Conformalized Survival Counterfactuals Prediction for General Right-Censored Data
Authors: Sijie Ren
Meng Yan
Zhen Zhang
Xu Yinghui
Xinwei Sun
Venue: ICLR 2026 Poster (2026)
Keywords: Conformalized survival analysis
Counterfactual inference
General right-censored data
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
TL;DR: We propose a conformalized survival counterfactuals prediction method under the potential outcome framework with exact miscoverage guarantee.

Abstract: This paper aims to develop a lower prediction bound (LPB) for survival time across different treatments in the general right-censored setting. Although previous methods have utilized conformal prediction to construct the LPB, their resulting prediction sets provide only probably approximately correct (PAC)–type miscoverage guarantees rather than exact ones. To address this problem, we propose a new calibration procedure under the potential outcome framework. Under the strong ignorability assumption, we propose a reweighting scheme that can transform the problem into a weighted conformal inference problem, allowing an LPB to be obtained via quantile regression with an exact miscoverage guarantee. Furthermore, our procedure is doubly robust against model misspecification. Empirical evaluations on synthetic and real-world clinical data demonstrate the validity and informativeness of our constructed LPBs, which indicate the potential of our analytical benchmark for comparing and selecting personalized treatments.

Published as a conference paper at ICLR 2026
CONFORMALIZED SURVIVAL COUNTERFACTUAL PRE- DICTION FOR GENERAL RIGHT-CENSORED DATA
Sijie Ren' 1,, Meng Yan Zhen Zhang 2.5 Yinghui Xu' Xinwei Sun?
Artificial Intelligence Innovation and Incubation Institute. Fudan University
2 School of Data Science, Fudan University
3 "Zhejiang Key Laboratory of Particle Radiotherapy Equipment, Zhejiang Cancer Hospital
*Department of Radiation Oncology, Tianjin Medical University Cancer Institute & Hospital 5 "Department of Radiation Oncology (Maastro), Maastricht University Medical Centre+
j ren23@m fudan. edu. en, yanmengl 9990 tmu edu. edu. en, zhen. zhang@maast:a nl, (xuyinghui, cunxinwell e fudan. edu. cn,
ABSTRACT
This paper aims to develop lower prediction bound (LPB) for survival time across different treatments in the general right-censored setting. Although pre- vious methods have utilized conformal prediction to construct the LPB,, their re- sulting prediction sets provide only probably approximately correct (PAC)-type miscoverage guarantees rather than exact ones. To address this problem, we pro- pose new calibration procedure under the potential outcome framework. Under the strong ignorability assumption, we propose a reweighting scheme that can transform the problem into weighted conformal inference problem, allowing an LPB to be obtained via quantile regression with an exact miscoverage guarantee. Furthermore, our procedure is doubly robust against model misspecification. Em- pirical evaluations on synthetic and real-world clinical data demonstrate the valid- ity and informativeness of our constructed LPBs, which indicate the potential of our analytical benchmark for comparing and selecting personalized treatments.
INTRODUCTION
Predicting survival time under specific treatment is of great importance in making correct predictions in high-stakes domains for healthcare OObermeyer et al. (2019); Navarro et al. (202 For example, predicting the survival time lung cancer under different heaitohhmmtherapyyoooo regimens crucial for tailoring therapies to individual patients Wilson et al. [222)> Horne al. (2024). optimizing out- comes while minimizing side effects. This problem can be formulated as predicting the conditional expectations of potential outcomes under different treatment regimes Rubin (2005). However, sur- vival data are often right-censsred & Moeschberger (2006); Wilson et al. (2021), meaning the true survival time is not fully observed for some patients. Unlike traditional causal inference, where outcomes are fully observed, right-censored data pose challenges due to the partial information on survival times.
While many previous methods have been proposed to predict the survival function (Cox 1972 Murphy et al. 1997 Tibshirani 1997: Gui Li 2005 Steyerberg 2016). they often rely on model assumptions that are difficult to verify, limiting their ability to provide reliable uncertainty quan- tification. contrast, providing a Lower Predictive Bound (LPB) for survival analysis, instead of predicting the entire survival function, offers enhanced reliability and robustness. LPBs are particu- larly effective at handling censoring and provide a conservative estimate, making them more suitable for high-risk decision- decision-making. This is especially crucial in clinical contexts, where overly optimistic predictions can lead to suboptimal or even harmful treatment choices.
To ensure reliable Lower Prediction Bounds (LPBs), conformal prediction methods Vovk et al. 2005) have been applied to right- segsorc survival data Candès et al. (002)). Recent works have im- proved the calibration process by incorporating additional data points. For instance, Qi et al. (2024)
Corresponding author.

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Published as a conference paper at ICLR 2026
assigned estimated values to censored data based on their true event times, while Gui et al. (2024) introduced a covariate-dependent, data-adaptive censoring time to account for the heterogeneity in the censoring mechanism for Type-I right-censored data with PAC-type guarantee. Additionally, Davidov et al. (2025) extended this framework to handle general right-censored data.
