of p 2 0 quantifies the level of unobserved confounding in the observational data, corresponding to violation of ff55]when p >> 0. The value of 7 2 0 quantifies the difference in potential outcomes between the RCT data and the observational data, corresponding to violation AA when y > O. Both terms report the level of violation as relative measures of the observed outcomes. For example, p 0.2 corresponds to the setting that unobserved confounding in the observational dataset affects outcomes by 20%. In Section 4, we formally define and y, expand on their interpretation, and employ them sensitivity parameters for partial identification of 7 and 7(x).
2.2 Breakdown Frontiers
Masten and Poirier (2020) introduce an approach for visualizing how conclusions about a parameter of interest vary set of assumptions are relaxed. We leverage this framework to plot how treatment effect conclusions change as we relax A5 and A6 via our sensitivity parameters p and y.
3

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Figure TamreTillne Tillustrates a breakdown frontier plot constructed
20%
Conclusive
by bounding the treatment effect for various (p. 7) pairs,
Violation in Study Exchangeability (v)
Tentative
The x-axis represents violations of A5 expressed as per- centages corresponding (100 x p)%. Likewise, the y-
Inconclusive
Incompatible
axis represents violations of A6 expressed as (100 x y) %..
As a result, the bottom-left of the plot corresponds to
10%
stronger assumptions (small p. 7), and the top-right to weaker assumptions (large p. 7). For each (p, 7) pair, we
estimate upper and lower bounds on the treatment effect. These estimates divide the plot into four regions: (i) Con- clusive: The point estimates of the upper and lower bounds are both positive (or both negative), and both confidence in- tervals exclude zero the chosen confidence level (equiv- alent to the robust region of Masten and Poirier 2020). (ii) Tentative: The point estimates of the upper and lower bounds are both positive (or both negative). but at least one of the corresponding confidence intervals includes zero. (iii) Inconclusive:
0%
0%
10%
20%
Violation in NUC (p)
Figure 1: Example breakdown frontier plot for and 7.
The point estimates of the upper and lower bounds are not the same sign. (iv) Incompatible: The sensitivity parameter values lie outside the admissible range for this dataset; they imply assumptions that contradict observed discrepancies between the study groups (see Section 4.. We note that Figure Tis is an example for illustration, and that the size and shape of the regions vary by dataset.
3 Relevant Literature
Data Fusion for Causal Inference. Data fusion methods leverage randomized trials to mitigate unmeasured confounding in observational data and have become central causal inference. Broadly, approaches differ based on whether treatments and outcomes are observed only in the trial (Degtiar) and Rose 2023) or in both datasets (Brantner al. 2023 Lin al. 2024). Our setting aligns with the latter. A more detailed review is provided in Appendix
Partial Identification. Partial identification (ID) and sensitivity analysis frameworks are widely used to assess robustness to assumption violations in observational studies (Cornfield et al.) 1959
Rosenbaum and Rubin 1983 Liu et al. [2033] Ding and Vander Weele, 2016 Bonvini and Kennedy
2022). In data fusion, most sensitivity approaches focus on violations of study exchangeability when
observational treatments outcomes are missing (Nguyen al. 2017 2018 Nie al. 2021 Colnet
ct al. 2022 Dahabreh et al. 2023
Huang 2024).
In settings where treatments and outcomes are available from the observational cohort (like ours), both non-unmeasured confounding (NUC) and study exchangcability must be addressed. Most existing
work assumes exchangeability and focuses on NUC violations (Lin et al. 2024 2025; Triantafilouu
ctt al. 2023 Chen et al. [202]] Oberst et al. 2022;) Kallus et al. 2018 Yang et al. 20244) Rosenman
2023 Yang et al.] 2023). Yang et al. (2023) and and Parikh et al. (2023) propose tests for assumption
violations: the former attributing test failures to NUC, the latter recognizing that failures may stem
from either assumption, though requiring knowledge of which one fails difficult task in practice.
Partial Id w/ Data Fusion. While partial ID and sensitivity methods are well-developed for fusion without observational treatments/outcomme
they remain sparse when these are available. Related work includes partial ID approaches in contextual bandits (Joshi et al. [024a] and structural causal modeling with qualitative knowledge (Zhang et al. (022). Most closely related, Yu et al. (2024a) develop a two-parameter sensitivity analysis for fusion settings, addressing NUC and study positivity, but assuming study exchangeability.
4 Partial Identification
In this section, we present a general framework for partial identification of treatment effects under si- multaneous violations of the no unmeasured confounding assumption (A5) and study exchangeability (A6). When either assumption fails, point identification of the average treatment effect (ATE) and conditional average treatment effect (CATE) becomes impossible.

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To quantify the degree of these violations, we introduce two interpretable parameters: p. capturing the extent of unmeasured confounding in the observational cohort, and 7, capturing the extent of study exchangeability violation between the experimental and observational populations. We define:
Er (t) x - S - 0, 1 d]
(x X - S-S=00 - 0j
= sup x,
7= sup x,
Ep (+) X = x, S = 0, = 8
Ep (f)
X = x, S S= 1
The blue denominator terms represent quantities point- ceittiddetifiible without assumptions A or A6 whereas the orange numerator terms involve counterfactual quantities that are not directly observed. If[A or o[A3]or[6]holee holds, then - or y = 0, respectively, and treatment effects are point-identifablee Conversely, dissimilarity between (t) X = S = 0, = -1- - t] and Ep (Y()) X = x,
= reflects unobserved confounding (i.e., a violation A5 and leads to nonzero values of p. Similarly, dissimilarity between [ X = S 0j and Ep [Y (1) X x,S 1] reflects study selection bias (i.e. a violation A6 and corresponds to nonzero values of y. Both p and y are relative measures of dissimilarity; for example, P -1 implies 100% difference between the observed outcome expectation and the counterfactual counterpart. Note that while p and y could be extended to functions of x and over all rovariate-treatment profiles (x, t). This enables tractable. worst-case sensitivity analyses.
conservatively treat and 7 as scalars, taking the supremum
We chose a ratio-based sensitivity model in part for its across different outcome scales and covariate profiles. While alternative approaches, such as a difference-in-nne mn-me model, could also be used, we see this as a modeling choice rather than a struc- tural limitation, and our framework could accommodate such a formulation with mild adjustments.
which allows it to generalize
We now focus on using and to bound т and rxx) when A5 and A6 are simultaneously violated. For parsimony, we define the following estimable quantities:
study selection score: g. (x) (T = X = x. S = s),
( = s X x), (ii) treatment propensity score: e, ex(x,5 s)
and (iii) expected outcome: p(x, s,t) = Ep[Y X= x, S s, T t). Using the law of iterated expectation over study selection, we express:
Ep (t) X = x 91 (x)EE X = x, 1] + (x) (t)) x 0]
(I)
= (x) 1,1) + (x (1) X x x, 0),
where we invoke A4 replace the potential outcome expectation
x with the observed outcome expectation (x. 1. f). since treatment is randomized in the experimental study
(S=1) 1). In contrast, without (A or A6 the term Y X = x, 5 0)] remains unidentifiable. However, we can leverage P and y to construct sharp, identifiable bounds on Ep X = xx x.
0), and thereby obtain bounds on the overall potential outcome EEp (L) X = x). Towards this, we
define two estimable functions that upper bound (E(((()) X x, 0]:
v(x,,, t, 7) == (1 + ( 1, t), w(xx == ex () 0, + (x, 0) + p) 0, t).
The function vxx, 7) is derived from experimental study data. is based on the relative deviation
of (L) X = S = 0) from the identifiable quantíty Ep [ (f) X = S = 1| =|= (xx, 1, t), as governed by the parameter 7. Conversely, the function w(x, L, p) is derived from observational study data. It combines the identifiable component Ep (1) X S 0,T = 4 = ,x 0, (), weighted by the treatment propensity, ea (x. 0), with term that inflates the counterfactual component (Y) X S, 0, T = 1 t] according to the parameter p.
In Lemma 1 we combine the upper bounds u(x, f, 7) and w((x t, p). along with their lower bound counterparts ((x,t,->) and w(x, f, -p), to construct identifiable bounds on Ep ( X = x). Specifically, we replace the unidentifiable term Ep (t() X = x, S = 0) in Equation ! with the min of the two upper bounds and the max of the two lower bounds.
Lemma (Conditional Potential Outcome Bounds), Suppose Al hold. Then for any t E {0,1)
and given x, if v(x,t, -7) I w(x,t, p) and w(x,t, pp) I t,), the conditional potential
outcome satisfies
Ep X = x] E (xx,,, 7), u(xx [),, where (x,t,,,)) = gi (x)x (x, 1,t)+ (x) max (w(xx t, pp), v(x, t, -))), (xx,,, = (x) (xx 1,t) + (x) min (w(x t, p), t(xxt,xx)))) t,,)
5

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A full derivation of the results leading to Lemmal I is provided in Appendix B.1 results, we next derive bounds on the conditional and average treatment effects.
Building on this
Theorem 1 (Treatment Effect Bounds). Suppose A/ A4 hold and that for each x and t E {0. 1}, v( -?) S w t. p) and w(x, t, - p) $ v(x, t, y). Then, the conditional average treatment effect satisfies: 1(x, p. u(x, 0. p. 7) I r(x) I u(x, p, 1(x, 0, p. P.
Further, if this holds for all x such that Pp (x = x) > 0, then the average treatment effect satisfies:
Ep[1(X,,,,)) 1, u(X, O, P. 7)] S S Ep (u p. - 1(X, p,T)]
We derive doubly robust estimators for the treatment effect bounds established in Theorem Din the next section. Before proceeding, we discuss infeasibility conditions in [
Remark 1 ((In)compatible p and 7.)). Every (p. 7) pair corresponds to a data-generating pro- cess that could, in principle, have produced the observed data. However, some values of (p,7) imply assumptions that conflict with what we observe. To illustrate this, consider a scenario where 3(x,t) t) such that the conditional expectations of the outcome differ across study groups, i.e. EP |X X =x =, =
Ep X == = = A(t) > 0. Pariikh et al. (0223) shows that (1) > 0 implies that A3 and/or/A6 is violated. Therefore, setting (p,7) (0,0)
which implies both assumptions hold contradicts the observed difference (1)) >
More broadly, the bounds in Lemma 7 and Theorem 1 are valid only when both v(x, t. 7)) S
w(x, p) and w.x,t,-p) S v(x, t, ?); a violation makes the parameters incompatible. Because checking the inequalities at every (x, t) is infeasible in most settings, we test them in expectation over X for each treatment arm. The null distribution is estimated with a resampling test that keeps the fitted propensity and outcome models fixed, as generating resamples that satisfy the null and re-estimate these models is non-trivial. This may label some pairs incompatible that a full bootstrap would not. Intuition behind and estimation of (injcompatibility are discussed further in Appendix D
5 Semiparametrically Efficient Estimation
We now turn to the problem of estimating the bounds identified in Section 4 Our goal to construct doubly robust estimators that offer both statistical efficiency and robustness to model misspecification (Chernozhukov et al. 2018). However, a key challenge arises: the presence of non-differentiable max and min operators in our estimands makes it intractable to directly derive the efficient influence functions (EIFs) needed for such estimators.
Recent work has shown that statistical functionals of this form arise in a range of modern causal inference problems. Some papers establish necessary and sufficient conditions under which such non-smooth functionals are pathwise differentiable and thus admit direct efficient estimation without smoothing (Luedtke and Van Der Laan 2016 [Bnvvini and Kennedy| 2022, Kennedy et al., 2019). Other work studies smooth approximation- based strategies that replace non- con-diiffered wit-ddifffeeeabble operators with smooth counterparts that converge to the target functional (Levis et al., 2025 Ben- 2025). We follow the latter approach and introduce a smooth approximation based on the Boltzmann operator (Section 5.T). which provides a unified and computationally tractable way to approximate the bounds while enabling efficient estimation. This smoothing step allows us to derive the EIFs and construct birs-coreected estimators in Section 5.2 applies to bounds T(x).
While we focus on bounds for т, the same strategy
5.1 Smooth Bounds
For any 31, 22 E RR the Boltzmann operator of the form 21 + d2 where exp(cx2) A := exp(aœ1) + exp(022)) Xa == exp(0>11 exp(a32)
exp(oa))
This operator is similar to the popular softmax function in that \ + A2 1 and their relative magnitudes are linked to x; and V2 However, it differs in its incorporation of the hyperparameter a which causes Xaæ1 + Agzz approach max(x1, x22) as a + and min(x1, as - -00.
In Lemma 2 we show that the Boltzmann operator can be used to construct smooth approximations of our partial identification bounds. Specifically, we replace the functions ((X, t, p. 7) and u(X,t, t, p, 7)

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from Lemma with b(X, P. at), which uses the Boltzmann operator in place of the max and min functions. We then establish treatment effect bounds using these smooth approximations anaallog and show that, a increases, these estimates converge to the max and min bounds.
to Theorem -
Lemma 2 (Smooth Bounds). Consider a setting where A/ A4 hold but A5 and A6 may not. Define
b(X, (x7, = : (1 (xX,,,111.. 1,t) + (x( (x t, P, P. a)u + X2 (X, P, p, a,a),, where exp(av)
exp(aw)
\1 (, exp(av) + exp(aw)
A2 ((X,t, (.
exp(av) + exp(ow))
and v = vXX, 4,7), w= (X, t, p). Then for any > 0, p. and 7y such that Vt E {0, and Vx for which Pp = b(x,t, 7 t, 7, -a). follows that
Ep [0)bbXX,1 1, -, 7; 11000=_-000000000000000000000030
< Ep
1, -(x, P; a),
and
lim |ggp,,/||_.11111.000000000001110 w(x, p, a-+00
1, -
lim =, 01+000
1(x,0, 0.
5.2 Efficient Estimators
Having established smooth, differentiable approximations of our bounds, we now derive their corresponding EIFs, which form the basis for the bias-corrected estimators. We begin by defining 0t..P:7,0) = Ep b X, t, P.
7, a)]. Using this, we can express the bounds on as
01,-p. -7, 0(0, p,7, a)) S S 011, P: 7, a) 0(0, -p, -7,,),
where each bound written as difference between two instances of with different parameter settings. Therefore, once we establish an EIF for the general form O(t, 7, a),, we can obtain EIFs for both the upper and lower bounds by leveraging the linearity of EIFs.
Following the approach of Schuler and van der Laan (024), we derive the EIF for A(t, p. 7 a), denoted by o(Z;t, p, y, a:) with Z = [X, SYY.Y. The full, centered EIF is given below with each term tagged by a superscript (.) for reference. For brevity, we omit the explicit arguments of v, w, A and X2, which match those of
SI(T= =t) (Y 1(X, 1,
((Zit.p.7.)) =
(1 + A2w}
+
ex(X,1)
(i)
(e(T))g
+ ((1+))(A++ 1aan+aaddlee - w)
(Y -0(x,111))
es (x,11)9
IT(
+ - (1-5){A2 + -addd2(w -
- -,(X,0, + pe1--(XX00)
ee (x( 0)
+ p -) - -(( 00t
(n)
-x)-(x,0))
The five superscripted terms correspond to: (i) A plug-in term from the experimental and observational samples. (ii) A correction term for şi (X))(X X, 1, using experimental samples. (iii) A correction term for (o) using experimental samples. (iv) A correction term for go (2) using observational samples. (v) centering term, (f p, y, a),, to ensure Ep(ø P: 7, a)] = 0.
We will use the EIF for the generic ((t.p.7,0, a) to obtain EIFs for the lower and upper bounds. Denoting the lower bound estimand as (Lp(p,7,0r
0(1, -p. -,,-,pp--000 a) 0(0,, p, -a), and the upper bound as By -o) -00,-,.-7,)) their EIFs are given by
a)),
6(;0 -(2
7

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We use these EIFs to construct bias corrected estimators for the lower and upper bounds. Let 9.. ét. and f denote the estimated study selection, treatment propensity, and outcome regression functions used to compute LLB and B. These are commonly referred to as nuisance functions, they are not themselves of interest but are necessary for estimation. We collectively denote them by v) (0.. ee. p),, where the hat symbol indicates an estimated quantity. The bias-corrected estimator allows these components to be estimated with flexible machine learning models, which helps protect against model misspecification, while still enabling valid inference (Chernozhukov et al. 2018). The form of the lower bound estimator
" t (P, (p,7, 7, f)) = = Optiainn (p. 7, a; + Σ (zB(Ziip7,,;;))
where @olugin (p: y, ax; f)) is the plug-in estimate and ΦI.B ( Zz Y, a; ii) is the corresponding centered EIF evaluated at each sample Z; (X;, S; Tú, Y;). The estimator for the upper bound is defined analogously using the corresponding plug-in and EIF components.
To ensure valid inference, we use K-fold cross-fitting and assume standard convergence rates on nuisance functions, leading to asymptotic normality of the bias-corrected estimators (Kennedy) 2024 Chernozhukov et al., 2018 Rudolph et al. 2025; Schuler and van der Laan 2024). We also enforce that, in the observational arm, the estimated treatment propensities sum to one (ég (X, 0) +0 (X, 0)
1). We present full implementation details in Algorithm ! (Appendix [E] and formally state the
asymptotic properties of our estimators in Theorem 2
Theorem 2 (Asymptotic Properties). Assume A7 A4 and bounded outcomes. Suppose we use cross-fittin, for all 8, E {0, }} the nuisance errors satisfy
- -
s, 1)l,,
IX..(X) 9s =x(()12=00(n1111)
op(nn1/1))
where ||- is the L2 norm, and êo(X,0) + ê1(X00) = 1. Let > 0 be fixed,
and take any (p. 7) such that Vt E {0,1] and Vx for which Fp((X = x) >
b(x,t, t, -P. a)) I b(x,t,, -a). Then, vūl (0 (LB(P.7,0)) 4 N(0,, and vn Qhe (p, ( 3 ((0,00)). where LLB = VarlØLB Z; p. 7,)) p. a))
and oig = VarlouB (Z:00uuBBZPP.000
Theorem leverages results on estimators derived from EIFs (Kennedy 2024 Chernozhukov et al. 2018) and establishes that our estimators for the partial identification bounds are asymptotically unbiased under standard regularity conditions. Variance be estimated using the sample variance of the estimated influence functions or via resampling methods such as the bootstrap, enabling valid confidence interval construction. We note that while larger values of a yield closer approximations to the non-smooth bounds, they may also lead to estimator instability in small samples due to the increasingly steep gradients of the smoothed function near the max/min crossover point.
Experimental Results
In this section, we bring together the partial identification bounds developed in Section 4 the estima- tors derived in Section and the breakdown frontier plots from Masten and Poirier (2020) introduced in Section 2.2 to demonstrate how sensitivity parameters, P and 7, enable comprehensive sen- sitivity analysis. We begin with synthetic data to illustrate key propertics of our framework under varying data generating processes. We then return to the Project STAR study from the Introduction, examining the robustness of treatment effect estimates in the presence of unobserved confounding. Relevant source code to implement algorithm and replicate these results can be found in the accompanying GitHub repository!! []
6.1 Simulation Study
We consider data generating process with unobserved confounder, U. which simultaneously affects study selection (5), treatment assignment among observational units (T = 0), and outcomes
-aataa--ttaaa--aaaaaaeee Fusion