However, these works have two main limitations. First, they do not provide LPB for the treat- ment effect on survival time. For example, while Candès et al. (2023) predicted counterfactual survival times under different conditions, this was only applicable cases where the censoring time exceeded a specific threshold. Furthermore, the LPB provided by these works Gui et al. (2024); Davidov al. (2025) only offers a Probably-Apee Approxiattpppriiittteyyy Colrrer (PAC)-type guarantee, which does not ensure that the prediction is accurate for the entire population. In contrast, the marginal coverage guarantees more reliable and safer predictions across the entire population, including rare and extreme cases, which is crucial in high-stakes clinical scenarios MacDonald et al. (2025).
To bridge this gap, this paper introduces conformalized survival counterfactuals prediction, a novel approach providing exact marginally valid LPB for counterfactual survival outcomes for general right-censored data. At its core, this approach provides an upper bound of a pre-specified nominal level a, through non-conformity score defined via the counterfactual quantile regression function. This upper bound is identifiable, and can be further calibrated using weighted conformal prediction (Lei & Candès 2 Jin et al. 2023). We can show that such an LPB can achieve the marginal coverage, provided that the weight function can be well estimated. We also provide the doubly robustness property. We demonstrate the validity and the effectiveness of our method synthetic data and an in-house lung cancer dataset. We observe that the LPB provided by our method is valid and less conservative than other methods. Besides, the LPB under different radiochemotherapy regimens varies across diverse patient populations in lung cancer, offering valuable insights into personalized treatment strategies.
To summarize, the contributions of this paper can be listed as follows:
Survival counterfactual prediction: We propose a new procedure for quantifying the un- certainty of counterfactual survival time predictions under different treatments in general right-censored data. Our procedure establishes an upper bound for the miscoverage rate that can be reliably identified and further calibrated through weighted conformal prediction.
Theoretical guarantee: We provide distribution- exact guarantee for the counterfac- tual prediction set and quantify the error from weight estimation. We also provide the doubly robustness property.
Empirical validation: We validate our procedure using both synthetic and real clinical data. On the synthetic data, we show that our calibration process yields less conservative LPBs while maintaining the desired coverage guarantee. On the clinical data, we demonstrate its effectiveness in identifying optimal treatments across diverse populations.
2 RELATED WORK
Conformal prediction Vovk al. (2055)
widely used for providing reliable LPB in survival
analysis(Candés Candès et al. (2023); ; Qi et al. ) Meixide el. (2024): Qin et al. (2025) and counterfac- tual inference Lei & Candes 2021 Jin et al. Candes al. (2023): Deshpande & Kuleshov
2024). The conformal prediction for survival analysis was first considered by Candès et al. (2023)
for the Type-I right-censored data. Q el (024) assigned "best-guess" (BG) value as a surrogate of censored data for their true event times, of which performance is heavily affected by the quality of the imputation. Although distribution free methods have been proposed in Meixide et al.] (2024) mid[Qin et al. (2025) for constructing LPBs in the general right-censored setting, the validity of their bootstrap-bases approaches depends on asymptotic results under specific regularity conditions. Gui al. (2024) offered more informative LPBs for Type-I right-censored data by employing the adap- tive cutoff method by tuning a hyperparameter for quantile regression using holdout calibration data to attain the desired coverage rate, and Davidov et al. (0255) extended this framework the gen- eral right-censored setting with a well-designed data selection strategy for calibration, but both with PAC-type guarantees.
However, none of the above works achieves an exact guarantee on general right-censored data. Candès et al. (2023) Gui al. (2024) addressed survival analysis under Type-I right-censored
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setting, which assumes that the censoring time C2 is known and relies on the availability of C₁- Davidov et al. (2023) considered the general right-censored setting, but with a PAC-type guarantee. In this work, we firstly establish conformalized procedure for survival counterfactuals prediction for general right-censored data with exact coverage guarantee of the LPB.
3 PRELIMINARY
Throughout the paper, we focus on the potential outcome framework |ubin ) Splawa-Neyman
al. (1990) with different treatments. In particular, this paper focuses on the under-treated general right-censored data setting. Specifically, suppose we have (W,, X;, Ti, es (=1 e= where W, X, T denotes the treatment, the vector of covariates, the observed censored survival time T min(T, C) and the binary indicator e 1(T < C), with T, C' respectively denoting the true survival time and the censoring time. This means that for each subject, we can observe either the censoring time or the survival time, but not both, depending which event occurs first. For simplicity, we consider binary treatments, i.e., W E (0, 1}.
Denote by (T(1), T(O)) the pair of potential outcomes. We assume that
TI) e4) (1d,
Under the stable unit treatment value assumption (SUTVA) condition Rubin (1990), the observed T; equals (1) W. = and (0) if W, for each proceed, we further require ignorability and overlap assumptions, which are standard in causal survival analysis Kalbfleisch & Prentice (2002): Candès al. 2023).
Assumption 3.1 (Ignorability). (()) (W,C)JX.
Remark 3.2. In addition to the treatment W, we assume that the potential outcome (a.k.a true vival time) is also independent of the censoring time. This has been similarly assumed in Kalbfleisch & Prentice (2022 achieve identifiability.
Assumption 3.3 (Overlap). 0 P(W, 1|x) <.
Given a nominal coverage level a e (0, 1), our goal is to provide the lower predictive bound (LPB) N (X) for any treatment w, ensuring that it satisfies the marginal coverage.
Px, (T(w) 2
(w) N.n. (X)) 2 1 a-a
where N, n are the sample sizes of the training and calibration data.
To provide the LPB, previous works Gui al. (2024); DDvidov et al. (2025) introduced adaptive cut-off methods. These methods offered LPBs that achieve
(PAC)- type coverage, which is approximately marginal coverage based on the available data. However, their approaches may fail to achieve exact marginal coverage that considers the average across the whole population, including extreme cases that are omitted by PAC-type coverage, yet are crucial in survival analysis.
Specifically, Gui et al. 2024 considered the type-T censoring scenario, where we can observe (X,, T. C. Given the estimated quantile regression function 7r (X), they propose to search to meet the coverage guarantee, where 4, is the true 7-th quantile we want to estimate. First, they define (TT) such that
a(T) P(T (X))
8 E[isT < T (X) S (X)
[[I C))+(X)
(1) Extea 86 (T) C A a((7)
Emtr T. (X,) (x.)
where Ical denotes the index set of calibration data, and (x) is chosen to be approximately equal to 1/F( (X) & C|X x). Then, they adaptively choose a cut-off value F, such that a == s7 e [0, sup,eer a(r') a). Later, Davidov al. (2025) extended this adaptive
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Algorithm 1 Conformal Survival Counterfactual Prediction.
Input: Data X,, Ti,e, ex). counterfactual quantile regression estimator 9. of of
I: and function (x)) to fit the weight function; level a, testing point
Split the data into two folds: the training fold Drr and the calibration fold Deal.
2: Define the non-conformity score function: v(w) = v(w) (X,, Ti) gr ( T,
3: Define I(ar) {i: (X,, W,, Tú 6;) Deal with W; = w, e, =
4: Compute the vi(w) for each i I(w)
5: Compute the weight (X;) (X; for each i e I(w)
6: Compute the weights (x) and Poo (x:) from 2
7: Compute the c1-d (T) Quantile a; E seztop ((()))y()) + +oo (2)
Output: The calibrated Conformal Survival Counterfactual Prediction LPB:
I(m) N.u (X,T)
cut-off to a general right-censored setting. However, this method can only achieve approximate marginal coverage due to "(1)" that the empirical average can approximate the population average with high probability. This motivates us to provide a new calibration procedure, which can achieve exact marginal coverage.
METHOD
In this section, we introduce our calibration procedure for survival counterfactual prediction. The core idea of our procedure lies in transforming the coverage probability into a reweighted expecta- tion, and then applying a reweighting scheme for calibration. Section 4.1 introduccs the procedure, and Section 4.2 provides the coverage guarantee.
4.1 CONFORMAL CALIBRATION
To begin, we split the dataset into a training data Drr and a holdout calibration data Deat- Let N [Drl. [Deat]- Correspondingly, we Ia and Teal respectively denote the index set of the training and calibration data.
(T") (x) the true 7-th quantile of Counterfactual quantile regression function. Denote by g.
T(w)JX = x and L(")( (x) the oracle LPB for the corresponding a-th counterfactual conditional quantile function. Under the ignorability condition, we have
P(T(w)|X = x) P(T(w)]X x W = w) PPTXX W w.
That means, the goal is to estimate the 7-th quantile of 7|X I, W w from observed samples (T,, W;} with T; == min(T, C,}. We can then apply censored quantile regression (CQR) meth- ods to estimate (w)) (z), including CQR Peng Huang 20088. CQR forest Li & Bradic (0200) and CQR neural networks Pearce et al. (002)).
Non-conformity score. We follow Romano et al. (2019) to define the non-conformity score as v((") := (w) (x,,T;) (w) (X.)- T; on Deat. indicating how atypical value of the outcome is given observed covariate values, and large value indicates lack of conformity to training data.
Let q (7) denotes the 1 - quantile of (1w) i.e., 2 (T) == inf (nV((P(V(())) (X,, c) a).
Calibration procedure. Note that the adaptive cut-off method Gui al. (2024) adaptively adjusted the 7 such that (<T) == P(T (X)), because ((T) may not necessarily equal (or even close) a when 7- (xX does not estimate the 7-th quantile of T|X (T)X, W in our scenario) well. Therefore, they used (T), the empirical version ((T) to learn the cut-off value. The gap between ā(r) and (T) makes their procedures achieve only approximate coverage.