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(Y). We generate baseline dataset with a positive treatment effect, as well as four variants where we (a) increase the treatment effect, (b) decrease the treatment effect, (c) increase the amount of unobserved confounding, and (d) decrease the amount of unobserved confounding. We plot the
breakdown frontiers for each of these datasets Figure
Conclusive
Tentative
Inconclusive
Incompatible
(
Base
Larger
(b) Smaller T (c) Larger
(d) Smaller U
Violation in Study
Exchangeabiing
10%
0%
10%
20%
10%
20% 0%
10%
20% 0%
10%
20%
10%
20%
Violation in NUC (p)
Figure 2: Breakdown frontier plots for various synthetic datasets. Figure titles indicate the relation between the data used to generate that plot to the data used to generate the Base plot. Conclusive and tentative regions are distinguished using 95% confidence intervals, computed from the sample variance of the efficient influence function.
observe that the conclusive region (dark green) expands as the magnitude the treatment effect increases. Conversely, weaker effects lead to smaller conclusive region. We also observe that the amount unobserved confounding shrinks, so does the incompatible region (black), while greater confounding enlarges it. This behavior demonstrates how the breakdown frontier plot effectively summarizes the strength of evidence for conclusive treatment effect by incorporating both the effect size and the observed discrepancies between observational and experimental data.
Specific details of the data generating process are provided in Appendix [F] and the algorithm used to construct the breakdown frontier plot is presented in Appendix E.2 The procedure involves handful of hyperparameters, including the minimum and maximum values and 7, the confidence level, and the a scale used in the Boltzmann operator. Guidance for selecting these hyperparameters is provided in Appendix E. Additional simulation results, examining different sample sizes, choices of the parameter, and cases where model assumptions are moderately violated, are reported in Appendix G
6.2 Project STAR
Project STAR was a large-scale study conducted Tennessee investigate the effect class size on student learning outcomes (Mosteller) 1995: Achilles et al. 2008). The experimental cohort included 11,601 students randomly assigned to one of two groups: small classes (13-17 students) (T 0) and regular classes (22-25 students) (T' 1). An observational cohort of 1,780 students ausigne-assignedd assigned to the same class size types but without randomization was also available. Demographic data, including gender, race, birth year, birth month, and free lunch eligibility, were collected for both groups. Learning outcomes were measured using standardized test administered from kindergarten through third grade. Our analysis focuses test scores from third grade.
Figure 3 a) presents breakdown frontier analysis of the Project STAR ATE, varying the sensitivity parameters and 7, The incompatible region at small values of both parameters aligns with prior findings on unmeasured confounding the dataset (von Hippel and Wagner 2018 Justman] 2018 Athey et al. 2020 Parikh et al. 2023). In contrast to existing estimation approaches, which require assuming either A6 or A5555r55olaa holds, our framework enables investigation causal effeets under simulta- neous violations of both. The analysis show that as long study exchangeability violations remain below 5%, there conclusive evidence
positive ATE-even under substantial NUC violations. Given that scores range from 486-745 (mean 618), this suggests that study selection bias would need to shift outcomes by over 30 points on average to render the results inconclusive.
Beyond the population ATE, our framework supports subgroup comparisons. Figure () shows breakdown frontier plots for students who enrolled kindergarten before age six (left) and or older (right). Consistent with simulation insights, the positive treatment effect is more robust to assumption violations for the older subgroup, suggesting larger benefit older entrants. Developmental Psychology describes significant changes in cognitive development around the typical

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kindergarten entry age (Piaget 1964), and education research has shown that students who begin kindergarten at an older age tend to experience early learning advantages (Datar (0066). While neither directly addresses class size, these findings provide context for why older students may be better positioned to benefit from the learning environment of smaller classes-a hypothesis further supported by our analysis. The older subgroup also exhibits a larger incompatible region, potentially reflecting additional unmeasured confounding related to delayed school entry (ages six to eight) and its influence study participation, class assignment, and outcomes.
(a)
(b)
20%
20%
olds
>= olds
Study Exchangeabilty (V)
Study Exchangeabilly bigggllblllyy.oo.o (V)
Violalion
Violation
10%
10%
0%%
10%
20%
0%
10%
20% 0%
10%
20%
Violation in NUC (o)
Violation NUC (p)
Conclusive
Tentative
Inconclusive
Incompatible
Figure 3: Breakdown frontier plots for Project STAR (a) population ATE and (b) subgroup-specific CATES for students enrolled before age six (left) and at age six or older (right), Conclusive and tentative regions are based on 95% confidence intervals computed via bootstrap resampling.
7 Conclusion
Causal inference methods for data fusion typically assume either study exchangeability or NUC. Our work addresses settings where both assumptions may be violated, filling gap in partial identification and sensitivity analysis. We introduce interpretable sensitivity parameters- ry for external validity violations and for unmeasured confounding- that enable transparent robustness assessments. We derive treatment effect bounds under these parameters and develop double machine learning estimators. We breakdown frontier plots to visualize regions where treatment effects remain conclusively positive or negative. Applications to synthetic data and Project STAR highlight our method's utility. In the Project STAR analysis, find that the positive effect of small class sizes is robust under substantial violation both an] Subgroup analysis further reveals heterogeneity in this robustness, with stronger conclusions for students who enrolled at older ages.
Alternative Estimands. We focused on the CATE over the combined RCT and observational population. This perspective aligns with the generalizability framework outlined by Parikh al. (2023). where the objective is to extend findings from an RCT to larger cohort, rather than transport them to external population. That said, in many settings the observational population (S 0) may represent the target of interest. Our methodology is flexible with respect to the estimand, and can be adapted this case. This involves adjusting the marginal covariate distribution in the estimand from P(X) to P(X = 0), with corresponding modifications to the breakdown frontier calculation.
Limitations & Future Work. Our framework currently does not accommodate multiple experimental or observational datasets, continuous treatments, or dynamic treatment regimes. Extending our methodology to these settings is an important direction for future research. In addition, sharper bounds may be attainable by incorporating known outcome support and we view this as a promising direction for future research. While the Boltzmann smoothing approach provides valid and practical approximations to the sharp bounds, with Lemma guarantecing convergence as a - 00, a formal characterization of the approximation error for finite > 0, similar to analyses in Levis et al. (025
and Ben- Michall 2025). would further strengthen theoretical results.
Our procedure for detecting incompatible (p.r) pairs relies resampling test that does not account for uncertainty in nuisance function estimation (Appendix [D). Improving this test is a direction for
future work. Finally, our framework supports sensitivity analysis across (p, 7) values but does not
prescribe how to select them. Although our use of relative measures helps, domain expertise needed to interpret plausible violation levels. Appendix E.2 discusses guidance and considerations here.
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Acknowledgments
The authors would like to thank the funders, particularly the NSF and NIH. for supporting the effort on this work. Harsh Parikh was supported by NIH NIDA RO1DA056407. Quinn Lanners, Alexander Volfovsky and Cynthia Rudin were supported by NSF and Amazon under grant NSF IIS-2147061.
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Our contributions are clearly outlined at the end of our Introduction. We introduce interpretable sensitivity parameters to enable partial identification of treatment effects in Section 4 We proceed to develop a double machine learning estimator for the We introduce breakdown frontier plots in Section 2.2 and operationalize them for our framework in Section
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Justification: While we do not use traditional error bars, we quantify uncertainty through confidence regions in the breakdown frontier plots. We clearly explain how variance of our estimators can be estimated in Section and Appendix H includes specific details on variance estimation for the breakdown frontier plots in Section 6
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Data Fusion for Partial Identification of Causal Effects: Appendix
A Extended Literature Review on Data Fusion for Causal Inference
In data fusion for causal inference, it is helpful to organize prior work by the information each source contributes, the assumptions that enable transport across sources, and the inferential target (point identification, partial identification, or decision-making under uncertainty). For broad syntheses on combining randomized and observational data, external validity, efficiencc/lheterogeneit gains, and
practical guidance, see Degtiar and Rose (2023): > Colnet et al. (2024b): Brantner et al. (2023)
et al. 2024).
In certain scenarios randomized trial (RCT) data provide treatment and outcome Y, while the real-world data (RWD) only has pre-treatment covariates X. Typically, the task, here, is extend internally valid trial findings to a target population whose --distribution may differ. Two closely related targets are generalization (trial sample drawn from the target) and transportation (trial-eligible differs from the target) (Degtiar and Rose 2023). Estimators typically fall into three families: swweighting/matchingg most often via inverse probability of sampling weighting (IPSW) built from inclusion propensities (Cole and Stuart, 2010 Stuart et al., [011]] Buchanan ct al. 2018 Colnet al. 2024a); rregrsssinrr plug-in in estimators fit in the trial and marginalized over the target X -idistribution (Kern et al., 2016 Lesko et al. (017);; and doubly-robust hybrids (augmented WPSW) that achieve consistency if either the inclusion model or the outcome model is correct and are efficient when both are well-specified (Glynn and Quinn [2100] Dahabreh et al. [020] Colnet et al.) (024b). Identification can also be characterized graphically via selection diagrams and transport formulas (Pearl and Bareinboim] 2014 Bareinboim and Pearl 2016). Practical diagnostics focus on limited
support/overlap -when the target X X-region is not covered by the trial (Huang et al. 2024] and on
sensitivity of transported estimates to unmeasured effect modifiers that differ across settings (Huang
2024 Nie et al. 202I).
Additionally, data fusion literature considers the case where both the RCT and the RWD contain T and Y. This unlocks efficiency gains and learning of heterogeneity, but raises the risk of unmeasured confounding in the observational cohort. One line of work combines separate estimators g.g.. shrinkage or entropy approaches that blend high-variance unbiased trial estimators with low-variance but potentially biased observational estimators (Rosenman et al.) [023] Oberst et al., [022] Chen jet al., 202T). A complementary line bias-corrects observational learnery using experimental in- aommation-exampple
include variants that learn CATES in RWD and estimate/adjust confounding bias using RCT data (Kallus et al. 2018 Yang et al., 2024 Hatt et al. 2022). third line irains integrative estimators jointly both sources, using experimental data for identification and observa- tional data for efficiency, c.g., the Adaptive-TMLL TMLE (van der Laan et al. 2024), integrative R-learner (Wu and Yang 2022) and power-likeline lower-oikel methods that temper the observational contribution via aata-adaptive learning rate (Lin et al. 2025). Several of these approaches also contribute tests that decide when and how to borrow from RWD Yang al. 2023 Parikh al. 2023).
Beyond this setup, another line integrates mismatched outcomes or time horizons. When long- term outcomes are unavailable in the RCT but short-term surrogates exist (or vice versa), one can combine experimental and observational information to identify or estimate effects on the long-term endpoint under additional structural assumptions (Athey et al. [0200 Ghassami et al. (022). Relatedly, the surrogate-inde -ndex literature develops conditions and estimators that use multiple short-term proxies to stand in for long-term causal target (Athey et al.) 2019 Kallus and Mao 2020). A recent cautionary note shows that integrating studies with disparate outcome measures can help only under strong cross- study measurement assumptions and may otherwise incur bias; articulating and stress-testing these assumptions is crucial (Parikh et al. 2025). Finally, when the aim is to predict heterogeneous effects for new target setting- rather than to point-identify them caii--multitiee trial integration with uncertainty- conscious prediction intervals helps translate trial-learned heterogeneity to new populations (Brantner et al. 2025).
complementary thread tackles preximal/egaaiie-ccorre identification, which measured proxies enable recovery of causal effects despite unmeasured confounding. Although conceptually distinct
23

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from transportabilityy, proximal methods can be paired with data fusion when proxy variables are available in at least one source (Miao al., 2018: Tchetgen Tchetgen al. 2020). These ideas intersect with "validation-study" designs that combine large main dataset lacking some confounders with a smaller validation set (Yang et al. 2024).
Spanning all of these settings is a growing literature on sensitivity analysis and partial identification. Historically, sensitivity analysis has focused on evaluating the robustness of causal conclusions in purely observational studies, for example to unmeasured confounding, while more recent work extends these ideas to settings with multiple datasets or other complex observational designs (Cornfield et al. 1959; Rosenbaum and Rubin 1983 Liu et al. 2013] Ding and Vander Weele 2016; Bonvini and Kennedy 2022; Schweisthal al.) 2024). In trial-to- -to-target settings without T. Y in RWD, most sensitivity frameworks quantity departures from conditional exchangeability between study and target (Nguyen al. 2017 2018, Dahabreh al. 2023 Colnet et al., [022 Nie et al. 202T]] Huang, 2024). For fusion with T.Y observed in the RWD, sensitivity analysis must simultaneously confront (NUC) violations in the observational sample and non-exchangeabllig
no-uuneeusured-coofounniii
across studies. Much of the efficiency-focussee work implicitly assumes exchangeability and models
violations of NUC (Lin et al. 2024, 2025 Triantafillou et al. 2023 Chen et al. 2021 Oberst et al.,
2022, Kallus et al.] 2018: Yang et al. 2024 Rosenman et al.
2023 Yang et al.
2023
). with recent
contributions proposing tests for whether borrowing from RWD is safe Yang et al.
2023 Parikh et al.
2023 Two-parameter sensitivity frameworks that jointly benchmark internal and external validity are emerging Yuu et al.] 2024b). In parallel, structural approaches provide partial counterfactual identification from arbitrary mixtures of observational and experimental data under qualitative model knowledge (Zhang et al.] 2022). Finally, drcisioonmmkingg under partial identifiability-ee e.g., safe policy learning that maximizes guaranteed performance across feasible models-offers principled way to act with bounds rather than point estimates (Joshi et al.) 2024b).
Our contribution sits here: when both NUC and study exchangeability may fail simultaneously, we introduce interpretable sensitivity parameters, sharp bounds, doubly-robust estimators for those bounds, and breakdown-frontee analyses that characterize how conclusions evolve as the two forms of violation increase.
B Theoretical Results
B.1 Section Proofs: Lemma & Theorem 1
Lemma (Conditional Potential Outcome Bounds). Suppose Al A4 hold. Then for any t E {0.1) and given x, if u(x,t, t, -y) I w(x,t,) p) and w(x,t, t, pp) S 1)), the conditional potential outcome satisfies
( X = x E [1xx, u(x, 1, p)1, where (xxt,,,7) = ( 1, (x) max (w(x,t, t, -p), v(x, t, -)), ((x, (x) + (x) min fwxx t, p), vxx, t,,)
Proof. Start by rewriting
Ep(((t) X x Ep -
+ X 1|
x
using the law of iterated expectation over study selection. By the internal calidity of the RCT treatment randomization (Assumption A4). we have that
(Y X x, S = 1] = EE x 1 E x= x =(.
Plugging this in, we get
Ep (1) x
Y
+ Ep E (1
xxxx) 0][1 (0l (x)),
= #xx 1, ())) (x) |
where /xx, 1,1) and 9 (x) are the shorthand notation for these identifiable quantities. Our attention
now turns to bounding (t()) X S = 0). Recall that y is defined
24

---

Ep sup x,
S
We again use Assumption A4 to replace PEp (E)(Y = =x, = 1] with its identifiable shorthand notation yxxx, L, Then, for any x and E {0, we have that
Mxx, 1,t) Ey(((() | x x=x,8=00
7 2
[xxx 1,t) 7- p(x 1,t) 2 1a(x, 1,1) (t) X xx
where we use the fact that the outcome space positive multiply each side by μ(x: 1, We can bound the quantity inside the absolute value using the positive and negative versions of the left-hand side, and then rearrange terms to establish identifiable lower and upper bounds on EEp (1) = x, S = 0). as shown below:
-Y ((,, S u(x, (t) X == x= x 2.
(1 - (1-,))(x,1,t s Ep = x 0] S (1 + (1+,)u(x11,t)
We then define v(x, (1 >) x 1, () and have that
[Y (t) X = x, S 0] E [u(x, t, -),, v(x, t, 7)]-
Now, we turn our objective to using p to bound Ep (Y = 5 = 0j. Recall that p is defined as
- (p)
-,
1- []
p= sup x,
S 0,T =
((x,0,1) - (E))) X x
sup x.
(x 0,
where u(x, 0, t) is the shorthand notation for the identifiable quantity Ep = x, X 0, = t).
Then, for any and E {0, 1}. using the fact that the outcome space is positive, we have that
p(x, 0,t) (t) X 0,
p
P. 1(x, 0,t)
1x 0, - +||
Just as before, we bound the quantity inside the absolute value using the positive and negative versions of the left-hand side and rearrange terms to establish identifiable lower and upper bounds on (I)) x, S 0,T =1T11- l).
P 0,0 & (xx =,, 1 11 p ,(x, 0, (1 - (1-p)((x00,) 0, S
= 0,
p)p((x0,1)
From here, we note that the bound established from is for the incorrect quantity. Namely, we have thus-far used bound Ep Y (t) x, S = 0, - t]. not E (t) x, 5= 0j. To reconcile this, first observe that we can use the iterated expectation over treatment selection in the RWD to decompose ( x, S as
Ep (t) = x, S = 0 = (t) X
+ EpllY +
= =EpYY [=cp://====3ss-0---0===x==-33=0
= = (x, 0)),
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---

where (x, 0) are the shorthand notation for the propensity score in the RWD. Then, we can simply
plug in our above bound for Ep (t) X = - = 0, T 1-t] #] to bound Ep(Y x, S 0]
((x,0,0)((xx 0) + (1- (1-1 (1 -
(1xxxx))) (x,
- (t) x=
$ (x 0, () (x, 0) + (1 + م) 0, (1 - ex(x, 0))
We then define w(x (x, P) = (1 + p)p( x, 0, (1 - (x, 0)) and have that
Ep X = S 0 [x, t, p), wxx, p)]
Now, we have used our sensitivity parameters y and p to construct two intervals around Ep(Y(t)
X = x, S = 0). Namely,
= 0] [o(x, -1),,,,-),000 v(x,t, 7)], and (p X x -
(V(((() | x x, 0] E [w](x, 1, - -P), woxx, 1, p)].
We can use these two separate interval to construct a tightest interval by simply taking the maximum of the two lower bounds and the minimum of the two upper bounds. In particular, we have that
Ep (t) X = x. S 0] E [max{v(x t, -7), w(x t, -p)], minfu(x t, 7), wxx, t. p))].
From here, we note that this interval is valid if and only if max(u(x, 4, -)), w=(x L, p)} S
min(v(x, L, 7), w(x, p)). Since v(x, (--) S v(x,4, 7) and w(x,,, L, -p) < w((x p) (because the outcome space strictly positive), max (, 17), w(xx I -p)} S min(v(x, 7), w0, ()
is equivalent to the condition in the lemma that =(x, t, -7) S w(x t) and w(x, t, -p) S exx,t,,
We can then use these bounds on Ey (t) X = x, S 0] to establish that Ep (t) X x e
[[x,, t, P, 7), u(x, t, p. 7)] where
1(x, (:77) gi (x))) x, 1,4 + 90(x) max (w((x, 1, -p), n(x,1, (7)) w(x,t, (:))) = -9 (x) 1, + (x) min (w(x t, p). v(x, t, 7,7)
Theorem (Treatment Effect Bounds), Suppose A/ A4 hold and that for each x and E (0, 1). w(x,,, 1, (,7) & w(xx 1, P) and w(x. L, -p) v(x, 1, 7) Then, the conditional average treatment effect satisfies: 1(x, 1, 1, - u(x, 0, p. 7) S Txx) S u(x, 1, P. 7) - 1(x, 0, p, 7)-
Further, if this holds for all x such that (P = x) > 0, then the average treatment effect satisfies: Ep 1, u(X, 0, p. 7)1 S + < Ep(u(X 1, p.7) (XX.0,
Proof. We first provide a proof for the bound on the CATE. Start by noting that
t(x) = Ep (1) (()) x = Ey((()) X = x] () X = x]
by the linearity of expectation. Given that for both t = 0 and = 1, v(xxx( 4, -7) S w(x, p) and
w(x, L, pp) S v(x,4, 7), we have from Lemma 1 that
(1(( X = x] E [[xx 1, 7), u(x, 1, p)], and Ep (0) X x| E [l(x, 0, u(x, p, 7)].
From here, we note that the maximum possible value of T(x) is the difference between the maximum value of EY (1) X = x] (i.e. u(x, 1, p. )))) and the minimum value of (0) X x (i.c. 1(x, (1 7)). And coversely, the minimum possible value of 7(x) is the difference between the minimum value of Ep (1 X= x) (i.e. 1(x, 1, 7)), and the maximum value of Epp (Y()) X =
x] (i.e. u(x, 0, p. 7)). Summarizing, we can bound the conditional average treatment effect at x as
1(x, 1, u(x, p,7)) S T(x) < u(xx 1, 1, 7) - 1(x, p,
26