To achieve the exact marginal coverage, instead of ((T) := P(T < Tr (X)), we provide the upper bound for ((V(") (x, s (T)).. which is exactly equals and therefore avoid the need for
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Table 1: Experiment results with different within reasonable range, with the corresponding optimized r", average coverage rate. and the corresponding optimal LPB. The results are evaluated on 10 independent trials on setting with the same ratio as shown in Figure
0.05
0.10
0.15
0.20
T
0.16
0.16
0.26
0.21
Average coverage rate
0.958
0.914
0.872
0.845
LPB with T
0.411
0.778
1.19
1.57
Optimal LPB
0.503
0.803
1.25
1.64
adaptively adjustment. Specifically, we have:
a= PPV( (XTT)
P(T(w) (e (X) (w) (x)
[P[P(T(W) =4 (w) (x)
(1) (r)
@ Ex[P(T s _ (z) - 9- ( =z, = w=
Ex
P(T s )1") (1) 10d (T) =x W w)) 1,,, W w
1
ple = 2 W w)
(w0)
s
Ex
P(T th
(7), || x, W w)
p(e 1/ W = w)
(1))
Ex
[(T
=
1
P(T
w|X
pl== 1, W = w(w)
E
I [
La
w0) (x) (x) (T), W w. e= (xx)
(w))
[ (7) IW -0 1)
=
W w. e
(1)
(xx)
where (i) follows from Assumption 3.1 and the SUTVA condition, (ii) comes from the tower prop-
erty, (iii) derived by the proof of Lemma A.I conditional on X x. W w, and (xx) == p W
w,e = 1|r). Through the upper bound in (iv), note that it is sufficient for the LPB I (1) N, (X,
to satisfy the coverage guarantee for Px x P Fr -i,. (since T given 1). Denote
wxx) <(W= Since [xip=
[w ( = w 1] 1, we can employ the weighted
>(x)
conformal prediction Lei & Candès 2021 to ensure that
E [[((((")(TT)
(T. 4nd (T) w(X)WW w. w,
To this end, for each e Ioai with W and 1, compute ( (W--we-=l) For the test 7(X,)
data ø, we compute (xx) == (W-nog o- 1) Following Lei Candès 2021) , we can take
(w)
(w) (T) Quantile 1- a; Σ ( + Poo (2)
sez!"))
where
/ (X = x2))
((x)
(x(x)= Σ Exer(m)
(x) =
(2)
(X 244 (xx
Firz ox 22) (xx)
ma
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Setting
Setting 2
Setting 3
Setting 4
Setting 5
Setting 6
0.34
0.54
0.94
0.94
Coverage
0.92
0.90
0.90
0.92
0.90
0.92
0.90
0.92
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0.92
0.00
0.88
0.00
0.20
0.86
0.58
0.86
0.86
0.86
0.56
0.80
Focus
0.325
0.30
0.10
Ours
0.22
Relative LPB
0.30
0.20
0.275
0.00
0.14
0.38
0.a
0.38
0.200
0.20
0.32
0.7
0.14
0.6
0.04
0.30
0.00
Figure Comparison of the different calibration methods in different settings on the synthetic datasets. Top: empirical coverage rate, with a red dashed line indicating the nominal 90% level. Bottom: relative LPB obtained by different methods. higher relative LPB is better. The results are evaluated on 50 independent trials, each consisting of newly sampled train. validation, calibration, and test sets with ratios of 50%, 10%, 30%, and 10% of the synthetic datasets, respectively, totaling 3000 samples.
Since p(W- w = 1) can be canceled out, it is sufficient to only estimate 7(x). Therefore, with a bit of abuse of notation, we denote @ (x) -(2) The LPB can be then given by: I(w) (X, ==
()) (X) (r) The whole procedure is summarized in Algorithm
Zr
LPB optimization. As we will demonstrate the next subsection, our procedure yields a prediction
set that satisfies the coverage guarantee for any (0, To ensure the LPB is informative as possible, we choose T to maximize (((v) (X,+) r) for any test data X. Specifically, for each = x, we obtain:
7" (x) arg max re(0.1)
(wr) ( (z) (T)
The optimized LPB the test data is given by N,,
c/ (x) (rm
4.2 THEORETICAL GUARANTEE
Our theoretical analysis builds on Lei & Candès (2021): ): Candès et al.
2023 Specifically, by (iv)
in T the problem reduces to constructing the LPB for the distribution Px HeW--wee1.pr from the data
(X T:) LeT W, e- PFiws,eel.X.
To achieve calibration, the weight function w(x) is introduced such that
dP
(x) pWW 1)
w(x) dP xI.
(x)
We then propose to estimate (xx from the training data. Theorem 4.1 establishes a distribution-
free exact guarantee for counterfactual prediction intervals under covariate shift, in which
(x)) is the estimated density ratio quantifying distribution shift via Radon-Nikodym derivative (dIP x/ x/TI x
Theorem 4.1. Let (X;, iT.. i.i.d Px] X Pr(w)]X.e=-- Set N = |Drl and (x; be an estimate of the a-th conditional quantile (w) (x)
[Deat) Further, let (w) w)) (x) (w)
Qo
of x, ((x) (x:; be an estimate of (dT x /II x (x). and P((w) (z) be the counterfactual LPB resulting from Algorithm 7 Assume that E D[r) < co, where E. denotes expectation over X IP -- Redefine ((x) as w(x) /R @ (x) (Dr,] that E( (x() (D.)) Then we have:
2 P Pxx (T(w) 2 [(w) (X)) 2 1 a Ex-P sp [[x(X) w (X)]]. (4)
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w/o Treatment 1.
Treatment
Treatment
w/o Treatment
Treatment
Treatment 2
Coverage Rate
Rolottve LPB
0.8
1.0
1.0
0.7 0.7 0.7
Figure 2: LPB comparison from the methods with coverage guarantee of 0.1 for regimes with the same ratios to each other on setting 4, with 10 independent trials.
This bound quantifies how estimation error in the density ratio (#x(z)-w(2)) affects the coverage probability, with the enror term diminishing as the density ratio estimation improves. The weight estimator is normalized to ensure ( [2tr]
which stabilizes the variance and enables tractable theoretical analysis. The result holds without strong distributional assumptions, maintain- ing the non- parameerreiic spirit of conformal methods while extending them counterfactual settings.
The following theorem provides the doubly robust property for counterfactual prediction intervals. This theoretical guarantee ensures that our method maintains valid coverage even when either weights function (ax) or the counterfactual quantile estimator 7()) (x) is misspecified, provided one of them is consistently estimated. The doubly robust mechanism provides mutual compensa- tion between the two estimation approaches: when the weights function is inaccurate, the quantile estimation compensates through Assumption AI, and vice versa, when the quantile estimation is misspecified, the weights function provides robustness through Assumption A2.
Theorem 4.2. Let N (Durl |Deail, and qo (1) (1:) qa")) (xr(Der) denote the estimate of a-th conditional quantile qa (x) T( = z. Further, let 7(x) = 7x((D.. denote the estimate of (z), and I N.n.
(()) (x) denote the corresponding counterfactual LPB resulting from Algorithm 1 and [E[1/<(X)] co. Assume that one of the following holds:
Assume that (X)
AI lim E ((m) T(x) 00;
No000
A2 (i) there exists b,b b2 0 such that P(T(w) 4X = =x) [bi, b2] uniformly over all (x, L)
with & [ga (w)
(x) -r, ra qa (x) + r].
(w)
(ii) let (X) 1o (x)) 93
(w) (x)), there exist 8 > 0 such that
Ex)(
[x(()] 7(x)
DDr < 00,
lim N-
lim N0000
(5)
(x)
FN
Then under SUTVA and the strong ignorability assumption.
lim P ((X.T( (T(w) > I (w))
N.n (x)
(6)
Furthermore, if A2 holds, then for any
O,
(w) lim P Px-Bxin-w.c.> (P(T(w) 2 N.n 5 s1--)))
Theorem 4.2 is a special case of Corollary B.4 in Appendix applies to counterfactual predictions across all treatments. Moreover, property (7) implies that con- formal survival counterfactual analysis has approximately guaranteed conditional coverage for coun- terfactuals if the conditional quantiles estimated well.
easy to see that Theorem 4.2
EXPERIMENT
We evaluate both the coverage rate and the average LPB over the test set synthetic data and real clinical data, with the desired coverage level 1-a to 90%. In Section 5.1 we compare different different scenarios with simulation for censoring and treatment rates in
calibration procedures real-world clinical trials. We further validate our method using some real clinical cases collected from cancer hospital in Section 5.2

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Original: outlier
with N(1.2)
Outllers with
Outliers with N(20,2)
1.50
1.00
1.0
3.00
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0.85
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Coverage Ratts
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9.65
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Uncab
Navee
Rocus
Rroed
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Our
12
1.2-
0.7
1.1
0.4
Relative
1.0
0.3
0.9
0.2
07
0.8
Figure 3: The outliers of survival time are simulated by subtracting the corresponding normal noise for 10% data randomly selected. The simulated outliers, with increasingly smaller values from the left to the right column, with details listed Appendix D.2
5.1
SIMULATION
Data. Simulation is crucial for verifying the efficiency of performance, as the survival time T is often lacking. We test our method on different settings designed to mimic the censorship in real- world clinical trials, as Candès et al. 20033 Gui et al. (2024) Davidov et (2025). With the details described in Appendix
Models. The model utilized multilayer perceptron (MLP) with only one hidden layer to reduce the potential overfitting, implemented in PyTorch. And we fit Random Forest classifiers to estimate the weights function w()). For more details, please refer to Appendix D
Results. All methods aim to achieve coverage rate of 1
90%, corresponding to the red dashed line shown Figure [ The larger the relative LPB, the more informative it is. The results evaluate our method against the uncab
of synthetic data shown in Figure [ In comparison, method applied without calibration, the naive calibrated method, the focused calibration method Davidov al. (2025), and the fused calibration method in Davidov al. (2025).