---

We now turn to proving the bound on the ATE. We start by using the law of iterated expectation to write
ET(Y(1) (())
/ (
/ T(x)dIPx(x
where I is the probability measure induced by X over P.
Then, for any x where (xx > O, because for both / = and = 1, 0(,, 17) S w(xx 1, and
w((x -P) u(x,,,7) (),, we have from the above result that
((x,1,p77 - u(x, 0,7,) S T(x) S ((x,1,p,)) - 1(x,0,
Thus,
/ (1(x,1,p:) - u(x, 0, p, 7)) dFx(x) ST / (u(x O, (xxxxx))))))))(x()))
express the bounds in Since, for E {0, 13, lxx,t, t, 7) and uxx, (,7,) depend expectation notation as
Ep
1(X,0, p.
B.2 Section [JrtionsSPProfs:: Lemma 2
Theorem
Lemma 2 (Smooth Bounds). Consider setting where A/ A4 hold bui A5 and A6 may not. Define
b(Xtt, (xtt,%%7,) := 91 XX) (x) 90 (X, t, p. (, avv + X2 (X, t, p. 7, awww, where
exp(are
exp(ow)
Xx (,777 exp(av) exp()w))
A (,,,,,)) exp(av) + exp(ont)
and ,,,X,,,7 w == w0- p). Then for any 0, p. and y such that VI E {0, 1} and Vx for -a), it follows that
which Pp(X = x 0, -1
b1,, 7,
Ep
- [1l0a-/0-----00000000000--33333-
6(X, 0, - -7, a)).
and
lim Ep(b(X,1 1, - -.7,,) = p.7) uXX,0, P)1, a-+00
lim Ep [[X, a0000
b(X, -.A.00,-P-7.000 -P-7,
Ep (x, ((x,0,
Proof. As did using the hard max and min functions in Lemma Boltzmann operator quantities to bound
use the weighted
Ep Y(t) X x, x,5 = =0 E L, -2, -)) ex, - -0
so long as
(x,,,, t, -p, -p. o(x, + de(x t, -P: w(x, & (x p)
+ (x -c) w 7).
Because x and are constant in this inequality, this is equivalent to the condition that
b(x, t, P. y, a) < b(xx p. -a).
Then, under these conditions we have that
1, (Y) X x E t, - - ( b..
27

---

Then, we can proceed as we did in the proof for Theorem 1 to bound T(x), and subsequently, т, Namely, we note that we can bound r(x) between (i) the difference of the lower bound on (1(Y(1) x = x] and the upper bound on (Y) X x and (ii) the difference between the upper bound on (Y( X x] and the lower bound on E (Y() X = x]. Concretely, we have that
-
b(x - S T(x) - b(x, 1, -0) - P,-,,,)
Then, since for any x where [xx > O and / E {0, 1J. b(x, 1, 1-,, y,
-,, a) < b(x, L, P. -x), we
take the expectation over the upper and lower T rX) bounds as we did in the proof for Theorem 1 to conclude that
Ep [b 1, p. 7. - b(X, O, P.7. -]]] S Ep|b(X, p,7, -a) 6(x 0, -p. a)]
00. By We now prove the convergence of these smooth bounds to their sharp counterparts as - 00. the properties of the Boltzmann operator, we have that for any 7, x, and E {0,
(0) 4 (,7) max(v(x, wx, (1 P, a)) Au(x, (ok + min(v(x,t,,). wx, as a -00
00,
Then, note that
S (X, (x,,
+ 12 a) S go ma 4,1)
and
(X) min(v(x, t, p), w(x,t, 7))] S Elgo(X max &, p), wx, [,,, < 00
because the outcome space is bounded. We then have by linearity of expectation and the dominated convergence theorem that
lim Ep 1, -.. -00
a a+000
lim Ep 0
+ 1
-0 (XX (x,,,,,, -x
+ (X, - 0, 7)}]
= (X)x 1, + Ep lim go X) [A
a0000
+ (x 1, p, a, w X,, 1, -
-
1)] - Ep lim 90 (X) (XI -
+ (x,x,,,,77 X, 0, 7)
(X) 1, Ep (0 max (u(x; 1, p), w(x -
- Eplgi (X) Ep (x) 0, p) w(x 0, 7)}
Ep 1,(X, - Ep 0,
= Ep
- -(X,
The same steps can be done to show that
lim 5ppbb(X11,,.1 aa) b(X, 0, - a,,) Eppu(XX ((0r)))-((X0,,,,0))))00 0, ((X,0, a-000
-
Theorem 2 (Asymptotic Properties). Assume A4 and bounded outcomes. Suppose we use cross-fitting, for all s, E {0, 1} the nuisance errors satisfy
[1, - u(X, s, 222)) [|ε((X,s) s) (X, s)]l2, lXgx(X) gx((1) op 1/4),
28

---

where || Il is the L2 norm, and êo(X.0) + e)(x,0) = 1. Let > 0 be fixed, and take any (p, 7) such that vt E {0,1] and VX for which Pp(X x) > b(x, 1, -p; a) S b(xx P,7,, a)). Then, vūi (0,77) (,)(),7,00) 4 o() A and v 0ir (p, - 4 o(),ota), where oi, VarloLB (.:777,)) and LtB 2
Proof. We prove the result for 0% and note that the case of Oec analogous. We start by writing the standard decomposition 0 10-0p OLB = PROLB + (P,, ( - ΦLB) + Rg where P is the true distribution and Pn, the empirical distribution (Schuler and van der Laan 2024). By the central limit theorem, / (P,ØLB) 3 (N(0,, Therefore, what is left to show is that the remaining two terms are 0g(1 (n/2))
The second term, (P,, -P)LBB .LBB. is referned the empirical process term. Previous results have shown that this term is always 0, (n_12) when K-fold cross-fitting is used (Chernozhukov et al. 2018 Schuler and van der Laan 2024).
Lastly, RI. is referred as the second-order remainder. This term can be decomposed as (f)
Ookugin OLB + P (LBB for nuisance estimates ý - (Øs> é,, fi). In Appendix B.2.1 we decompose LB
RLB(1) and show that under bounded outcomes, study positivity (Assumption A3). treatment (X, 0), this term is sum of produets/squaress of nuisance errors. Then using the assumption that for all s, 8,6 E {0, 1 the nuisance errors satisfy
positivity (Assumption A2), and the condition (X,
| s,t) - (XX, t)ll2,
[|)((X,)) ex(X,
ll 9o (x)]1 op(n
we obtain (r.((i) 0p(n-1/2)
B.2.1 Decomposition of the second-order remainder term
we focus here on the decomposition of the second-order remainder term for golugin The steps are analogous for ğolugin
Similar to the proof for Theorem 2
Recall that the second-order remainder term is of the form
gnlugin (p,7, (LB((, Ep [ LB
We can expand this as
Gotugin (1, -1,-p- -7- -1,-p:-7,0)
golugin (0, p. -a1
-10(1, -,l.-p-7,00 0, 0(0, -a)}
Ep (012;,1,-,-74 -,, 3(2: 0, p.7, -a;
and then group the treatment arm specify terms, with the 1 = terms being
@otugin (1,-p-7;,0) 0(1, -; -p. 0) + Ep [0(Z: - P, -7, ~,,0;
and = terms being
Golugin (0p aa) Ep [0(
Note that if we can show each group of terms are (n.1))), then the sum of them is also op (n 11/2). The derivation of the second-order remainder term for both treatment arms is similar, so we show the form for t and note that similar steps can be taken for t 0. Towards this, we plug in the full
29

---

form of 0(2;1,- -p. -,,;()),
@pkugin
- 0(1, pp, -7,a)
(TT(T)
+E
(Y ((X,1,1))
éi (x 1)
11 (T)) (X)
8(1
(()))
2I 1)91 (X) (Y- 1X,,
(TT) (V (x,,0, -
(1 - ())2)) (x)( [(X))}
(x, 0)
-A(X -,x
Golugin p1, [.)
We dropped the P subscript from the expectation notation and omit the t, P. 7, and parameter values from v, w, \1 X2 (given they are constant throughout) for simplicity going forward. We cancel out Gotugin (1, P: -7, a) and write the full form of 0(1 -p. -7,-y,1) a)):
+ (T) é1 (x
E
Si(X, - (1- +(()))))
(X))((()
(Y ((X...1)
(T))n X)
+ 8(1 (x)- +Q ((x))
( xx,,1,1))
è1 (X, 1)91
[TT
+
(1 - (1-)))12(()))) + -XX1)))/000)) 6(1)}}
(V - p(x, (11
(i(X, 0)
- (X 0))
(g 1, + (X) [d (X) +
Let's group these terms into four groups for easier analysis.
(T)(T) (Y A(X. 91 (X), 1.1) (X, 1)
E
101X,1,1)
+ (1 - (X)0(())) + (x)x((( - 9x( (X)x + (x())x(X))
(i))
8(1 - +x))))))(())++))()x)))))) -x()) (T) (X)
(Y ( (xX,
(x, 1)91 (X)
(T)
+ (1 (1-11)11 + ( 6(X)1}
(Y A(X 1)}(1 -el--ppeeoiiml 0))
(x, 0)
} (M)
-A 17(T(T) 6,(X,
For readability, going forward we suppress explicit arguments from the nuisance functions when analyzing each block. In particular, let 9s 9, (X), e e (X, s). us u((X, 8, 1), v =(X). w = w: [), and X₂ = (X) (and similarly for all equivalents). Starting with (i), note that by the law of iterated expectation have that
E =
(g121}
30

---

and
(T)
E
(
mm)
(1)
E
E
[
E
(T)
X,S 1] (E
1|- (1)
gie (1)
in)
E
(41
Therefore, we have that the second-order remainder of this portion can be expressed as
916 (1)
( - (#1 pi))
B{(} = E 91 +
E(") {u1 P1}
91
E
Turning our attention term (ii), we continue to use the law of iterated expectation, but show less steps for brevity's sake. We can rewrite (ii) as:
fao0h m)}
where dιv + dz and h = R + Λ2w. From here, define w - c, which allows us to write h m()) where m(0) = X20 = T+expl-ax Further, define Av - Aw = -
and A6 8 6 Aw Av. Then, Ar + m66 + 26) m()).
rewrite h h (0 v) (m(6) mm()))
Using Taylor's theorem with remainder, for some E (8,5 + A6),
m66 + Ao) - -m(6) = (5) + 2 =m (A)
where m'()) X2 + A2 and m() 2adi1d2 + 221122 2)2)ç. Therefore, write
h - h = Au m (8)25 + (A)
2
= + aa w)) Ao + D22 + -)) + = ())
B,Ar B& 10:0++mmmS (A)(()))
where B₁ = b1 + - - w) and B, = de + aadid2(w - v). The expectation of (ii) is then
1 ((i)) E (go (BiAr + B2A1))
2 g00m (() (Aw Am)²]
The first expectation above contains two linear components, Bio and E Baww, that we will subsequently show cancel out from components of terms (iii) and (iv), leaving with
E ((i) = ( (gom (Aw - A
Focusing now on (iii), we apply the law of iterated expectation to rewrite this portion as
E
91 - () + adiis
t))
90 e I1 e
(ur - m1) 1in-mn
mi]
91 - 8)B, 30 e
80 cl B ugg
E
ग1 (1) {#z
Ar2
=
91 (1)
31

---

where the last equality comes from the fact that (1 1-))[[1 three components of this term as
Ai] =
Av. We can rewrite the last
(1)
(1)
(1)
90 1
90
91
91 1
(1)
91
91
90
90
(1)
91
6
90
(1)
(1)
91
1
1
90
+
91
- 1
2(1)
90
(1)
- 1
Plugging this in, we get
E
Bo
91
91 -
B
90
((())
6
1
Bi BlAyy )-
c -1)
E Bo 90 (Bs (Bi- Bi) Au -
91
1
ei
B1 Ar 90
1
B1 Du }
(
2
01
The first term then cancels with E { (ib) from term (ii), leaving us with
(1)
[(iii)) = -E 90 [1 - B1 Au +
91
B1 Au + go
9 (1)
-1
Bi
91
Now, applying the law of iterated expectation to term (iv), we get
E { 00 L + adids(i0 - })}
]
c Ipo - Aol [1 pE! - pii [ (0)
co - 200
90
Bo A wuo - nol + pii se! [0)]
E
( where A 1= - [1- and Del() = = -ct
With the objective of canceling out E (go B2 from term (ii), define k = e + (1 p)ep (0)
and A = + (1 p))(0) and note that w KWgg and v RāO- Then, -A KHo RuO (0)
k(uo- - (0)) AA - k) = k(po- - -An) (o + +(1 - -)))) From here, add and subtract
s(yo (O) get
[((iv)] = IE { B2 B2 (M(HO - A0) + (A <Ho - po) + pÑoSe (00 }
(0) and then replace x(M0- - (a) with - -Aw + Aoo (ee (0) + (1 2) to get
B((iv) E B2 Aw + (n (e (0) +(1 (1 p) | (0)) + (A r:) - (io) + p
E
{ go Bu + go B2 | (0) + )ol + (A x)(10
- x))(o- }
= BoBrAw go (Bz B2)Aw + B2 |o (ee + = + (A- } }
As before, the first term cancels with E. (go Brdmr from term (ii), leaving us with (1))
[{(vv) = E { - B2 - Aw) + go B2 [ + " + (A k)(40 - } }
go
Further observe that because ep (0) (1 10 and E00) = (1 - E(")), ). we have that Ae + A (0)
0. Therefore,
E((iv) E { {-90 (B2 - B2) + 90 (A x)) - m00
32

---

Gathering all of these terms together, we have that the second-order remainder term can be expressed
as
(0)
E
91 (ef)) - El") (11 (11-1) -i-))
+
(00mm(s) (s) (Aw Au)?
( 91
(1)
(1)
(ii)
90
Bi Bi) Au
-1
91
1
1
(1)
Bi
Ao
90
Bi Au
(iv)
-90(B2 B2) + (2 - x)) Po)
+
We will show that each of these components is 0p(n" 1/2) under our assumption that for 8, t E {0, 13
each nuisance function satisfies
|i(X, s,t) (XX, l)l,,, ||ē((X,s) ex(X,, 3)1[2, (X) Io (X)]l2 on
We will repeatedly use the following consequences of boundednessspoosiiiityyy
a - 1112 - Gl g-ll2 on
11l2 clle! -e(
- <l21
(n(1)114)
(2)
Avlla & C1AA - mill2 (n 11/4),
(3)
|| |Awll2 1 C(1 - Holla + ||e - - ( 1|2 + |1 - (l) = (n 1/1).
(4)
for j E (1,2), || B, 112 s C(1Ar1122 + ||All1)) 0p(n 11/1) 1/1),
(5)
mm) C,
(6)
||A- xl12 S C(1e)) - e ll + | -0000 l12) = 114).
(7)
For (i), by Schwarz and treatment positivity (Assumption A2).
E 91 (e(1)
- a pm]] s Clle(") - e( 1|2 ll -1l12 = 0p(n (1-/2))
For (ii), using
(4) and (6) along with the fact that (a - b)2 - 2(a² + +b²). we get
[[gomm()) 2o)2]
s C(IAw)] + ||A0)13] 11/2).
2
For (iii). there are three pieces. By and
El [go [3 B1)
S i B1 |Avlla S C(IAr|12 + (1 op -1/2).
Then, by ( treatment positivity (Assumption A2 p. and study positivity (Assumption A3
R [
B Ar]
g
*11
B Aull s CC 11 ||1l22 = (n(2)112)
and
E [ -
1)
< C - e(t 12 ||lll2 (122).
Finally, for (iv), there are two pieces. By
and
go B. Bz) $ B| Aul22||A||2 I C(IAr|12 + ||a]||)]| | = =ppr (112))) and, then using
El go Ba ( (A x)(uo - [o0) S Cl - polla (126g) - e 2
||2 + || - (0)112) = =0,(n-11 -1/2).
33

---

Combining (i) (iv) gives |andd 2 C,R, + tp 1/2) with
Rn = || 11 M1l12 + llão - mol1 + |lē ei eill2 + 11e0 coll2 + ll gill2 - on(n $),
hence R2nd = 0pp 1/2)..
Assume: (a) bounded outcomes so |x| Σ M; (b) positivity, i.e., for some c E (0,1/2), c I
(x s), (X) < 1 a.s.; (c) cross pross-fitting; (d) propensity normalization in the observational arm: én (X = é 0); (e) fixed a.
C EIF Derivation
C.1 Setup Recap
We start by reiterating our setup and relevant notation. Recall that have the following variables,
X: A vector of pretreatment covariates.
S E [0, A binary variable indicating study assignment (RWD vs RCT data).
. 7 E {0, }]: A binary treatment indicator.
. Y E IR+: The outcome of interest.
The distribution of the population is denoted by P over (X, which we assume admits a probability density function, denoted by s(X,,,
We then defined the quantities (i) study selection score: (x(x) TP x x). (ii)
treatment propensity score: (x, Pp t X x,5 s), and (iii) expected outcome: u(x, 8, t) EY
x, S s, t].
And, using our sensitivity parameters y and we constructed the following terms:
(xxtt,p): = ex (X,,XXXXXXX))))) + ex 01) (),,,,,,1),
o(X,t,7) (1 (1+7))1 (X,,) 1,t),
exp(au(X,1,)
(x(,,,,,77)) exp(au(X, + exp nww X, p)) (X,
>e(t,,, = exp(av(X,tt,)) + exp aww X, t, p))
Our goal is to derive an efficient influence function for 0l1, 7, a), where
0tt,,,,a)
Ep [gi /AA.
Note that we can write
0(1,),,,)) = 00(0++00()))))))
where:
(1( = 9 (X) 1.t) and
(z p, 7, = go [ (MK
C.2 Efficient Influence Function
We use point mass contamination to derive candidate EIF, (Z;4, P. 7; a),, for our estimand, (t, p, a). We proceed to verify that the candidate is indeed valid influence function, and
34