Our method consistently achieves a more informative median-value LPB while ensuring it is closest to the desired coverage rates across all six experimental settings. Although the average coverage rate of our method slightly falls below 1 - a in setting 6, it remains remarkably close to the target level. It simultaneously demonstrates the highest LPB among all methods that guarantee the desired erage rates (including naive and focused methods). In settings 3, 4, and 5, our method achieves valid coverage rates comparable the fused approach while yielding significantly larger LPB values. Al- though the resulting prediction intervals are wider, our method provides exact statistical guarantees for the coverage probability.
Our method also achieves desired coverage with outliers. To verify the robustness of our method, we introduce outlier data (details in Appendix D.2) into Setting 4 and report the resulting coverage rates and L.BB in Figure The results show that our method consistently maintains the desired coverage guarantee, whereas the compared baselines fail to so. particular, those gethods-"Focus and "Fused" Davidov al. -2thh PAC-type guarantee do not necessarily achieve the marginal coverage in the presence of outlier data.
Additional Experiment. Besides the comparative experiments above demonstrating the effective- ness and robustness of our method, some additional experiments are conducted in Appendix We first verify in Appendix [.Il that once the sample sizes reach certain threshold, our method can stably achieve the coverage guarantee while producing less conservative LPB. In Appendix [.2] we further validate that the I.BB of our method effectively captures adaptiveness to different covariates.
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Radicheerpyyy Techniques
Concurtent Chemstherapy
Inductioo Chemscherapy
Consoldation Chemocherapy
1.000
1.00
0.075
0.9G
0.50
0.000
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0.95
5.7-
0.50
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0.45
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0.40
VMAT
Figure 4: Performance in real data of different kinds of radiochemotherapy regimens: radiother- apy technique, concurrent chemotherapy, induction chemotherapy. and consolidation chemotherapy. Top: empirical coverage rate, with a red dashed line indicating the nominal 90% level. Bottom: LPB (years) obtained by our method. A higher relative L.B indicates the corresponding treatment better. The results are evaluated on 10 independent trials, each consisting of newly split train, validation, calibration, and test sets with ratios of 50%, 10%, 30%, and 10% of the clinical dataset. The ratios of the four above radiochemotherapies are listed in Table
In Appendix E.3 we explore the impact of p(W = e = 1) on our method, with results consistent with the expectations derived from (3. Besides, Appendix [..]and and Appendix E.5 present sensitiv- ity analysis with respect to the regression algorithm and weight function, respectively. The results show that our procedure consistently attains the desired coverage. In addition, report the value of 7" selected by our optimization procedure, along with the corresponding coverage rate and IPB in Table 1 and Figure 11 As shown in Figure 11 the LPB achieved y* is comparable to that at T = a. indicating that the quantile regression model is well trained. Finally, we expand setting 4 to multi-treatment scenario. Figure 2
shows that the LPB varies across treatments but consistently satisfies the coverage guarantee. Please refer to Appendix D.I for additional details.
5.2 APPLICATION ON REAL DATA
Data. We evaluate our method on a real-world dataset of 541 non-small cell lung cancer patients from cancer hospital. Four different radiochemotherapy regimens in real data are examined, and the details of radiochemotherapy and the proportion of patients receiving different treatments is shown in Table 4 The dataset includes 124 clinical and quantitative radiomic features of lung cancer. Detail of the dataset is provided in Appendix C.2
Models. The model trained for real data is a multilayer perceptron (MLP) with three hidden layers, implemented in PyTorch. And we also fit Random Forest classifiers to estimate the weights function (xx). For more details, please refer to Appendix D
Results. To validate our method's performance and reliability in reality, we apply it to real- world dataset stratified by different treatment regimens. First, we investigate the effects of four distinct radiochemotherapy regimens on survival time. As shown in the Figure 4 the result shows a higher median L.BB than those treated under intensity modulated radiation therapy (IMRT), which consistent with the VMAT's better clinical benefits in lung cancer Hunte et al. (002)) Besides, as for the three chemotherapics, the addition of induction chemotherapy and concurrent chemotherapy shows higher LPBs, consistent with prior studies Curran et al. Agguddo et al. (2022). Our analysis also reveals a higher LPB under consolidation chemotherapy. potentially due to more favorable baseline characteristics Liu al.]
Then, we explore the adaptiveness of the LPB on cases under the VMAT technique. We select a set of covariates from three categories of known prognostic factors for survival time, including three

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Szace
sage
Stage
Nax3
Figure 5: Adaptiveness analysis on clinical dataset of the stage, T-Stage, N-Stage, KPS, Max3D- Diameter, and Voxel- Volume. A higher LPB indicates that cases within the corresponding variable subgroup have longer survival times when receiving the current treatment. The details of the above six covariates are described in Table
tumor-related clinical factors (i.e., overall stage, T-stage, and N-stage), one host-related clinical fac- tor (i.e., Karnofsky Performance Status (KPS)), and two quantitative radiomic features of the tumor (i.e., Max3D-Diameter, Voxel-Volume). To better assess how these covariates influence LPB, all covariates are binarized, listed in Table As shown in the Figure S patients with more advanced stages, larger quantitative radiomic features tend to have larger tumor burdens and greater lymph node involvement /Aerts et al. (2014); Amiin et al.) (2017). and shorter survival time. The patients with better body performance status (i.e., larger KPS value) usually have better functional status with longer survival time Quinten et al. (2009). Additionally, Appendix E.6 implements baseline methods that satisfy coverage guarantees in the simulation settings, and reports the corresponding results in Figure I0 As shown, our method can produce higher LPB, suggesting the informativeness of our procedure.
The LPB of our method in real data consistently correlates with important factors while maintaining the desired coverage rate, which demonstrates the applicability of our approach in complex, hetero- geneous real-world scenarios and the potential in supporting personalized clinical decision for tumor treatment.
DISCUSSION
We introduce uncertainty quantification procedure of counterfactuals with exact LPB coverage guarantees for general right-censored survival data. Under the SUTVA and strong ignorability as- sumptions, achieve counterfactual prediction for new test point by training counterfactual quantile regressor and performing a weighted adjustment of the non-conformity score using the counterfactual calibration set. Although the assumptions employed in this paper commonly used in causal inference, isn't easy to guarantee that all these assumptions hold completely during real-world data collection processes. Therefore, in practical applications, appropriately constraining data that violate these assumptions in real datasets could enhance the robustness of our method in counterfactual prediction Oliveeia al. (2024); Feldman Romano (2024). In actual follow-up studies, extreme scenarios such as imbalanced treatment usage proportions and high censoring rates may lead to inaceurate estimation of (z), consequently increasing the uncertainty in counterfactual outcome predictions under such treatments. Hence, mitigating the impact of data imbalance is cru- cial Gui al. (2024). Furthermore, beyond demonstrating differences between treatments, accurate quantitative estimation of the causal effects between different treatment outcomes holds significant importance for decision- maaiing Lopez Gutman (2017): Hu & Gu (002]: Lei Candès (20II)
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ACKNOWLEDGEMENT
This work was supported in part by the State Key Program of National Natural Science Foundation of China (Grant No. 1233)009), Young Scientists Fund of the National Natural Science Foundation of China (Grant No. k..KKHH230088), Funded by Tianjin Key Medical Discipline Construction Project
(Grant No. TITXZDXK-3-004/ and National Natural Science Foundation of China NN..82303672 No.82573437).
ETHICS STATEMENT
The institutional ethics committee approves the usage of the clinical dataset, and all the cases are anonymized in this work. No specific information provided to preserve anonymity.
REPRODUCIBILITY STATEMENT
The proof Theorem and Theorem provided in Appendix A.I and Appendix B.3 respec- tively. The generation details of synthetic data are provided in Appendix C. The model parameters and implementation are provided in Appendix D Our code available conformalized survival counterfactual code

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A PROOF OF THEORETICAL GUARANTEE
Lemma A.1. Suppose that I E X, q( : + Rt. Under Assumption 3.1 and if S qta (x)
4 X xr W w) > O, then we have:
P(T S La (x) x = I, W w) 12, W = w)) s PP caac) (2) ciaa 1|x z, W w),
(w:)
(8)
where the probability is taken over P(W;XX.T..)
Proof. Suppose x X. All derivations in this proof are conditional on X z, W w, and so we denote (x)(wo))) = P( = z, W = w) and La = La w)) (x) C-a here for simplicity.
(w)
We begin by developing P(T N CIT's s La):
(T<C T < I.)) Pm)(w) (T < II, a La)
+ (T<
< T Tran C Ca T 2 (C &Tra)
-e L.)
PT(w))T < C,T Lr, C C(
P (C
(T.(1)((T <La; Ita))
(o) La)
(9)
P Pa((w) (T < C,, < Tra, Cxtraa)
P Pz,(w) (C Lt)
Px((w) (T La) Px))P L(a)
P Pa((w) (T' < <,, < <La, C <<
(C. +
P2,(w) (T <(aa
= = (C2 La) + P (TC,
PA Pa.(w) (T L.))
where the first transition is by the law of total probability and the third is by Assumption 3.1 Then we develop Pa,(w) (T < C): P Pa (T< Pa, (w) <T< <C,C<< + Po) (T CC La)
(10)
By subtraction 1 and TTp. then we get:
P (w.(w) (T C TT.) Pa,(w) (T (C))== (C2 T.a)) (T < <C
P Pa((w)(T (T (T< La)
(P2,(1)) (T CCC LT)) Pa( (T C, Lw))
Pan 2 T.a) - Pr,, (/),(w))( (T CT<C,C2
P Par(w) (T' < <C, < <ba)
- -Pr,(w) (T <C,C<La))
Pz,(w) (T < Lt)
P Pr C,C LTa)
1 P2((()) (T Tea)
Pa((w) (T < Laa)
0,
(1I)
then, multiply 11 by Pa, (a( (T < [T) we have that:
Pa,(w) (T <C, T.n.) (2.(w)) < CC) (T < <Ta)) 20.