---

subsequently the efficient influence function because we are in a fully saturated model space, by confirming that for any generic score, h, we have
= Ep(0 (z,,
We start by letting & be EIF operator which "takes a parameter and returns its efficient influence function" (Schuler and van der Laan 2024). In our case, ((0(1, P, 7, a)) - o (2; L, P. 7; ax)
Using this notation, we can break our estimand of interest into separate components for easier derivation. We start by noting that thanks to the linearity property of the EIF
( P. 7, (00 (t)) + ((02(t,):)))))
Before proceeding to derive ((0,(t)) (t)) and (022(t, P. 7, a)), we first compute o(-) for various sub- components that make up our estimand.
(p((x) = (X) p(x),
(X)
(gx(x))
p(x) (La(S)) - 96 (x)],
(S) [I((T) cx(x, s)],
(xx(
p(x, s)
(SII, [Y s,t)
(p((x,stt)
p(x, s,t)
In the equations above, and throughout the remainder of this section, (AA denotes an indicator function that equals when the random variable takes the value a, and 0 otherwise.
C.2.1 Candidate EIF for 01
Taking the point mass contamination approach, we first rewrite 0, (1) assuming all of our covariates discrete, Namely,
(t) Σ 91 (x))((xx1,t)pp33
We apply the product rule and plug in values to calculate (0 (1) below.
Σ
(X))
p(x) |S -
SIX (X) I8 - p(x, 1() + [Ix(X) (xx
1,0,
(XT,,
1,1) 1,,
(x 1)
(T) [Y
p(X, 1.t) + 1,(X,1,t) (1)).
1X,
In summary, the candidate EIF for (t) is:
SI(T = Y (X. + Sp(X, (t).. de, ex (X,1)
We drop the arguments for and simply denote the portion of EIF for (1()) as de, for simplicity.
35

---

C.2.2 Candidate EIF for 82
Since 7, A. and a are predefined hyperparameters, while deriving the candidate EIF for 02 we omit them as arguments to w, v, Y and X2 for brevity. Importantly, these hyperparameters will become relevant when deriving the EIF, so they are still present in each function just omitteed for notational brevity.
We again start by rewriting 82 assuming all of our covariates are discrete.
82 Σ go (x) (AI(x))(() + x2(x)u(x)) Σ 90(x)) {u(x) + 22(x)) [w(xx) - v(x)]} p(x).
x
Then, can apply the product rule to see that
2(12) Σ (xx(x)))((x)) fu(x) + (2(x) [xo(x) x(x)]}
(xxxx)) {1 X2(x))
+ ((x(x)))0(()))()))xx
+ Σ (((A2(x))90 (w(x) v(x))
Σ ((p(x))9o(x)) (u(x) + X2(x) [w(xx - u(x)]}
+
We already know (x)) and (p(x)). But need to calculate ((v(x)), (ww(x)), and (A2(x)).
First, note that we can rewrite
w(x) p( = ex (=(k))))/k.........::000401////0 (xx, + pe1-1(x,0)) 0)).
Then,
(((()) =0(1(xx(,7))
= ((1 + 7)x(x, 1,t))
=(1 + 7) (1(xx,1,1)
=(1+7) STx 3Xx(XX)Tr (X)T,
(Y 1x,,(x,1,1))
p(x, 1,0)
(w(()) = =x(xxt, tp
= (n(x, 0, 0, (1 + px,-)(xx0))
(M(x, 0, 0,()) [1 + pe) (x, 0)] + pu(x, 0, (x),0((-xxxxxxx
(1- (X) (V
= pe
-xx,(xx,,1))
p(x,0,1 0,1)
(X)
+ +
[T'- - -1-((xx,))
p(x,0 0)
and
36

---

exp(ow((x)
(()2(x)) =b exp(ou(x))) + exp(aw(x)) (xxp(ox)
=xxpp(ox(x) (exp(ax(x)) + xxp] -2
((v(x)) (ax
=a exp(aw(x)) (exp(axx))) <xp(ax)))))
x [x)((x)) (x)((())))))())))))))))))))))))) - (xx()) {
=x)2(x) (1 (x)) [
(xxx (x)(x) - ((((x))]
Now, we plug (x)(x)) in to the full (02)) and, because (x)( is composed of (v((x)) and (xx())), combine like terms.
(02) = ((1o(x))((x) (x(x) (2(x) [xx(x) exx)]
+ (xx(x)))9(x)x((x {1 A2(x))
+ Σ (x)(x))))(()))
+ {ad (x) (Lw(()) (v(())) (x)x(()) (w(x) - v(x))
+ (((()))00 (u(x) + A2(x) [xx()) - v(x)]}
((x(()))((x) (x(x) A2(x) [xx(x) wx)]]
(xx(x)))9(x))x(() (x))x (u(x) w(x)]}
+ ((x (x(x)) + ad1 (x) (w(x) e(x)]}
+ Σ (p(()))g0 (u(x) + X2 [ux(x) - v(x)]}
We now plug in the corresponding values for the remaining 4--)'s in 4(02).
(x)
((0))
p(x) [1 1 5 - 80(x)) -S-00(00] p(x)
(u(x) + (2(x)) -
(xx() v(x))}
(1 + 7) (T)(( (V 1-(=m((x)1 1, 220 (00099900)))0 (0(x))((
p(x, 1,t)
x (x1 + add1 (x) [e(x) - mxx))
(1 (I-S)Ix(XXIL (X) -V- - p(x 0.01)
[1+ pe) 0)]
p(x, 0, t)
()
[ 11-e(( (T) - ex-r(xx, 0j] }
+
p(x,0) 0)
x xo(())((x) (x2(x) + ad aii (x) [uxx)) v(x)]}
(X) -(x) (x(x) xx x2(x)) vxx)]]
37

---

By applying the indicator function and simplifying, we get
((2)) = [1 S (u( (x()) + ( [X(X - (x)]}
ST,(T)
+ +1++)) e, (x, 1)91 (X) [Y - 1,))
(A1 (X) + aA a (X)) [x()) (X)]]
+ 8) {22 (X) + a (X) [x() 2 (X)]
[()
x
ex(X, 0) [ - M4 X 0, pe 0)]
+ X, 0, 1) (7) - -X,x((x0))
90 X) [x(X) X [ix(X) (X)I]
82.
Then, reorganizing the order of terms and canceling, we arrive at the final form of a candidate EIF for 02:
(x(X) (X))}
LeTT) Y (,
(x 1)91 (X)
+ [1 5] (AI + A2
+ [1 5] (A2(X) + ( [x) v (X)]}
I(T) ( - 0, 1) + 1- 0)
(x(X 0)
+ pp(X, 0,t) [T) ex
82.
As we did for do, we drop the arguments and simply denote the EIF for 0g as Φo, for simplicity.
C.2.3 Checking Candidate EIF for 01
We start by rewriting
91 [
{ ( yp(y)x, 1, t)dy p(1)x))(x)ddr
by replacing (x(, 1, with S, sp(p)x /x, 1, f)dy and 91 (x) with (x).
Recall that we need to show that VxO = Ep(øs, h) for any generic score, h. To compute the directional derivative for Og and ht V, we introduce the notation P. (1 + eh)p. Then, for any generic h we have that
/
de yp. 1,tdd (1 (2))).(()dd) (x)dr
For the expectation E, (0 hi] we forego denoting the P for the remainder of this section for brevity. Then, we take the following steps to rewrite this expectation.
38

---

E H0B h] -B](<do + +11)A K|
ΣΣ Σ / SSU(t' =t)
ex(x, 1) ( ((xx 1, Su((x 1,
hppy/xx 8, d'ydy p(t' xx,
Iy u(xx 1, hply) 1, f)y p (x)
(1xx,1 1,t) Σ hp 1tpl/lx,1, p( |x, ()
In the above, we leveraged the fact that [[ for any valid score function
It is casier to prove this equality by decomposing h = hy + hrixx + hxii + fix and showing that
S O1 Vhr x. O1 Vasix 01 Vax 01
E Blo, hylx, + Eloo,
sl + Eløe, hs|x + Blde, hx]
In particular, will show that
hx 81 Eióo, hx),
k 8 = Riio, hg(x);
hrixx 01 = Elde, /T|x..s)
Nyix.s,7 01 Elde, hy hy!x.s
We start with Vng 01 = Eiso, lx]. For Vux 0j we replace the corresponding term the factorized distribution function, P. (x) (1 + ehx (x) (x)), and set the other conditional probability density functions their normal p form.
θ de / yp(1/x, 1,t) p1 + chx(x)))((x)dr
V
/. up(y1x, x, 1, fy ( (x))x)xxxxxx)(()))))))))
((x, 1, (x]]xx)xx(x)))(x)d
= EE| 1, 1,9)
Then we are left show that RR(0,, lx| Fl[u[ 1, (X)lnxxx)
[./. (xxxxx...//xxxxxxx/xk...... p(1 ((1)))))
Elbo, 1xx
+ ((xx E // (xx())((xxx p(t /x, 1)x((1x)p(x)dz
[ hxx()) (f m(x M(x
1,d p(1)x)p(x)dr
+ S [xx(x))(xxx1))) 1,8) (E/ pluix, 1, 1, p((1xx 1) ((()))((111 p(1xx)(xx)d
-0 [ [x(())x(x, 1, 1100)))))1
=
39

---

Similarly, for V hhix8 81 = Elée, hajx], we first simplify V Vasix 81 by replacing the corresponding
term in the factorized distribution function, Pe (1(x) (1 + chsix (1 (1))(()((())))) and set the other
conditional probability density functions to their normal p form.
① / aplylx,,,t)dd
Vasix
[.]. up(u|x, 1, t)dy (1xx((1)x)))(11)))xxx)
1. ,(x, 1, t) (1x)) Elu( t) (X
We then show that EE(oo, hs[x] = [[,(X, 1,
[ -
Bioo, h
u(x, 1,1) E hxxx(sx))p())x p(t 1)p((12)))(xd
| h1)x((lx)) f,10 f -
p p(1(x)((x)dx
Σ plulx; 1, dyy ( 1((1x))))))))) p(1xx)((x) / [ 1,0)
-0+ /
E (X)
0 E[ hrTx.s), we start by simplifying Vhrix, 01 by replacing the corresponding term in the factorized distribution function and setting the other conditional probability density functions to their normal form. We note here that there is no p. (1|x, s) in the integral, so this term simplifies to zero.
Moving on to V fer x.s
a de
[ up(uix; 1, 1,t)dy ((1)x))(x)dr
V 01
-0.
We show too that Ei lrTx.ss s] is equal to zero.
40

---

Elibo hrixxs) = ( / (y 1(xx 1,4) s()x, s)p(ulx, 1,1)dy p(1 (x)(())
+ ((xx, 1, 1,xx,,,1)
Σ
(TTxx,(((1x, p(t (1x,11))11111 1)p(1()))(()))))))))
[ hmxx,s(tbxxs))
L // p(1()))(x)))
+ { n(x, 1(xx,,,1)
Σ hmxxs(((xxx)) (1 1, ) pl(lx ())(((x)p(x)dd
=0 + [ 1(xx 1,t) E //nxxstt/asss 1) pl p(1xx)p(x)dd
-0+ [(x, 1, () p(1 x))(x)dr
=0.
Similarly, for Vay x.
01 = E Ei hy we simplify V.y 01 by replacing the correspond-
ing term in the factorized distribution function, Pe (y/x, x, 1, t) = +1+ pe, xx.ssr(y/xx, 1, 1,t))p(y(x L,t), and set the other conditional probability density functions to their normal p form.
/ + ((LL+chwwkssrr//sk...000lss....0 ((1
Vhyix 01 =
de
= =1, 1 -Elgi (X)EEYY (y..E///...............
We finish validating the EIF for 0, by showing that
Elóo, hylx,s,]] Blgg
- 1,7 t]]
Elog, hy|x,sTT = [ ly-
+ / 1xxx, 1,t) Σ / /NNKKT/TUKss///004111110011 (1(p:///pppp(xxxxxxxxxxxx/////
= |X0
"ttp: "1 11]
- ||
= 3551===__||3333..........1131113 []]
= 11
-E[I(S = 1), 1,t) x 0]
+ (,[,(XX,,,,101 1) x 0]
= B[gi
C.2.4 Checking Candidate EIF for 02
We perform the same steps to check the candidate EIF for 0g. We first write 0g as shown below:
41

---

82 - 90(x) (x1(x))(x)) + (x)x)x((x) p(x)ddx
exp (a(1 + 7) J, sp(s)) 1, 1,t)dy)
=/6 Lexp (a(1 + +1) yp(y)x, 1, 1,)dd) + exp (a pplllxx 0, t)dy(1 pp(1 f/x, f/x,0)1 011
x (1 (1++)) () / sppllll 1, dyy p(0))))(())))
exp [a vp(u x, O, t)dul1 + pp(1 1/x, (xx, 03)
(((1 + +-)1, up(u/x, 1, 1, + exp a up(ulx, O, t)dy(1 + pp(1 - 1|x, 01)
exp
x / 0, ddyl + pp(1 4/x, 0)] p(0)x)p(r)drr
where we replace go(x) with p(0|x), 1,0) with Sg vp(ylx, x 1, 1,yd,, n(x, with Iy yp(y x. O, t)dy, (x, 0) with p[)]x, 0) and e1 0) with p(1 t|x, 0).
We then have that for any generic h that Vn02 can be written as
VnO2
exp (a(1 + 7) Jy vp. (y/x, (x1x,
De 1
exp (a(1 7) J, vp. (1/x 1, t)dy) + cxp a up. 0, t,dyy ppc - t|xx, 01)
x(1 7) / (u. 1,1xx,1,1000 1,1ddy pr (0
=
exp (a S, u.. (,, 0, + pB. f/x,, 011
+ De /- Texpl (a(l + 7) S, wp (//x, 1, f)dyy + exp a S, yp. (2/x, O, t)dy(1 + pp. (1 f/xx, 011
/ up 0, + pp (1 t/x, 0)] p.
and E[ hi] can be written as
Eldo, h] =E((do: + +22)hl - E[(2h]
s(1 7) (x((x) (x)(()) w(x)]}
Ie(t))
(x(xx, 1)g1 (x) - 1, 1(,
+ [1 s) (x)(ax)))t + xxxx)x(x))
+ [1 s] (xx(x) + (x)( [wx(x) - x(x)]}
x { It(t')
[y - ((x00,)) [1 - px 0)] + pF(x, 0,t) [I-)(t)) - ex (x((x, 0)
x hp(y/x, s, t'')(t'' (x. s)p(s)x)p(xxdddd
42

---

We can distribute terms to write this as
El / [ (1+ (x)))))))))))))))))))))))))))))) [v(x) w())]
ly n(x; 1p(l) (1000))))))))))))))))))0))))
+
+ h/w.:///sheauuu.............../0
+
(12(x) + (x)
-x))]
pel
x hp(p/xx 0, 0,)))(00xxxxxxxxxdddx
+
Σ
(2((x) + (x) [w(x) - u(x)]3 p 0, [U)-c(t)) ex,r(x,0)
X
x hp(y/x, 0, (tt x (0p(())))p(p))////)))))))))))) dx)ddda.
And then write in expectation notation as
Eioo, h] =(1 + 7) (X) (X) ax) (xX) (X)I
x E[(Y
r (X, 1,), 1,7
90 (X) xxxx EX) 01]
+ E
+ E
90 (X) [1 + phl ()) xx
(X))2 [w(X) - v(X))
x EI(Y - 1(X, 0, 0, :1]
+ E PX) 0, (A2 + aλ, (X)A2 [Ix(X) - oX)]}
x - -1- X 0) || 5 0]
We now show equality of the separate decomposed components of the generic score. In particular, just like did for 01, we will show that
V xx 82 Eigo, hx],
x 02 = Eloo, hspp],
hrixx O2 = Elgo,,rrix.ss hrix.s)
hyix.8,7 02 = Eldoo hy(x.s
Starting with Vn E(bo,,ix) we first simplify Vay O2 by replacing the corresponding term in the factorized distribution function, (x) = (1 + elx (x))p(xx, and set the other conditional probability density functions to their normal p form.
43

---

exp (a(1 7) Sy ",l0l,x,1,110 1,
Vxx 0/2 Fc o6 /-
a
eep (a(1 + 7) I, vply/x, 1, tjdy) + exp (a J, O,
O, t,dyy((++ + pp(1 - -1x,, 011)
x(1+7) / up(11x, 1, t)dy p(0)x)(( + +10 (x))())(xx
exp (a yy up(yi> x, 0, t)dy([ pp(l t|x, 0)]
0 / x exp ((1 + +7) I, sp(1/x x 1, ())d)) + +ex aa up(ulx, 0, ddyl + pp(1 - 11x, 01)
+ De
/ 0,, + pp(1 - 1/x 03] p(0)x)(L +
+ (x) ( xx00)xxxx)
=F -R (o(( (x)
We now show that Bloo,Axl = ||
E [loo, Ax] -(1 + (X) (0x JCC -
x - - o)
+ ) ( 01]
90 (X)
go
+ /e) pen- (x) ( (Xx (x)(X) - [X))
x BI(YY - O, (x o1]
+ 90 ( (P 0, (12 .. 12 X [xx - (X}i}
x (T - GI ( / 5 01]
We can pull the hx outside of all of the inner expectations write as
[((oo,1x]] -(1 + F
go (X) (A1 (X)
(XX)) (x)X - (X)]
x lix
X0) 1X -1,7
+ 1E
90 X {Ai (X)
E
90 (X) + pC: PX. (2x (XX)) |4 x(X)))
s]
1x (X)
0))
E2
90 (X) p) (X. t) + ad1 ax) (x( - v(X)]
hx o) 01]
X, 5
And then, applying the inner conditional expectations and cancelling like terms, we get
44

---

B 1 =(1 +
go (X) (2x))))) (x)i}
x -
90 (X) ( (w(X) x(X)]}
90 (X) pe1- (,0000))00000000000000000003,113
(x) 0. (xX,0,))
+
90 ( 0, (x2(X) ad [(X))
hx
0) e 0]]
(X) (X) (A1 (X)
90
X2 m (X) hx(x)
Next, we show Vhg 02 B h We start this by simplifying Vasix by replacing the
corresponding term in the factorized distribution function, (0)x) = (1 + (hss)x(0)))))00))))] and set the other conditional probability density functions to their normal p form.
and
exp (a(1 + 7) Jy yp(y/x 1, 1,4)dy
V Niy O2 = de /
Lexp (o(1 7) J, vp(yix, 1, 1,() exp a wp(plx, 0, 0,4) + pp(1 1|x, 0))
(1 chs]x (00) x(1 + 7) / vp(u)x, 1, t)dy
aa 0, pp(1 t/x 01)
06 / exp (a(1 + J, yp(y/x, 1, t)dy)
0
exp
exp (a vp(u/x, 0, t,dyy(1 + pP(1 t/2c, 031)
/ applvx,0,t)dde 0, + pp(1 f/xx 0>} (1 + (=xx
(Xn(X) A2 (nxx (00000000)HHHHHH,XXXXX)00))0XX13
E go (X)
+ (3)
We proceed to show that hs = E go (X) (2) (xx))))))))))))))))))))))))) (xs (0/x)) First, plugging in hss)x.
E Eloo, hsix] =(1 + (X)
B[(Y -II. 0]
go X) {Ai AA
g0 + |1 (w(X) v(X)))
0, ( 0, $]
x
90 (X))X))(XXXX 0, fA2 |1A (x(X) :(X))}
x xB([I1,(7) - (XX 0]
45

---

Then. pulling x out of the inner expectation where possible
Eloe,
=(1 7) | w(X)}
x hSi =
E
g0 ())
Ax XXX
E
go (X) [1 + (X) (1 ( [w(X) vx))])
h (X, (X), 0, |]
13
9o))) (x) (x( x(X)]
90
-1- OX X S 500
- 00.
And then applying the inner expectations, cancelling terms, and simplifying.
Esoo =(1 + 7) go (O) - - (x)]}
x h 1, (X, 1,
90
X) (X))
(wx (X) [XX_|11
v(X)1)
x (x) ((x00)))))(xX0)))
E
g0 (X,0) (X)x (w(X) - =(X)])
hss -X, 0]]
E
9o(X (X)(X) + de (X)) hSp (0/x
Nearly there, we now show Virxx, 02 = hrix,s) We simplify V VArix, Og by replacing the
corresponding term in the factorized distribution function, Pc(1 - f|x, 0) = (1 + chrix,s(1 I 1|x, (1)p( f|x, 0), and set the other conditional probability density functions to their normal form. We have to format V V Og slightly differently to allow it to stay within the page margins.
46

---

[al (a(1+))) +r) (p02l),,1,tdd)
V 02
dc
exp
exp a 0, p(1 + elrrix f/x, 0))p(1 f|xx, 011
+ cxp (a(1+ / uploll 1,)
x
(1 +
yp(ulx, 1, t)dy p p(0)x)p(z)dd
0
vrllll x, 0, 0)dyl p(1 + ehrixxs(1 4/x, 0))(( f/x,, 01]
+
exp
dc
/ yp(olx t)dyl1 p(1 + +erixx 1/x, 0))p(1 1/x,, 0)]
exp
(a(l+ 7) / up(1lx, 1, t)dyy
+ exp
( wollixx (dddl + p(1 + eh elrpxxs(1 f/x, 0))p(1 - d/x,011 -1)p]11)(xx00]
x ((0)))p(z)dx
To simplify, we plug in exp (av(x)) for exp (a(1 + 7) Sy up(p)xx,, 1,t)dy) and μ(x, 0, for Sg up(y x. O, t)y. We also replace p(1 t|x, 0) with e (x, 0) simply for clarity in the following steps. Then, Varix, O2 is equal to
( De exp (oxv(x)) + exp (ap(x, 0, ()(1 + p(1 + x0(x))(((x)p(()ddaa
0
exp (ax(x))
- (xx
exp (axn(x + p(1 + chyix,s (1x 0)))
+ de x exp exp (av(()) exp (ap 0, 1) + +
e l/x, (x) O)])
x,(x, 0,t() p(1 c f|x, ((x))))((0))))))))))))))))))))))
==-
This is rather complex partial derivative. To proceed, we will let
Wey (x) (, + p(1 chrix.s(1 +/x, (x) 0]]
and rewrite the above as
[. ②
v(x) (ar(x))
Wep (x) exp (awer (x))
+
p(0)x)p(x)dz.
de Lexp (aro()) exp (awee (x))
exp exp (av(x)) exp (arwez (x))
Note that
0 w.p (x) hTpx.s(1 t/x, 0)p -xx,)
47

---

and Wer (x)le=0 w(x). From here, we evaluate the partial derivative piece by piece. First the left fraction:
0
v(x) exp (aro(x))
((x exp (<<o(x) exp (aWWer (x)) de Wer (x)
de exp (aw(x)) + exp (aoWee (x)) c=
exp (ax(()) + exp (ower (x))
c==0
ax((x) de Wer (x) exp (xx(x)) exp (aww(x)
exp (axv(x) + exp (ano(x)) 3
Second the right fraction:
0 Wer (x) exp (a (x))
ae exp (x)( + exp (oeWer (x)) c=0
exp (av(()) + exp (orWer (x))
@Wenn(X (x)
cxp (awer (x)) + wer (x) exp (awer (x)) x a Swer (x)
x
de
de
Ower (x)
-Wer (x) exp (2awez (x)) x a dE
[exp (av(x)) + exp (awee (x))
Bwey (x)
exp (xco(x)) exp (aww(x)) + oow (x) exp (xx((x)) exp (aw((x) + exp (2ow(xx))
(exp (a<o(x)) exp (x))]
Putting these terms together, have
@wer (x)
exp (av(()) exp (x)( + exp (x))
x
[exp (ax((x) + exp (x))] +
de
(xt(x) exp (arv(x) exp (x)) - (x)( exp (ar((x)) exp (aw((x)
[exp (av(()) + exp (oww(x)))
Ower (x)
x (2(x)) + ad) [w(x) - u(x)] uxx)]
de
hrix,s(1 ()xxO)),(xx O, (xx, 0) d2(x) + adi (x))2(x) [ux(x) (w(x))
n(x)]
Now, we can plug this partial derivative evaluated at e = 0 back into our integral, to get that
0, (x) 0) V Vrrxx O2 / hr - f/x,
x (M2(X) + add [m(x) - u(x)] p(0)x)p(x)ddr
(00)))p((xx (,/ [w(xx) vxx]]
x (x 1)/ t|x, 0)x)))
9o (X) (X, 0,
(22 X an) (X) (X))2 (x(X) .(X))
x 61 1 00 1|X,
48

---

We proceed to show that Bl equals the above value. First, plugging in hrpx,s
10 El /vy s] =(1 + (o (X) (A1 (OX
x E[( - (|)
+ E[go (X)
+ Ligo () + 1,0 0]
+
x IE{(Y (xX, O, (XXX0///X0xXXXXXX))5550-00000 -,
+ Elgo (X) pp ( [w(X) - v(X))
[[ - ( (x) X,5 OJ].
Then we pull x out of the inner expectation where possible
1100, =(1 + Eigu (X) (X) + (X) (X) w(X]}
x X
X, -
+ Eigo X) (A1 (X) + (X) 7X
0]
+ Elgo X) + [| (x))
x hrpxx XX 0) | 0, 0,
+ E
(0)))) 0, (X) ad (X) [x( - e()]]
00
XX =o)]
and use the mean zero property of the score function to set all terms multiplied by (7),) = 0) equal to zero:
Blos,hrixx,s =(1 + E(go X) (X))) -X(I)
+
x hTiX s (1) X, -X
+ Elgo (X) [1 + pel - (X, (Xx) v()))
x h ()) - (X,,,, -
+ (xx
[w(X) - v(X))}
(T)x S)|X, S= Oj]. :
Then applying the inner expectations and canceling like terms we get
hTix.s) =(1 + E(gu X) + (xX (X)]}
X (X
+ E[go (X) + [1 [x: xX)]]
x (XX,,
+ E (X) X, 0,t) (o)) [x(X) - v(X)}}
EE (II
= EEgo (xx
[w(X) x(X))
x E (7
Now, notice that
49

---

[|L= =E
Σ 11-((t))rTIxss (t),, (x, 0)
(1E(111-1)
=E(e1- (0,)(((X))))xssssll - 0)]
Therefore,
E 1100,11 hT x =E 9o (p(X)
0,8) (2) (X, [x) - v(X)}
x (1
and we have shown that V Vrrix O2 = k(op,t
Finally, we show that "Avix,s. A2 hlo0, As always, we start by simplifying x,,,7 O2 by replacing the corresponding terms the factorized distribution function. This time we set (3)x, 1, = (1 + ehy|x,s. (vix, x, 1, ())p(, 1, and pe (,)x,,0, thyix.s.r(yle) 0, t)))((((xx 0, t). We set the other conditional probability density functions to their normal form. We again have format V 02 slightly differently to allow it to stay within the page margins.
(1 -
0 de
V VITIX
02
exp
y(1 + chyyi..sr((x t)dy +2) [((1+))
a(1
exp
y(1 + txxxxxx.rrrxxx.x00000)00...... pp(1 f|x, 0)]
+ +exp (a(l+)) / + 1
(1 (1+ / / p(0)))p(z))
1-=0
+ 0
dc { exp a ([lttllxxsrrkkkk0)))k00000 + pp(1 -|/x,00 0)]
a S y(1 wtw.xx.srtllxx.:)))))xxxxx
x
exp
+ Pp(1 - f/x, 0)]
(a(1 (()++)) / 1 1,
+ texp
(Lvi) 1( + wtw.x..s7((xx 0,0dg/ + pP(1 (|x,0)]
p(())))(())dr
50

---

We define the following terms to substitute in above:
Vey (x) = (1)==11+) 7) / 811 t.w.........kkkk..............
=x(x) + c11 + plyxx..sskkk 1, ,,)d, and
Wey (x) 1, y11 + wtw.xx.ssr(/lxx....
pp(1 f/x,
ww() + e[l + p=1-((x.0) 0)) / www.xx.../xxxx.c.. 0, O, l)dy.
Note that
(1 + +7) / shy 1,tppl 1,tdd, Oucey (x)
Ovey (x)
de
[] pel-1 yhrix.s: O, t))l/x,, 0, tydy
dE
and Bey (x)le=o = v(x), Wey (x)le=o = w(x). Then,
Vey (x) exp (avey (x)) + wey (x) exp (awwer (x))
02 V2
p(0)x)p(z)dd.
de
exp (erVey (x)) + exp (erwey (x))
Like above, we evaluate the partial derivative piece by piece for clarity. First, the left term of the fraction:
0 Pey (x) exp (avley (x))
de exp (av.y (x)) exp (aewe (x))
- (exp (awey (x)) exp (arwey (x))
(vve (x)
Dvey (x) de
de exp (owey (x)) vey (x) ex (x))
Quey(x)
exp (away (x)) Owey (x) de
vey (x) exp (avte (x)) exp (aver (x))
de
lexp (aVey (x)) + exp (artey (x)))
==
00.y (x) de
exp exp (av(x)) exp (xw(x)) (x) exp (av(x)) exp (aw(x)) exp (2am(x))
(exp (av(xx) exp (==)(x)))²
Bway (x) x (xxxx exp (x)(x)) exp (aw(())
(exp (xv( + exp (owr())))²
51

---

The right fraction partial derivative looks similar:
0 Wey (x) exp (aeWee (x))
de exp (arvey (x)) + exp (orwer (x)) ==
(exp (arvey (x)) + exp (owee (x))
Ower (x) de
Owey (x)
x
exp (arWer (x)) + wer (x) exp (awer (x)) x a
de
(x) + +oxp (awer (x)) x a dwer (x)
}
- Weey (x) exp (orwer (x)) oxp (aver (x)) x a
de
de
x (exp (attey (x)) + exp (arwey (x)) -2
Dwry (x)
exp (<x)(x)) exp (amo(x)) + ax)(x) exp (an((x) exp (aoo(x)) + exp (20w(x) (x))
de
(exp (xre(x)) exp (aw(x)))²
OvLy (x)
x (x) exp (x)((x) exp (x))(x)
de
(exp (am((x) + exp (xw)
Now, combine terms and group by those multiplied to Dory (x)
and Dwry (xc) First,
Dvyy (x),
de
de
de
Dvey (x)
exp (a<v(x)) exp (aww(x)) + exp (2)v(x))
OE
[exp (av(x)) + exp (xw(()))
(x(xx) exp (av(x)) exp (aw((x) cxw(x) exp (au(x)) exp (aw(x))
(exp (ax((x) + exp (xxw(()))
Dvey (x)
de x (21(0)) + adi (x(x) -(x))
(1 +1) why.x.srr//xx 1pullx 1,d)dy x A1(x) + (x)()))2(()xl (u(x) m((x)]
Hw Bw (x). dc
Next,
Owey (x)
exp (xpo())) exp (aw(x)) + exp (2aw (x))
x
+
de
(exp (an((x)) exp (aw (x)
axo(x) exp (<r(()) exp (aw(()) - oer exp (av(()) exp (xw(())
[exp (av(x)) exp (oow(xx)))
dwey (x) >x(x) + axd((x))2(x)) [wx(x) [(x))
de
+ per (x)-((xx0) 0]] / yhyix.s.. (v)x, Dllll f)dy >a(x) axd(()) [wx(x o(x)]
We plug both of these components that make up the partial derivative evaluated e back into integral:
52

---

Vhyixss,00 = / (1 77) 1 / 1)dyy
x
X1(x) (xx))))))) - w(x)] p(0)x)p(x)dd
x
+ pei 0) / w t)p(e/x O, t)dy
x Xx(x) + axx(()))((())))))))))))))))))))) (xxx) u(x)) (())x)p(z)dd
Reorganizing terms and writing in expectation form, we get
x O2 =(1 + m) (Λ - ((X) wx()))
x K| (
+ E go [1 + (X. (22 10
- e(X)}}
x SyYYhy - 0,7 $1
We finish by showing that this equals
First, setting it up,
El(o,, (ol0,,yyxxsssr) -(1
F[ (g (X) (
E[[ - [U
E[go {>i (X)e (4
E[go [X) [1 +
x [{ - -
+ (Xgo X, 0,t) (,)))) + -
x E T - - (X,
0]].
Then, we use linearity of expectation to separate terms,
Eloo, H1oo,hylx.s.rl -(1 T)
E (T 1,T =
((X, 1, (X
0j
go (x)) X) [1 (M0 [x (X]]]
VI T,T t|
(XX, O.:.
(w) (x vXX) Elgo X pp X, 0, (A2 +X)((X)
x 11 - T -1- 0) hy S,T) X, S = 0j]
Now, we use the mean zero property of the score function to remove several of the terms.
53

---

Elo0,/111X,s.7/ =(1 + B) gol (X) () ( (u(X) w(()]}
x 1 =(]
+ E 90 (X) P61 (e) (XX -xxx]
This confirms that Vhyix.s, 02 hy The verification of the candidate EIF for 0g
complete.
C.3 EIF for Pt,))
Having derived and validated the candidate EIFs for 0 and 02. we simply the linearity property of EIFs to write the full form the EIF for ((t, p. a)) as:
ST(T = Y m(X, 1,0] + 1,)++ ol (Z:t,P,7,))
c((X1)) 1)
S(1 7) (x1(X) + (XX)]
LeTT)
ex, (X) IY - -1X,,,,,00000
x
[1 5] (XI(X) +XX)))
[1 5] (x2 + + [w] - eX))}
L(T) (V 1xx, 0, (X,1-1((00)
x
(X,,0)
+ p (7) - -1 (X,,
0tt,P,7,a).
D (In)compatible p and
In Remark 1 briefly discussed the reasoning behind the Lemma and Theorem conditions that v(x, f, 7) I w(x, t, p) and w(x, t, pp) I v(x, t, r). We referred to (p.7) pairs that lead to violations in these conditions incompatible, which we visually represented as one of the four regions in our breakdown frontier plots. In this Appendix section, we aim to provide more context on this concept and discuss how estimate the (in)compatibility of a given (p. in practice.
D.1 Further Discussion of Incompatibility
We define pair (p+7) as incompatible they do not sufficiently relax the assumptions to allow overlap between the bounds produced by 7 and p. We elaborate on this idea in this subsection.
Incompatibility stems from the fact that there must be some source of unmeasured confounding affecting either study selection or treatment assignment if the observed potential outcomes in the experimental and observational studies are different.
other words, as discussed Remark [
if 3 f) such that [Ep Y X x, S = T. = t| Ep[Y X = x, = O,T = t] A(t) > U.
then Assumption A6 and/or Assumption A5 must be violated.
Recall that in our partial identification framework
exx,t,n) == (1 7)x(x, 1,t),
w(x,t, ex(x, (x, e1-1(x, 0) p))((x, 0,t),
54

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serve as two upper bounds on Ep x =xx, x, = 0]. Analogously, e(x, t. -2) and w(x, t. -P) serve as two lower bounds. particular, showed in the proof of Lemma I in Appendix B. that
(x(((() X 0] [o(x, -2), u(,, 7)), and = 0] [x t, p), wxx,t, p)]
We then took the max over the two lower bounds and the min over the two upper bounds to get the tightest possible bounds on PPp (L) X x, = 0] that we subsequently use to upper bound Ep | x].
Consider the case that and are both set to zero and (tt > 0. In this case, we have that the two intervals bounding Ep ( (t) X x, S 0] are
(t) x = 0] E [o(x, t, - 0). xx , t. 0)] = p(x, 1, 1, and
(t) x x, 5 0] [w(xx t, -0), ( t,0) = 0x,00,t)
But, as we established earlier, if (tt) 0, then p(x, 1,t) * pxx, 0, This leads a contradiction as the bounds based or treatment groups. But (xx. 1.) & p(x, 0,t) observed, then some violation of assumptions must be present. Hence, (p.7) = (0, 0) is incompatible with the data.
0 and
assume no differences in potential outcomes across studies
y bounds
p bounds
ELY(e))
6
10
x
Figure 4: Bounds on (() |
x. S
from (0), v(x, 0) and
[w(x, f, -0,, (x, 1, 0)] when /x(x, 1, & p(x, 0,
Towards building up an understanding through a series of visuals, we depict this scenario a clear that the bounds on
toy example using Figure 4 where μ(x, 1,t) μ(x, 0, t). Here, incompatible choice for Ep(Y(t) (t) X x, S 0J do not intersect-leading us to call this an
but increase 7 sufficiently so that v(x,t, f, -7) (1 ()))(xx,, 1,t) f) < w(x,t, t, 0) = p(x. 0, t). This scenario is depicted in Figure 5 Here, we see that the bounds on (t) X x, 0] do intersect- leading us call this choice of p and y compatible pair of parameters.
and 7. Conversely, consider the scenario where keep p
Connecting this back to the conditions in Lemma [ and Theorem in Figure 4 we violated the condition that v(x, t,, -7) & (xxxt,)) whereas Figure both cx,,,, ->) S w((x f, p) and w(xxt, -p) t))
When 7 and p are both greater than zero, they can still be incompatible either v(x, t, -)) > w(x, t, p) w(x, L, -p) v(x,t, 7). Continuing with our example, consider the case depicted in Figure ((a) where the lower bound on the 7 bound is larger than the upper bound on the p bound, not
ie. v(x,t, - > w(x, p). This is example where A A(t) O and the values of p and and r are
sufficiently large to explain this discrepency. Conversely, increased, making v(x, - < w(x, t, p). and leading to compatible parameter values. In this plot, we also show how we take the max of the two lower bounds and the min the two upper bounds to
Figure b).. the values
get the tightest bounds-corresppoteo corresponding how constructed the tightest bounds in Lemma I and Theorem
D.2 Estimating Incompatibility
In practice, we do not observe the true values of v(x, l, 7x,x), 4, p) and w(x, (, and must estimate them from data. As result, determining whether a given (p,7) pair incompatible must account for estimation uncertainty.
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bounds
p bounds
11
EIX(O)
8
10
x
Figure 5:
Bounds on Ep|Y(t) X x, 0) from [u(x, t, -7), t,)x, and
[unxx, f, -0), t)) when (xx, 1) + p(x, 0, and y is large enough that t(x,t, -7) <
((x, 0).
(a)
(b)
bounds
bounds
bounds
p bounds
Tightest bounds
X
E[Y(E)
5
8
2
6
10
X
x
Figure 6: Bounds on [Y X x,S 0 from [u(x, L, -7), v(x, l, 7)] and [w(x, f, -p), w(xx t, p)] when μxxx 1,t) * p(x,0,0 t). In (a), and y not large enough for the bounds to intersect. Whereas, in (b), P and y are made large enough for the bounds to intersect.
Ideally, for any given (p, 7), would test each (x, () whether the estimated bounds based and y intersect, indicating compatibility. However, this intractable with finite samples in high- dimensional covariate spaces. To address this challenge, target estimand, rather than at individual covariate profiles. Specifically, we evaluate whether the bounds intersect on average over X, separately for each treatment arm t {0, 1}. When focusing on the CATE, this expectation is taken over the subpopulation of interest, whereas for the ATE, this expectation is taken over the full population.
assess compatibility at the level of the
As shown in the proof of Lemmal iin Appendix B.I the condition maxfu(x,t, -7), w(x, t, pp] s minffr(x,)) t, 7), w(x, t, p}} is equivalent to checking that both v(x,t,-7) w(x,4, 1, pp) & this condition testable, tive. Specifically,, we check whether the expected difference of min min(f((x,,,)) (,), w(x,,,)) max(u(x,t, -7), w(x,
w(x,t,)) and Both conditions ensure that the and intervals overlap. To make assess whether the average size of the overlap region is nonnega-
(,xxtt--0 t, -p)}
0. We define G(t)
BImin(u(xe f, f) w(x,t, p))
max(v(x -7), w(x t, H. This expectation is taken over the relevant covariate distribution and is evaluated separately for each treatment arm E {0, 1}.
Because generating resamples that both satisfy the null and re-estimate the nuisance functions is non-trivial, adopt simplified resampling approach that treats these functions fixed. We describe the full procedure below and return to its limitations at the end of this section.
For each treatment arm we conduct a one-sided hypothesis test where the null hypothesis is that the average size of the overlap region nonnegative, Hg G()) 2 0, and the alternative that the average size is negative, HA G(t) 0. Rejecting the null corresponds to concluding that the
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specified parameter pair is incompatible. Given dataset Dr,, parameter values (p',²") and treatment arm t', we construct a resampling-basee test as follows. We first compute the observed test statistic as the sample mean of overlap sizes:
T Σ min(u(X; 7'), w(xx max (2)) wo -pp'}--
To simulate the null distribution where G(t")= = O, subtract Tobs from each individual overlap value, producing a centered dataset that has mean zero. We denote these centered values as o min(v(x,, L', 7'), w(x, p')} max (xx,, , w(x,,'' f', - Tohe. We then perform R resampling iterations. the centered overlap size set .o..... )n), and compute the mean of the resampled values T, 1 EEE,, oj, where J, is the set indices sampled iteration r. Finally, we compute the one-sided p- 1
each iteration, т, we randomly draw n values with replacement from
value as the proportion T, values that are greater than or equal to Tohs: p - πi Σ I[T, 2 Tona].
We write this testing procedure as an algorithm in Appendix E.3 We also note two primary limitations of this approach.
First, as stated above, we do not test this condition at each (x, pair. This is primarily a practical limitation, checking compatibilityy pointwise across the full covariate-treatment domain is infeasible in high dimensions or with continuous covariates. However, in settings with only discrete covariates and sufficiently large datasets, such pointwise test could be applied to each covariate profile.
Second, as briefly mentioned above, our resampling- procedure operates on estimated overlap values rather than resampling at the dataset level. This differs from a formal bootstrap test, which would involve resampling full observations and re-estimating the functions e, (x, 0) and μ(x, 8, I) each time, propagating the uncertainty in estimating these quantities. In setting it is extremely difficult to generate new datasets that satisfy the null G(t') = while preserving the original data generation process. We therefore center the empirical distribution of the overlap values satisfy the null and perform hypothesis testing relative to that.
While our approach captures part of the sampling uncertainty nnamlly the variance in the derived overlap values-it treats the estimates of cx 0) and pxx, for E {0, 1} as fixed. Ignoring the uncertainty in these nuisance functions understates the true variability of G()). Consequently, the reference distribution of G(t) under the null too narrow, causing the test to reject the null, and classify (p.7) pairs incompatible, more often than fully bootstrap-based procedure that re-estimates (x, 0) and (uxx s, t) in every resample. For our sensitivity analysis framework. this potential for inflated incompatible region is acceptable (albeit not desirable) behavior. We prefer to label (p. 7) pairs as incompatible that may, in fact, be compatible rather than risk retaining ones that do not sufficiently explain differences between the study types. Nevertheless, developing testing scheme that fully accounts for quisance-estimatt uncertainty is an important direction for future work and would strengthen the overall framework.
E Algorithms
In this section, we include algorithms for the various components of our paper. We begin with the bias corrected estimator from Section [ followed be procedures for constructing breakdown frontier plots, like those in Section 6 and for estimating the (in)compatibility of parameter pairs, described in Appendix D For each algorithm, we also discuss hyperparameter choice considerations and note relevant computational considerations.
E.1 Bias corrected Estimator
Algorithm T outlines the cross-fitting procedure used to implement the bias-corrected estimators defined in Section 5) When estimating population-level ATE, the algorithm is applied to the full dataset. For CATE estimation, the dataset is first filtered to include only those samples whose covariate profiles X, fall within the subgroup of interest.
There are two hyperparameters beyond our sensitivity parameters p and y, namely and the number of folds k. Larger values of a will more closely approximate the hard minimum and maximum operators. However, especially in small sample sizes, large values can lead to instability in the
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bias-correcti correction term. This occurs in regions where the maximum or minimum function switches between its arguments, as the Boltzmann operator exhibits steep gradients at these transitions. For this paper, we use a moderate value of a = 10, which balances stability and approximation quality. This choice may be increased in larger datasets.
The value of l controls the number of splits to use for the cross-fitting procedure. Larger k values provide more training data per fold but also increase computational demand. A suitable choice should consider the type of model used to estimate the nuisance functions, the dataset size, and the available computational resources.
Lastly, the choice of model used to estimate the nuisance functions ii0)) is another implicit hyper- parameter. Flexible machine learning models are commonly used, though choices should consider sample size and computational constraints.
In Algorithm I we omit the specifics of calculating the uncentered EIF value ós.e",g' "t.tr_p'."''.a "',''"'"' for a given sample i, treatment indicator t' and parameter values ,', and a' given the length and complexity of the term. We include that full form in Equation 8 where we dropped the arguments from X() X o60), and w65 for brevity, but note that they correspond to those functions evaluated at x; and parameter values (f', p,,,)a))
8; I(t, = =t')
1,t) + + +.s.ppl0 (2,0 1,))
2/7 (x,, 1)
+ s (1 + 7) (2() + a' 1217)100)
(00) - =00ii}
20) (xx, 1) (1) (x,) U/1 -
I[t= =t
(x) 1, 1,) (xx))
x
+ (1 si) { (1 ((1) + +1670 11)100())}
(8)
+ (1 s:) {x()) + a' a2xi1)) ((1)
(()))))))) [w0()) - $5001
l1(4 = (')
Ui (x), 0,'' [1 p' e 0]]
x
2l7) (x,, 0)
+ p' -100 (x,, 0,z') [TtC: = 1 - +) - -2 22,
E.2 Breakdown Fontier Plot
Algorithm Joutlines the procedure for constructing breakdown frontier plot like those in Section As with the algorithm for the bias-corrected estimator, when estimating population-level ATE, the algorithm is applied to the full dataset. Whereas, for CATE estimation, the dataset is first filtered to include only those samples whose covariate profiles X; fall within the subgroup of interest.
The hyperparameters used are as follows: a and are inputs to the bias-corrected estimator, R specifies the number of resampling iterations in the incompatibility test, and e denotes the confidence level used throughout the algorithm. The most critical design choice is the grid of (p, 7) values. Large grids that include many (p.7) pairs yield more detailed breakdown frontiers, but increase computational cost. A reasonable approach is select maximum values of p and ? based on what domain expert deems plausible, and then construct a grid of evenly spaced pairs of (p. 7) between (0, 0) and those maximum values. In our experiments, we consistently cap p and 7 at 0.2 corresponding to a 20% relative violation. While this choice reflects a substantial but interpretable level of assumption violation, selecting plausible range involves subjective judgment based on the context and domain knowledge.
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Algorithm 2 Breakdown Frontier Plot Construction
Require: Dataset D,, grid of (p.7) values, confidence level c E (0, 1). Boltzmann smoothing
parameter a 2 O, number of cross-fitting folds k, number of resampling iterations
Ensure: Heatmap assigning each (e, 7) pair to one of four regions
1: Estimate nuisance functions once via cross-fitting (see Algorithm Steps 1-7) to obtain
f) (g. ét. (.)
2: for each (p,7) 7) in the grid do
3: Compute bias-corrected bounds (0h, (bp) and variances (o).: dDB) using Algorithm
4: Construct 100(1- -c)% confidence intervals:
T 21-0/2
ØLB vn
ØLB Vn
CluB
21 21-0/2"
CluB
Bir - 21/-
ðUB 0 + 21 2--0/2
ōun B
Vn
5:
Run incompatibility test (Algorithm) 3) for and f. using resamples
6:
if p-value < for either t then
7:
Assign region: Incompatible
8:
else if both bounds are strictly > 0 or strictly < 0, and both intervals exclude 0 then
9:
Assign region: Conclusive
10:
else if both point estimates have the same sign, but at least one CI includes 0 then
11:
Assign region: Tentative
12:
else if point estimates have opposite signs then
13:
Assign region: Inconclusive
14:
end if
15:
end for
Render heatmap over the grid with regions color-coded
E.3
(In)compatible and Test
Finally, we include include Algorithm 3 to outline the steps of the (in)compatibility test. We refer to Appendix Soaddixxfoor for detailed explanation and discussion of this procedure.
Algorithm 3 Resampling-bassee based Test for Parameter Compatibility
Require: Dataset D, = {(X,, Tú, St, Y2 parameter values (W/,7), treatment arm 1' number of resamples R
Ensure: p-value for testing Ho G(t)) 2 0 vs. HA G(t') <0
1: Compute observed test statistic:
Tobs Σ
[min wX,, w(x,, ,')) max(u(((,(',-/))) ((,,',
2: Center overlap values to simulate null:
0 min(v(X,,t', 7), w(,,,t' E', max)_ w(X,, -Toss for I....,2
3: for k = 1 to R do
4: Sample n indices J, = (f.....,. fn) with replacement from f1,....
5: Compute resampled statistie:
T- Σ 0j
n jEJ
6: end for
7: Compute p-value:
1
p
[T, Tote
R
8: return
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Simulation Setup
Each synthetic dataset in Section 6. is composed of 2500 i.i.d. samples where for each sample i, we start by generating three observable covariates and one unobserved confounder,
X1,,, X22, X1, N(1,), U. ~11,1.
We then define C, = -1 + BU,, where 3 is a hyperparameter passed to the data generating process (DGP). This parameter controls the level of unobserved confounding, with larger values of corresponding to more unobserved confounding. Using this, we generate the study and treatment indicators
Beenouli(exppi((CC) (expit(-Ci)). and
S: ~ Bernoulli
T: ~ Bernoulli (; x 0.5 + S;) x expir(C:)).
where expit is the logistic sigmoid: expit(x) then generated as
The potential outcomes and observed outcome
Ite
Y2(0) 100 + X (ıL) Y,()) 12C; 10X,,, + T, and
Y = T, x (1)) (1 (1-7,) x (0) + Es.
where 6; ~ (0, 1) and T is another hyperparameter passed to the DGP that controls the size of the constant treatment effect. Therefore, larger corresponds to larger treatment effect.
There are five different breakdown frontier plots Section 6.1] Each plot is generated from a different dataset that is created from the above DGP by varying the two hyperparameters, /3 and r. In particular,
Base: 3 and 7 = 5.
Larger т: 3 0.4 and 8.
Smaller т: 0.4 and 5 2.
Larger U: 8 0.6 and 5.
Smaller U: 8 0.2 and r r= 5.
Note that in the plots, "Larger U" and "Smaller U" refer higher a lower levels of unobserved confounding induced by the hyperparameter B, not the values of the unobserved confounder U itself. With full knowledge the DGP, more accurate labels would be "Larger 3" and "Smaller 8 But given that the full DGP was not introduced in the main text, use "Larger U" and "Smaller U for simplicity and accessibility.
G Additional Simulation Results
We extend the analysis in Section 6.1 presenting breakdown frontier plots under variety of data generating scenarios. Specifically, we highlight
Sample size performance
250, 500, 1000, 5000, and 10000.
Sensitivity to the a parameter in the Boltzmann estimator, examining how its effects vary with sample size.
Robustness to moderate violations model assumptions.
Sample Size Figure presents breakdown frontier plots across these sample sizes. The DGP corresponds to the Base case described in Section with 3 = 0.4 and 5.
The results show the instability of the estimator at smaller sample sizes, reflected in wider and inconsistent tentative regions. However, the general takeaways from the breakdown frontier plot remain consistent, and estimator performance stabilizes quickly the sample size increases.
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Conclusive
Tentative
Inconclusive
Incompatible
Exchangeabiity (7)
250
500
1,000
5,000
10.000
Violation in Study
20%
10%
0%
0%
10%
20% 0%
10%
20% 0%
10%
20% 0%
10%
20% 0%
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Violation in NUC (p)
Figure 7: Breakdown frontier plots across various sample sizes (n). Figure titles indicate the number of samples in the dataset. All plots are generated under the Base DGP settings. Conclusive and tentative regions are distinguished using 95% confidence intervals, computed from the sample variance of the efficient influence function.
values Figure presents breakdown frontier plots across different sample sizes and choices the parameter in the Boltzmann estimator. Again, the DGP corresponds to the Base case described in Section F with B = 0.4 and
The results show that larger values of a tend to produce wider conclusive regions, which is consistent with the Boltzmann operator more closely approximating the max and min operators increases. The estimates stabilize for a > 10, suggesting this as generally reliable default choice.
Model misspecification To evaluate the robustness of methodology when model assumptions violated, we consider nonlinear DGP but continue to estimate the nuisance functions using linear models. Specifically, we modify the data generating process in Section ?? as follows:
X1,1, X2,, X1, ~ (0.5,
~ (0.5, 1.5),
C = (1 8)X;,1 + 3U, * #2 1,
S ~ Bernoulli (expi((- (x)pi(-C.)): and
T. Bernoulli S, x 0.5 + (1 S,) x expit(C;)) Y((0) 100 + Xx,2 * +2,
Y,(1) Y,(0) C, X *2 and T
T:) V2(0) 6, ~ N(.11,
and, as before, we vary B and 7 to generate five variants of this DGP.
Figure 9 displays breakdown frontier plots under this misspecified setting. As expected, the fron- tiers are less stable near the boundaries. indicating increased noise due model misspecification. Nonetheless, the plots still recover informative structure as they delineate regions of compatible and incompatible confounding levels and preserve the qualitative shape of the conclusive and inconclu- sive regions. This demonstrates that the breakdown frontier can remain informative even when the nuisance models misspecified.
Figure 10 presents the same designs but with nuisance functions estimated using nonparametric machine learning models (specifically, random forests). In this case, the boundaries are noticcably more stable and the conclusive region expands, consistent with improved approximation of the nonlinear nuisance components. These results suggest that flexible models are advantageous when model assumptions are uncertain and sufficient sample size available. However, when the analyst is willing to impose parametric structure and sample sizes limited, parametric models may be preferred due to improved efficiency.
In summary, our findings highlight that the breakdown frontier approach is robust to moderate misspecification and that the choice of nuisance estimator provides a practical trade-off between robustness and efficiency.
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Conclusive
Tentative
Inconclusive
Incompatible
Exchangeability (c)
100
Violation in Study
20%
10%
0%
10%
20%
0%
10%
20%
0%
10%
20% 0%
10%
20%
0%
10%
20%
Violation in NUC (p)
Sample size
500
Conclusive
Tentative
Inconclusive
Incompatible
Exchangeability (7)
0.1
10
25
100
Violation in Study
10%
0%
10%
Violation in NUC (p)
(b)
1000
Conclusive
Tentative
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Incom
Exchangeabiltty (n)
10
25
100
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20%
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20%
0%
10%
20% 0%
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10% 20% 0%
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(c) 5, 000
Figure 8: Breakdown frontier plots across various sample sizes (n) and a values. Subplot captions indicate the number of samples and individual figure titles indicate the value of used to generate that plot. All plots are generated under the Base DGP settings. Conclusive and tentative regions are distinguished using 95% confidence intervals, computed from the sample variance of the efficient influence function.
H Experimental Details
This section provides implementation specifies for the results presented in Section 6 We outline the hyperparameters and other relevant experimental settings used to generate each of the breakdown frontier plots shown in Section 6 Appendix E includes the algorithm for constructing breakdown frontier plots (Algorithm 2), as well as algorithms for the double machine learning estimator (Al- gorithm I and the procedure for determining (in)compatible sensitivity parameters (Algorithm 33). These components together form the full procedure used to construct breakdown frontier plot.
All experiments conducted in Python. We reference relevant packages and classes sary. The source code can be found in our GitHub repo: https::/ /tap::://gthub.. httpp:///ggthbb....///aaaaaaaaaa
Datasets. For details on the data-generating process used to create the synthetic datasets in Sec- tion 6. see Appendix FF the Project STAR dataset, the outcome of interest was defined as the average across standardized tests math, reading, language, and social science. The objective
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Conclusive
Tentative
Inconclusive
Incompatible
Exchangeabiity (7)
Base
Larger
(b) Smaller
Larger
(d) Smaller U
Violation Study
20%
10%
0%
0%
10%
20% 0%
10%
20% 0%
10%
20% 0%
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20% 0%
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Figure 9: Breakdown frontier plots for various non-linear synthetic datasets generated using linear models to estimate the nuisance functions. Figure titles indicate the relation between the data used to generate that plot to the data used to generate the Base plot. Conclusive and tentative regions are distinguished using 95% confidence intervals, computed from the sample variance of the efficient influence function.
Conclusive
Tentative
Inconclusive
Incompatible
Exchangeabiiig (x)
Base
(a) Larger
(b) Smaller
(c) Larger U/
(d) Smaller U
Violation Study
20%
0%
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10%
20% 0%
10%
20% 0%
10%
20% 0%
10%
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Figure 10: Breakdown frontier plots for various non-linear synthetic datasets generated using nonparametric models to estimate the nuisance functions. Figure titles indicate the relation between the data used to generate that plot to the data used generate the Base plot. Conclusive and tentative regions are distinguished using 95% confidence intervals, computed from the sample variance of the efficient influence function.
was to assess the effect of small class sizes from kindergarten through third grade on this outcome. Measured covariates included gender, race, and age at the start of kindergarten.
For the ATE plot in Figure 3(),, the analysis was conducted on the full dataset. For the CATE plots in Figure b), restricted the dataset two subgroups: students who began kindergarten before age six, and students who were at least six years old at the start of kindergarten.
Further details the Project STAR dataset, including the raw dataset files and cleaning scripts, can be found in the accompanying code.
Hyperparameter Settings. Several hyperparameters are held constant across all breakdown frontier plots. Specifically, the following settings are used throughout:
Grid of (p.7): We construct grid over the (p.7) parameter space by taking all pairwise combinations of values sampled uniformly from the intervals [0, 0.2). Specifically, we define:
E linspace(O, 0.2, 50), E linspace 0.2,
where linspace(a, n) denotes sequence of n evenly spaced values from a b, inclusive. The resulting grid contains 50 50 = 2500 (p, 7) pairs.
Confidence level c: 0.95
Boltzmann smoothing parameter 10
Models used estimate the nuisance functions: ( (x): (x): sklearn. linear Linear_modelll.eeeeeeer
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(x, s): sklearn. wwwimm.ninearraae Liineer_moeel.Lioe httppp/wwww...lgoa...c.eeeeeeeee (x, sklearn. wwnearinnoose mod.l. RidgeCV()
The remaining nayerparaaeeles-snanlll the number of cross-fitting folds to for the double machine learning estimators, k; and the number of resampling iterations to use for the (in)compatible test, R-aae according the data type. For all breakdown frontier plots based on simulated (Section 6.1). we use l = and R = 100. For plots based on the Project STAR data (Section 6.2). we use h and R 1000. The larger values for the Project STAR dataset were chosen generate more precise results.
Uncertainty Estimation. As discussed at the end of Section and in the double machine learning estimator Algorithm[] the variance of the bounds can be estimated either using the sample variance of the estimated efficient influence functions (EIF) or through resampling methods. For the simulated datasets, we used the sample variance of the EIFs. For the Project STAR dataset, however, we used a bootstrap resampling procedure with 1000 iterations.
The bootstrap approach was chosen for Project STAR due to the relatively small dataset size, specifically in the observational study arm, and the extreme nuisance function estimates produced- which led to large and unstable variance estimates when relying solely on the EIFs, In contrast, the bootstrap procedure produced more consistent and stable variance estimates.
Computational Setting and Details. All experiments were run on Slurm-managed cluster using VMware virtual machines, each equipped with Intel(R) Xeon(R) CPU ES-2699 v4 @ 2.20GHz. No GPU or specialized hardware was used.
For the simulated datasets, constructed breakdown frontier results by submitting single Slurm job with CPU core and 32 GB of RAM. Each breakdown frontier plot dataset was generated in a just few minutes, and compatible region tests (Algorithm) were within the same job.
For the Project STAR dataset, where we estimated uncertainty using 1000 bootstrap resamples, we distributed the work across 20 Slurm jobs, each allocated the same compute resources the simulated dataset setup. Each job ran 50 iterations with distinct random seeds. The compatible region tests for Project STAR were
separate Slurm job using the same resources.
Each job completed in approximately 1-2 minutes, and the overall compute cost was low. The primary limiting factor was the number of bootstrap iterations. All scripts can be run on a local machine with sufficient memory to store the breakdown frontier grids.
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Algorithm 1 Bias-corrected estimators for lower and upper bounds
Require: Dataset D, with n samples, sensitivity parameters p. 2 0, Boltzmann operator hyperpa- rameter 2 O, number of folds k E N. with (2 S k S nt).
1: Split data into h folds [E, where E is the set of indices for the samples in the hold-out set for fold and T (1....,n) \ & is the corresponding training set indices.
2: for 1i.,..... do
Estimate the set of nuisance function, ff - (g e ee,,,, ,))), using training data Ty:
3:
4:
go (x): study propensity function
ef (x,s); treatment propensity functions for {0, 1}
5:
6:
( (x,8,)) expected outcome functions for (s,t) E {0, 1] i & do
7:
8:
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65


Reviews:
[{"review_id": "lBzxPqmOkp", "reviewer": "Reviewer_Vp4r", "summary": "This paper contributes to the growing literature on data fusion /\ntransportability / generalizability, in which data from a randomized\ncontrolled trial (RCT; \"internally valid\") and observational study\n(OS; \"externally valid\") are to be combined to generalize to a target\npopulation (typically the OS population). The proposed approach does\nnot require that all confounders are observed in the OS (with\nviolation of the usual assumption quantified through a sensitivity\nparameter $\\rho$), and importantly relaxes the very common assumption\nthat the outcome has the same distribution in the RCT and OS,\nconditional on measured covariates and treatment (with violation\nquantified through the sensitivity parameter $\\gamma$). The\nmethodology is based on nonparametric partial identification /\nsensitivity analysis, where under relaxed assumptions, bounds on\ntarget causal quantities are characterized and identified in terms of\nthe observed data distribution, and subsequently estimated. The\nresulting bound functionals are non-smooth, and the proposed\nestimators are *doubly robust* estimators of **smooth approximations**\nto these challenging estimands. Finally, the authors demonstrate their\napproach in simulations and in a real study of classroom size on\nlearning outcomes.", "questions": "- **Sensitivity Model & Incompatibility**. Can the authors expand on\n  why they used a ratio-based sensitivity model, as opposed to simply\n  bounding the differences in means? For instance, one might have\n  chosen $\\rho = \\sup_{\\mathbf{x},\n  t}|\\mathbb{E}[Y(t) \\mid \\mathbf{X} = \\mathbf{x},\n  S = 0, T = 1 - t] - \\mu(\\mathbf{x}, 0, t)|$. This would result\n  in simpler bounds, but perhaps the authors might argue that this\n  model is less interpretable. More importantly, it is unclear why the\n  chosen parametrization of sensitivity to study exchangeability\n  ($\\gamma$) is the appropriate choice. Note that the numerator for\n  this quantity, $\\mathbb{E}[Y(t) \\mid \\mathbf{X} =\n  \\mathbf{x}, S = 0]$, is partially identified. It seems to me rather\n  more natural to only parametrize deviations from the unidentified\n  component, e.g., $\\gamma = \\sup_{\\mathbf{x}, t}|1 -\n  \\frac{\\mathbb{E}[Y(t) \\mid \\mathbf{X} = \\mathbf{x}, S = 0, T = 1 -\n  t]}{\\mathbb{E}[Y(t) \\mid \\mathbf{X} = \\mathbf{x}, S = 1, T = 1 -\n  t]}|$, where the denominator quantity is identified via\n  $\\mu(\\mathbf{x}, 1, t)$ under the Assumptions A1-A3. This would\n  result in a modified bounding scheme, i.e., $v(\\mathbf{x}, t,\n  \\gamma) = e_t(\\mathbf{x}, 0)\\mu(\\mathbf{x}, 0, t) + e_{1 -\n  t}(\\mathbf{x}, 0)(1 + \\gamma)\\mu(\\mathbf{x}, 1, t)$. Moreover, while\n  I am not 100% certain, this parametrization appears more\n  compatible with $\\rho$ and $w(\\mathbf{x}, t, \\rho)$, whereby the\n  incompatibility issues discussed in Remark 1 and Appendix D could \n  be simplified. Any clarity on these issues that the authors can provide I'm\n  confident would be appreciated by readers.\n- **Sharpness & Bounded Outcomes**. Are the bounds sharp / tight?\n  Before the statement of Lemma 1, it is claimed that the bounds are\n  \"tight\". However, it seems that only validity of the bounds is\n  proven. In fact, it seems to me that the bounds can be sharpened\n  under the assumptions of the paper. Concretely, suppose $b, B > 0$\n  is such that $Y \\in [b, B]$---the authors assume that \"the outcome\n  space is _bounded_ and _positive_\". Shouldn't we have $v(\\mathbf{x},\n  t, \\gamma) = \\min[(1 + \\gamma)\\mu(\\mathbf{x}, 1, t), B]$, and\n  $w(\\mathbf{x}, t, \\rho) = e_t(\\mathbf{x}, 0) \\mu(\\mathbf{x}, 0, t) +\n  e_{1 - t}\\min{[(1 + \\rho)\\mu(\\mathbf{x}, 0, t), B]}$? Analogous\n  lower bound adjustments could be made on the basis of $b > 0$ (there\n  are potential issues with the sensitivity model if the outcome could\n  take values equal to or very near zero). It would greatly stregthen\n  the result if sharpness could be proven. Regardless, a more precise,\n  careful discussion of sharpness / tightness will help clarify the\n  contribution.\n- **Estimation Considerations**. The functionals that identify the\n  bounds are challenging, non-smooth (i.e., not pathwise\n  differentiable) parameters. The authors choose an interesting smooth\n  approximation-based approach, coupled with influence function-based\n  estimators. While appropriate, I have some concerns and questions.\n  1) **Existing literature**. Perhaps most importantly, examples\n     belonging to the class of statistical problems studied here have\n     been investigated in recent years, and important works have\n     seemingly not been acknowledged---see *References* below. For\n     example, Luedtke & van der Laan (2016) studied the functional\n     $\\mathbb{E}(\\max{[\\tau(\\mathbf{X}), 0]})$, for\n     which similar issues arise. They discuss necessary and sufficient\n     conditions for pathwise differentiability, and propose procedures\n     for efficient confidence interval construction. Bonvini & Kennedy\n     (2022) also encounter similar functionals, and impose *margin\n     conditions* sufficient to yield pathwise differentiability\n     (obviating the need for smooth approximations). Levis et\n     al. (2025) and Ben-Michael (2025) also encounter such non-smooth\n     functionals, and carefully study different smooth\n     approximation-based estimators as well as direct estimators under\n     margin conditions. These or related works should likely be cited.\n  2) **Approximation error**. For any choice $\\alpha > 0$, the\n     Boltzmann operator-based smooth approximation bounds exhibit\n     error compared to the originally derived non-smooth bounds. While\n     Lemma 2 establishes that the approximate bounds converge to the\n     original bounds as $\\alpha \\to \\infty$, the result would be\n     greatly strengthened by *quantifying* the approximation error for\n     any finite $\\alpha > 0$---see also Levis et al. (2025) and\n     Ben-Michael (2025) for examples of such analyses. There is likely\n     a formal trade-off / tension between approximation error and the\n     impact of the tuning parameter $\\alpha$ on the second order\n     remainder for the smooth bounds.\n  3) **Nuisance error rates**. The statement of Theorem 2 is, in my\n     view, highly misleading, as asymptotic normality is seemingly\n     achieved without any conditions on the rates of convergence of\n     nuisance estimators $\\widehat{g}$, $\\widehat{e}$,\n     $\\widehat{\\mu}$. While mentioned in passing in the proof in the\n     appendix, these necessary rate conditions need to be explicitly\n     stated in the Theorem itself. Moreover, in their proof, the\n     authors mention an \"empirical process term\" and a \"second order\n     remainder term\", but do not actually state what these are. This\n     needs to be laid out in greater detail. Importantly, I do not see\n     a derivation of the second order bias anywhere in the\n     appendices. It is crucial that the form of this bias is\n     determined in order to conclude that the given nuisance error\n     rates are sufficient for this term to be asymptotically\n     negligible.\n- **Target Estimand**. Can the authors provide some context for why they\n  prefer the CATE, $\\mathbb{E}(Y(1) - Y(0) \\mid \\mathbf{X} =\n  \\mathbf{x})$, which mixes the RCT and OS populations, rather than\n  the effect in the OS population ($S = 0$, often deemed \"externally\n  valid\") on its own? This is not a huge issue, since the proposed\n  methodology implicitly bounds this effect, but a more in-depth\n  discussion of the chosen estimand would help.\n\n\n*References*\n1. Luedtke AR, van der Laan MJ. Statistical inference for the mean\n   outcome under a possibly non-unique optimal treatment\n   strategy. Annals of statistics. 2016 Mar 17;44(2):713.\n2. Bonvini M, Kennedy EH. Sensitivity analysis via the proportion of\n   unmeasured confounding. Journal of the American Statistical\n   Association. 2022 Sep 14;117(539):1540-50.\n3. Levis AW, Bonvini M, Zeng Z, Keele L, Kennedy\n   EH. Covariate-assisted bounds on causal effects with instrumental\n   variables. Journal of the Royal Statistical Society Series B:\n   Statistical Methodology. 2025 May 27:qkaf028.\n4. Ben-Michael E. Partial identification via conditional linear\n   programs: estimation and policy learning. arXiv preprint\n   arXiv:2506.12215. 2025 Jun 13.", "limitations": "Yes", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n- The paper is well written and clear.\n- This work tackles a very important problem, and the goal to relax\n  both the unconfoundedness assumption and the study exchangeability\n  assumption is commendable.\n- The general use of nonparametric partial identification and\n  influence function-based estimation is compelling and convincing.\n- The breakdown frontier approach is interesting, and well\n  illustrated.\n- Empirical illustrations (simulations and the Project STAR\n  application) demonstrate the approach nicely.\n\n**Weaknesses** (see **Questions** below for full detail on each point)\n- The incompatibility issue (Remark 1) resulting from the pair of\n  sensitivity models is quite confusing as currently presented, and\n  possibly improved with a different sensitivity model\n  parametrization.\n- Recent relevant literature on non-smooth functional estimation in\n  related causal problems has not be acknowledged or cited. These\n  works have many insights that could improve this paper.\n- The statement of the main estimation result (Theorem 2) is\n  misleading and, relatedly, its proof is seemingly incomplete.\n- Certain important conceptual and theoretical issues (target\n  estimand, sharpness/tightness) are inadequately or imprecisely\n  discussed.", "was_revised": true, "final_justification": "The paper, even as originally submitted, tackles an important problem using a rigorous, theoretically solid approach. I believe the more serious weaknesses identified by me and the other reviewers will be satisfactorily addressed by the authors' proposed revisions. Overall, the paper is solid and should have important implications for causal inference & sensitivity analysis.", "rebuttal": "Thank you for the extremely detailed and thoughtful review. We appreciate the time you took to read the paper closely and provide such substantive feedback.\n\n**Sensitivity Model & Incompatibility:** \n\nWe chose a ratio-based sensitivity model in part for its scale-invariance, which allows it to generalize across different outcome scales and covariate profiles. While a difference-in-means model (as you suggest) could also be used, we see this as a modeling choice rather than a structural limitation, and our framework could accommodate such a formulation with mild adjustments. We agree that briefly justifying this design decision in the main text would strengthen the paper.\n\nRegarding our parametrization of study exchangeability violations via $\\gamma$, we appreciate your proposed alternative of bounding deviations only from the unidentified component of the observational distribution. However, we chose to define $\\gamma$ via a contrast that is marginalized over $T$ because it more directly reflects violations of study exchangeability as differences in potential outcomes across study populations. \n\nTo illustrate, suppose that for some covariate-treatment pair $(\\mathbf{x}, t)$, there is a nonzero difference in expected potential outcomes between experimental and observational units who received treatment t: $\\left| \\mathbb{E}[Y(t) \\mid \\mathbf{X}=\\mathbf{x}, S=1, T=t] - \\mathbb{E}[Y(t) \\mid \\mathbf{X}=\\mathbf{x}, S=0, T=t] \\right| > 0.$ Now, under your alternative parametrization, $\\gamma = 0$ would correspond to equality in the unobserved counterfactual group (T = 1−t). But this would not imply overall study exchangeability. The marginal difference, $\\left| \\mathbb{E}[Y(t) \\mid \\mathbf{X}=\\mathbf{x}, S=1] - \\mathbb{E}[Y(t) \\mid \\mathbf{X}=\\mathbf{x}, S=0] \\right|$, would remain nonzero (as long as treatment propensities are nonzero). Thus, this alternative $\\gamma$ may appear to label settings as “no study exchangeability violation” even when outcome distributions differ between the samples.\n\nWe agree that your suggestion is principled and could support a valid alternative model. We chose our $\\gamma$ specification because it provides a transparent interpretation of study exchangeability failure that integrates naturally with the interpretation of the rest of our sensitivity framework.\n\n**Target estimand:** \nThis is a good question, and a point we plan to clarify further in our final version. Our decision to target the CATE over the combined population was deliberate. This estimand reflects the common goal in data fusion problems to combine complementary information from both the RCT and observational datasets to estimate effects across a broader population. This perspective aligns with the “generalizability” framework outlined by Parikh et al. (2023), where the objective is to extend findings from a randomized study to a larger or more representative cohort, rather than transporting them to an external population.\n\nThat said, we fully agree that in many settings the observational population ($S=0$) may represent the primary target of interest. Our methodology is flexible with respect to the target estimand, and can be adapted to this case. Specifically, this involves adjusting the marginal covariate distribution in the estimand from $P(\\mathbf{X})$ to $P(\\mathbf{X}|S=0)$, with corresponding modifications to the breakdown frontier calculation. All core components of the sensitivity analysis remain applicable. In the revised manuscript, we will discuss both estimand options (combined population and observational-only), clarify their respective use cases, and explain how our framework supports either goal. We believe this addition will better highlight the flexibility of our approach and address the important point you’ve raised.\n\n\n**Estimation Considerations:**\n\nWe really appreciate the depth of this feedback and the opportunity to address these estimation considerations. We address each below:\n- (1) *Related literature.*\nThank you for the helpful references. We agree that these works are highly relevant, and we will cite them and expand our discussion of related approaches to handling non-smooth functionals. In particular, we will better position our contribution in relation to both smooth approximation–based strategies and alternative approaches that avoid smoothing through structural or margin assumptions. We chose a smooth approximation framework (via Boltzmann operators) to preserve pathwise differentiability and enable influence function–based inference in a principled and generalizable way. One of these references, Levis et al. (2023), adopts a similar approximation strategy using the LSE function. We will incorporate a discussion of our approach and highlight the tradeoffs between it, and similar smooth approximation approaches, compared to other approaches that do not employ smooth approximations.\n- (2) *Approximation error.* \nWe agree that quantifying finite-$\\alpha$ approximation error is an important theoretical question. Our current work focuses on establishing the validity and practical utility of the smooth bounds using Boltzmann approximations, and Lemma 2 guarantees their convergence to the sharp bounds as $\\alpha \\to \\infty$. In practice, we found moderate values of $\\alpha$ (i.e. \\alpha=10) offered a favorable tradeoff between smoothness (for stable estimation) and tightness (for informative bounds), and we demonstrate their empirical behavior in Section 6. We will also include additional simulation results that explore the sensitivity of our estimator to different $\\alpha$ values across varying sample sizes.\nWe recognize that a more formal characterization of the bias–variance tradeoff introduced by finite $\\alpha > 0$ would strengthen the theoretical guarantees. Prior work (like those you cited) offers promising techniques in this direction, and we will cite these in the final version and highlight this as a compelling direction for future work. However, we view this refinement as complementary to the current contribution, which introduces a general sensitivity framework with valid and efficiently estimable bounds. We believe our approach adds meaningful insight even without a full approximation error analysis, and hope it lays the foundation for future extensions that can incorporate these refinements.\n- (3) *Theorem 2 and Nuisance parameter rates.*\nWe thank the reviewer for suggesting that the statement of Theorem 2 should include the regularity conditions and the convergence rate conditions for the nuisance parameter estimators. In the revised version, we will add the necessary rate conditions on $\\hat{g}$, $\\hat{e}$, and $\\hat{\\mu}$ into the theorem statement itself. These regularity conditions are analogous to the ones described in Appendix B of Robins, Rotnitzky and Zhao (1994), and Section 3 of Rudolph et al (2025). In the proof, we will also explicitly define the empirical process term, $(\\mathbb{P}_n - P)(\\hat{\\phi} - \\phi)$, and the decomposition of the second-order remainder term, $R_n$. We will further clarify the conditions under which the empirical process term is $o_p(n^{-1/2})$, and that the second-order remainder is $o_p(n^{-1/2})$ provided the nuisance estimators converge at sufficient rates (i.e., standard conditions such as $|\\hat{\\mu} - \\mu| \\cdot |\\hat{g} - g| = o_p(n^{-1/2})$, etc.). While this is expected to follow under standard regularity conditions for EIF-derived estimators of smooth approximations, we agree that a formal derivation of the second-order term is necessary to justify these conditions rigorously, and we will include this in the revised version.\n\n**Sharpness & Bounded Outcomes:**\n\nYou are correct that Lemma 1 establishes the validity of our bounds, but not sharpness. This was an error on our part. We will revise the text to reflect this more precisely.\nWe appreciate the suggestion to incorporate outcome bounds directly into the sensitivity model, by truncating $v(\\mathbf{x}, t, \\gamma)$ and $w(\\mathbf{x}, t, \\rho)$ with known support $[b, B]$, to yield sharper bounds under the model assumptions. That said, we chose not to incorporate additional min/max operations because it would add further non-smooth structure to an already challenging estimation problem. While we acknowledge this could be remedied using a similar smoothing approach to the one we already use (or one of the other existing approaches you reference), our current framework already applies Boltzmann smoothing between the functions $v(\\mathbf{x}, t, \\gamma)$ and $w(\\mathbf{x}, t, \\rho)$ that are at the heart of our partial identification method. Introducing further smooth approximations for support truncation would require additional derivations and tuning, which we viewed as beyond the scope of this initial contribution.\nWe agree that this is a promising direction, and we will add a discussion in the revised paper clarifying that sharper bounds may be attainable using known outcome support, but that doing so would involve additional modeling and inference challenges we chose not to pursue here. We hope future work can build on our framework to incorporate this work.\n\nReferences:\n- Parikh, H., Morucci, M., Orlandi, V., Roy, S., Rudin, C., & Volfovsky, A. (2023). A double machine learning approach to combining experimental and observational data. *arXiv preprint arXiv:2307.01449*.\n- Robins, J. M., Rotnitzky, A., & Zhao, L. P. (1994). Estimation of regression coefficients when some regressors are not always observed. *Journal of the American statistical Association, 89*(427), 846-866.\n- Rudolph, K. E., Williams, N. T., Stuart, E. A., & Diaz, I. (2025). Improving efficiency in transporting average treatment effects. *Biometrika*, asaf027."}, {"review_id": "iGVPea8foC", "reviewer": "Reviewer_WjYm", "summary": "This paper introduces an approach to sensitivity analysis in the context of estimating effects by combining RCT and observational studies. In essence, there are two main sources of non-identifiability here: unobserved confounding in the observational data, and non-transferability between the experimental and observational settings. The paper introduces a model where each of these can be characterized by a single value, and then applies this to visualize how sensitive study conclusions are to these parameters.", "questions": "See strengths and weaknesses. My main hesitation are about the significance of the work, and the degree to which conclusions from the sensitivity analysis rely on an unassessable sensitivity model.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "The problem seems well-motivated, the paper is clear, and the visualization apparently easy to interpret. I thought the experimental application was also reasonably well executed.\n\nThe main weaknesses in my view are both the relevance of the setting, which seems rather specialized, and the robustness of the sensitivity model.", "was_revised": true, "final_justification": "I continue to support acceptance", "rebuttal": "We appreciate your review and would like to address your concerns about the work's significance and the sensitivity model.\n\n**Regarding significance and relevance**\n\nWe respectfully disagree that our setting is highly specialized. Data fusion causal inference methods are increasingly crucial across healthcare, economics, and policy research, where combining experimental and observational data is necessary but fraught with challenges. For instance, consider the case of drug repurposing Wegovy or Ozempic for weight loss. Any time one has an experiment to examine a treatment that already exists, we will probably have both.\nOur approach addresses a significant methodological gap: while sensitivity analysis has a rich history in causal inference and data fusion methods are growing rapidly, robust sensitivity frameworks specifically for data fusion settings have been notably absent. Our contribution thus serves an important and expanding research area rather than a niche application.\n\n**Regarding the sensitivity model**\n\nThe critique about \"unassessable sensitivity models\" applies to virtually all sensitivity analysis methods, as their fundamental purpose is to explore the implications of untestable assumptions. The strength of our approach lies in its transparent parameterization:\n1. We use interpretable parameters ($\\rho$ and $\\gamma$) that quantify relative deviations in intuitive terms (e.g., $\\rho = 0.2$ means confounding shifts outcomes by at most 20%)\n2. Our parameters are scale-invariant, facilitating meaningful cross-study comparisons\n3. We take a conservative worst-case bounds approach that avoids optimistic modeling choices\n4. Our visualization framework enables practitioners to reason clearly about robustness thresholds\n\nRather than claiming to solve the inherent untestability of assumptions, we provide a principled framework for researchers to assess how strong violations would need to be to invalidate their conclusions. We believe this aligns with the core purpose of sensitivity analysis by enabling researchers to reason transparently about the robustness of their findings."}, {"review_id": "PN08uerf8W", "reviewer": "Reviewer_bEkz", "summary": "The paper proposes a new type of sensitivity analysis for causal data fusion to infer bounds on the CATE / ATE when both an unconfounded trial (potentially misrepresenting the population of interest) and a confounded observational dataset are available. Sensitivity models are imposed, bounds are derived, and estimators based on semiparametric efficiency theory are proposed. The authors evaluate their method using both synthetic and real-world data.", "questions": "Can the authors provide an example where sensitivity parameters can be chosen by domain knowledge for their sensitivity model?\nCan the authors justify their choice of sensitivity model as compared to using an established model such as the marginal sensitivity model (MSM)? \nCan the authors compare their modeling choices and results to related works and elaborate on what is novel or similar/ adapted from existing literature?\nCan the authors extend their results to CATE/ ATE on the observational dataset?", "limitations": "Limitations are discussed in Sec. 7. I would suggest adding limitations regarding the interpretability of the sensitivity parameters, which is a common problem in sensitivity analysis.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths\n- Rigorous work with theoretically valid bounds that incorporates semiparametric estimation for increased statistical efficiency\n- Data fusion in causal inference is an interesting problem with various potential applications\n\nWeaknesses\n- Consistency assumptions are missing in A1-5\n- The target quantity of interest seems to be the CATE or ATE averaged over both datasets. That does not seem to be the quantity one would usually be interested in, as it may depend heavily on the selection probability $P(S = 1 | X=x)$. Indeed, the CATE in the RCT and the observational data are not the same (if they were, we would achieve point identification). I suspect that in practice, one would be more interested in the quantity $E[Y(1) - Y(0) | X=x, S=0] (i.e., the CATE on the observational dataset) as the observational dataset generalizes better to the population of interest.\n- The related work could be expanded to include recent works on data fusion and partial identification, e.g., https://arxiv.org/abs/2406.02873, https://arxiv.org/abs/2406.02464, https://arxiv.org/abs/2412.11511, https://arxiv.org/abs/2405.07186\n- The novelty as compared to related work is not entirely clear. For example, are the proposed sensitivity models novel or adapted from other streams of sensitivity analysis (e.g., on unobserved confounding)? Are the bounds novel or adapted from existing literature? There is a large literature on both sensitivity models and bounds for unobserved confounding. Can the authors justify their choice of sensitivity model as compared to using an established model such as the marginal sensitivity model (MSM)? \n- The authors claim that their sensitivity parameters are interpretable, but it is not entirely clear to me how one would interpret a specific choice of sensitivity parameter in practice. Can the authors provide an example where sensitivity parameters can be chosen by domain knowledge for their sensitivity model?", "was_revised": true, "final_justification": "After the rebuttal I decided to raise my score to 4 (see may responses to the author rebuttal for details).", "rebuttal": "Thank you for your detailed review and critical questions. We hope the following notes help clarify some misunderstandings as well as elaborate on feedback we will incorporate into our final version.\n\n**Regarding sensitivity parameter selection by domain experts**\n\nThank you for this important question. We designed our framework with interpretability in mind, particularly for domain experts who may not be statisticians. In Section 6.2 with the Project STAR dataset, we demonstrate how our parameters translate to meaningful quantities: when $\\gamma = 0.05$, this means a confounder would need to shift student scores by more than 30 points on average (on a scale where scores range from 486-745) to invalidate our findings. Such concrete interpretations make the framework accessible to education researchers.\n\nThe breakdown frontier approach specifically addresses the challenge of parameter selection by visualizing robustness across a range of values rather than requiring precise point estimates. In practice, domain knowledge informs reasonable parameter ranges—for instance, when observational and RCT participants come from the same hospital system, limiting $\\gamma$ to 0-0.05 may be reasonable, while cross-system comparisons might warrant $\\gamma$ in 0-0.25 to account for greater heterogeneity.\n\n\n**Choice of sensitivity model and comparison to alternative approaches**\n\nWhile we build on established sensitivity analysis principles, our approach offers several key innovations for the data fusion context:\n\n1. First, we developed a dual-parameter framework that simultaneously captures both unmeasured confounding ($\\rho$) and study exchangeability violations ($\\gamma$) in a unified model. This enables analysis of how these distinct sources of bias interact—revealing that they can either compound or counteract each other, an insight not captured by applying single-dataset sensitivity approaches separately to each data source.\n\n2. Second, regarding our choice not to use MSM: while MSM effectively parameterizes propensity score sensitivity for a single observational dataset, this approach becomes problematic in multi-dataset settings where propensity scores have fundamentally different interpretations (true randomization probabilities in RCTs versus estimated probabilities in observational data). Our direct parameterization of outcome deviations provides a more consistent interpretation across the fused datasets.\n\n3. Third, our parameters are deliberately designed as scale-invariant, relative magnitude measures. This allows meaningful interpretation regardless of the outcome scale—a critical feature when combining datasets that may have different measurement properties. For instance, $\\gamma = 0.1$ consistently means \"up to 10% relative deviation in outcomes\" whether we're measuring blood pressure, test scores, or income.\nOur approach thus represents a principled extension of sensitivity analysis to the data fusion setting rather than a direct application of existing single-dataset methods.\n\n**Choice of Target estimand**\n\nThank you for this important observation. Our target estimand (ATE over the combined populations) was deliberately chosen because it addresses the common data fusion objective of leveraging complementary strengths of both datasets to estimate treatment effects for a broader population. This aligns with the paradigm described by Parikh et al (2023), where RCT findings are extended to wider populations rather than transported to a specific target. \n\nWe fully agree that in many applications, the observational population (S=0) may be the primary population of interest. Our framework is mathematically flexible to accommodate this alternative estimand. The adaptation requires modifying the marginal covariate distribution in our estimand from $P(\\mathbf{X})$ to $P(\\mathbf{X}|S=0)$ and adjusting the breakdown frontier calculations accordingly. This maintains all the sensitivity analysis benefits while targeting the population highlighted in your comment. In the revised manuscript, we explicitly present both estimand options (combined population and observational-only population), discuss their appropriate use cases, and demonstrate how our sensitivity analysis framework applies to each. This will better highlight the flexibility of our approach while addressing the valid concern you've raised about target population selection.\n\n**Missing Consistency assumptions**\n\nThank you for the suggestion. We note that the standard SUTVA and consistency assumptions are included in the first paragraph of Section 2. Following your recommendation, we have updated our manuscript and added them to our list of assumptions for completeness.\n\n**Literature review expansion**\n\nThank you for these helpful resources. We agree that these are relevant to our approach and have added them to our literature review in the updated manuscript. In particular, Schweisthal et al. (2024) is similar to our work in that it proposes a partial ID framework for combining multiple datasets, albeit it focuses on combining only observational datasets and considers a different subset of assumption violations. We will clarify this distinction and note the complementarity between these approaches in the final version. Van der Laan et al. (2025) and Wang et al. (2024) are good examples of approaches for facilitating valid inference in our setting, and we will be sure to reference them in our final version. Although, they do not appear to focus on partial identification or sensitivity analysis. We have included an expanded literature review on data fusion methods for causal inference in our appendix, and thank the reviewer for providing these sources as additional relevant literature to include.\n\nReferences:\n- Parikh, H., Morucci, M., Orlandi, V., Roy, S., Rudin, C., & Volfovsky, A. (2023). A double machine learning approach to combining experimental and observational data. *arXiv preprint arXiv:2307.01449*."}, {"review_id": "39HDlK0HGp", "reviewer": "Reviewer_Faet", "summary": "The authors consider a setting where we have collected both randomized and observational data related to some target population for which we would like to estimate a treatment effect. Here, the observational data may have unmeasured confounding and there is also a question whether transportability holds between the populations underlying both the two datasets and the target population. The authors propose in this case an approach to estimate bounds for the ATE and CATE when two sensitivity parameters controls the violations of the no unmeasured confounding assumption and the transportability assumption. They further propose an efficient estimator for the bounds of the ATE. They evaluate their estimator on a set of simulations and demonstrate its use on a real-world dataset.", "questions": "- I would like to ask if they can reply to my comments on the study design from my second bullet point under \"Weaknesses\"?\n- Given the claims on line 242-243, can you outline the similarities/differences in applying the same steps as in section 5 to deriving efficient estimators for the CATE? Somehow I would expect that estimating the CATE compared to the ATE should be significantly more difficult, especially if we want to obtain e.g. confidence intervals as you get in Theorem 2.\n- I might have missed this point in the paper, but how should we go about selecting the parameter $\\alpha$ in practice for the Boltzmann operator and have you looked into how sensitive this choice is for your estimator's performance?", "limitations": "The authors seem to address all important limitations.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strengths\n- The paper is overall well-written. I found most things easy to understand, especially Section 1 to 4.\n- I find the approach well-motivated, especially this setting (partial identification in data fusion problems) is definitely overlooked in my opinion and it is nice to see a method like this being proposed. The underlying idea to come up with the bound is relatively simple which I also appreciate.\n- I can not judge on the correctness of the proof for deriving the efficient estimator, but assuming that they are correct, I think these are all important properties to have for the estimator and it is nice to see them being derived theoretically.\n\n### Weaknesses\n- The simulations are relatively simple, focussing only on how the bounds change across various DGPs. It would have also been interesting better understand some of the statistical properties of the proposed estimator such as sample size efficiency, the robustness of the asymptotic properties in Theorem 2, or its sensitivity to choosing the scale in the Boltzmann estimator.  \n- There is one thing in the initial setup of the problem which I don't understand, which I think makes it hard for me to fully appreciate in what kind of study designs this method is applicable. In particular, if I interpret things correctly, we have three populations: the population underlying the RCT ($S=1$), the one underlying the observational study $S=0$, and then the target population.  On one hand, the authors say on lines 44-46 that we can get the ATE if transportability holds **or** if there is no unmeasured confounding in the observational study. This makes me think that the population underlying the observational cohort and the target population are the same, because then we would not need transportability if the observational study is unconfounded. But on the other hand, when reading section 2, I interpret it as being that the assumption A5 (which to me is the transportability condition) is needed together jointly with assumption 4 (NUC) if we would identify and estimate the ATE in the target population using the observational data. There seems to be some sort of contradiction here, I would like to hear the authors' thought on this, and in particular if they could explain, for instance using the STAR dataset as an example, how the experimental/observational populations are related to the target population.\n\nOverall, I like this paper. The reason for giving a borderline accept are some of my confusion on the second bullet point under Weaknesses. I'd be happy to re-evaluate my score if the authors are willing to address this point.", "was_revised": true, "final_justification": "Having read the response from the authors, especially their clarification on the study designs I mentioned under \"Weaknesses\" I feel confident about increasing the score from 4 to 5. The authors also addressed my other concerns (see rebuttal) and I think this is a solid paper.", "rebuttal": "Thank you for the thoughtful and constructive review. We’re glad to hear you enjoyed the paper and appreciate your openness to revisiting your score. Below, we address your main point of confusion and clarify the intended interpretation of our setup and assumptions. We also respond to your additional comments and outline how we will strengthen the final version of the paper.\n\n**Problem Setup and Population Relationships**\n\nThank you for raising this important clarification. Our setup involves just two source populations: the RCT cohort (S=1) and the observational cohort (S=0). Our target estimand is defined over the combined population (the mixture of these two), weighted by their respective covariate distributions. However, Our framework is mathematically flexible to accommodate an alternative estimand that chooses either observational (S=0), RCT (S=1) or a separate third population (S=2) as target population. The adaptation requires modifying the marginal covariate distribution in our estimand from $P(\\mathbf{X})$ to $P(\\mathbf{X}|S=s)$ and adjusting the breakdown frontier calculations accordingly. This maintains all the sensitivity analysis benefits while targeting the population highlighted in your comment. In the revised manuscript, we explicitly present both estimand options (combined populations and specific target population), discuss their appropriate use cases, and demonstrate how our sensitivity analysis framework applies to each. This will better highlight the flexibility of our approach while addressing the valid concern you've raised about target population selection.\n\nUsing Project STAR as a concrete example: both cohorts were drawn from Tennessee school districts—the RCT population consisting of students randomized to different class sizes, and the observational population consisting of students assigned non-randomly. Our target population are all students, not just within one cohort.\n\nWith this clarification, we can explain the statement in lines 44-46: The ATE over this combined population is point-identified if either:\n1. The observational data satisfies no unmeasured confounding (NUC), or\n2. The experimental and observational samples satisfy study exchangeability\n\nThis is because we can decompose the CATE as:\n$\\mathbb{E}[Y(1) - Y(0) \\mid \\mathbf{X} = \\mathbf{x}] = \\sum_{s \\in \\{0,1\\}} \\left[ \\mathbb{E}[Y(1) \\mid \\mathbf{X} = \\mathbf{x}, S = s] - \\mathbb{E}[Y(0) \\mid \\mathbf{X} = \\mathbf{x}, S = s] \\right] P(S = s \\mid \\mathbf{X} = \\mathbf{x})$\n\nThe only non-identifiable components are the potential outcomes for $S=0$. Either NUC or transportability suffices to identify these terms, which is why we develop sensitivity analysis for scenarios where neither assumption fully holds.\n\n**CATE vs. ATE Estimation Differences**\n\nYou are right that precise estimation of bounds for CATE is harder compared to ATE. Regarding lines 242-243 and CATE estimation: The same efficient influence function (EIF) derivation approach applies to both ATE and CATE, but with key differences in implementation. For CATE estimation (for a prespecified $\\mathbf{x}$):\n\n1. We must estimate conditional expectation functions at the covariate value $\\mathbf{x}$ of interest rather than averaging across the entire population\n2. This introduces additional estimation challenges, particularly for high-dimensional $\\mathbf{X}$. In particular, it reduces the resultant effective sample size and just inflates the confidence intervals.\n\nWhile our theoretical framework extends to CATE estimation, practical implementation would rely on stronger smoothness assumptions for tighter uncertainty quantification. \n\n**Selecting $\\alpha$ for the Boltzmann Operator**\n\nYou've raised an important practical question. We discuss this briefly in Appendix E, but note that it is not referenced clearly in the main text. In our implementation, we found $\\alpha$=10 provides a good balance between approximation quality and numerical stability across datasets of moderate size (n = 1000 to 5000). For larger datasets (n>10,000), higher values ($\\alpha$=25+) can be used without numerical instability concerns.\n\nPractically, we recommend:\n- Starting with $\\alpha$=10 for most applications\n- Testing sensitivity to $\\alpha$ by comparing results with $\\alpha$=1 and $\\alpha$=50\n- For very large datasets, increasing to $\\alpha$=50+\n- For small datasets (n<500), using $\\alpha$=0.1-1 to prioritize numerical stability\n\nFor most practical purposes, we found that our performance is not overly sensitive to moderate changes in $\\alpha$, but extremely large values and/or small sample sizes can introduce numerical instability due to the steep gradients in the smooth approximation. In the revision, we will bring this guidance to the main text or add a more explicit reference to Appendix E to ensure it’s easy to find.\n\n**Simulation Study Extensions**\n\nWe appreciate your suggestion to expand our simulations. While we focused on demonstrating bound behavior across DGPs, we agree that examining additional statistical properties would strengthen the paper. In the revision, we will add:\n\n1. Sample size efficiency analysis showing estimator performance at n=250, 500, 1000, 5000, and 10000.\n2. Robustness checks of the asymptotic properties when model assumptions are moderately violated\n3. Sensitivity analysis for the $\\alpha$ parameter in the Boltzmann estimator, specifically looking at behavior across different sample sizes.\n\nThese additional analyses will provide stronger empirical validation of our theoretical results and offer more practical guidance for implementation."}]

Author rebuttal:

Meta-review: My recommendation is to accept the paper.

The paper proposes a sensitivity analysis and bound estimation framework for inference about ATEs and CATEs in a target population, when an unconfounded trial and a potentially confounded observational study are available on distinct populations, and there are potential violations to exchangeability. The estimation framework targets smooth approximations to non-smooth bound estimands. The authors demonstrate the methodology in simulations and on the STAR study.

Reviewers raised some concerns about choices in the sensitivity model, and, most importantly, connections to the existing literature. These issues were addressed well enough in the rebuttal period to reach consensus for acceptance. However, I would strongly encourage the authors to follow the advice of reviewer bEkz to draw explicit connections between the sensitivity model proposed here and other sensitivity models. This is not just about drawing distinctions to show novelty, but also about drawing deeper conceptual connections or distinctions that could guide future work on sensitivity models.
Decision: Accept (poster)