(12)
We finish the proof with replacing (T < C,T < Lnm in 12 by (e= 1, T 2 ww)
(x) (w)
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Let Quantile F) denote the 3-th quantile of a distribution function F, i.e.
Quantile F) inf(= z: F(=) 2 3) sup(=: F(z) < (3--
Lemma A.2. (Equation (2) in Lemma 1 from Tibshirani et al. (2019) J. Let .i..... Un+1 E R. and (P1.....PAo Pn-1) be non-negative reals summing I. Then for any f [0, 1] and
Un-1 & Quantile 8.
Σ pióes = Un+1 Quantile
B. Σ prou; + Prt
Proof. Consider the following useful fact about quantiles of a discrete distribution F, with support point 01,..., Ok R: denoting q= Quantile F). if we reassign the points a, > q arbitrary val- strictly larger than q. yielding Quantile(B: (3; F) Quantile F)). Using this fact,
distribution F, then the level 3 quantile remains unchanged,
Vn Quantile 3; Vin u ((o)) =
Vn+1 > Quantile V1:(n+1));
equivalently, we have that
Vn= Quantile($; Vin ((oo) = Vn-1 s Quantile(B; (Biiteile(B:iisnee)))
Lemma A.3. (Equation (10) from Berrett et al. (2020)). Let drv (FFxxPP2x denote the total- variation distance between distributions Pix and P2x. Then
drv (Pxx x Xr. P2x Pv|x) drv (Pix,
Proof. For any (P, and if = y) $ (P'IQ'= = y) for any y, then by the definition of total-variation have:
(Iv(((.)))(PP))) / (()))())- - (z())))))
ʃ g(v)dy (p(x)) - p'x))dr
ʃ 1p(x)-1 -p' p'())
drv(P,P))
A.1 PROOF OF THEBRREM 4.1
Proof. Step I. Under Px-Py (w(X) < 00) = 1. We first consider the case where Px is absolutely continuous with respect to P xi i.e.
PX (X)x<)))
In this case, for any measurable function f.
Ex-rx[((X)] Ex-P Ex- an / [xX))((X)]
(13)
On the other hand, it always holds that Px-Px (w 00) = 1. In addition, the assumption
Ex-F awn [ [xr implies that IP X- am (5 X) 00) 1.. 13 we have
Px-Fx (#(X) oo) = Ex Xx
[w (X))T(((X) oo))
Since the integrand is non-negative,
Ex- en (xX) co)]
lim Ex-Pyi ( ( oo)]
(14)
lim KEX-P x em (x(X) co).
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Thus, also have
Px-Px (0(xx(x(X)))) < <00)= =1.
Index the counterfactual calibration fold (w) = {(X;, Wi, Ti, ez) Dent with W; w, ei = 1} by ..)...,)) and let (X++1,Tan Tn+1 Px
Write _7()) for (X,, Ti(w)) E ( the SUTVA assumption, and V(w) for (V((0) ;...., v(()). For notational convenience, we suppress the subscripts N and in T: W; I(w) well as in (2) and (-)())
((w))xx) (x)))PPxXX())
call under
(x)). Next, for any permutation
(w)+ a on (1...,m + }} and ((w)* E IR=*!, let en (w)= = ("((()) : (((+))) Further, let [(x(w) be the joint density of Z(w) (z("), , z(4)) and p(z(w)) be the density of Z(n) (with respect to dominating measure). Letting ((((w) denote the unordered set of p(w), it is easy to see that
(1))
(V(())((((())) (E(D(w))), Drr) "I1
(w)+
(15)
where I1 is a random permutation with
-(((()))) w w(Xa((+11) w w(Xx(x-1) n. E w(x))
P(n 7(n=)\D.t)
Σ, (z((())))) En w(x(((1))))
Note that this conditional probability well-defined because ((X) < oo almost surely under both ex] and Px. As result, for any E {1,2, n.,,
w(X,)
P(n(n+ 1) = D.. Σ w(X,) Ps (X
where Pn+1 denotes Pae for notational convenience. With (15). this gives
(V n+1 ((V())()("))== = =(((())), Dw.)
e (wr) "[(++1)" ~
n+1
Σ i-1
(16)
Note that ps involves the true likelihood ratio function wxx) and thus different from Pe. Let fpx be a measure with
dFx(x) = (x)) x|W (I))).e1(x).
Since [[0(=)] = 1, X- Xi 00)= == As result, Pxx is a probability measure. Consider now a new sample (X (w) px x x)(()))X, Let v-(w) denote the non-conformity a+1
score of (n-1: and set (ww) (V/(w) ,,, 1/(w) ,T(w)). Using the same argument as for 16... we have
(v(w)) n+1 (((()))((() = {()(w))) D,,)) ~
u+1
Σ pr (X
(17)
Note that each (() 1,..., n+ well-defined since (X.) is almost surely finite under both PP Ww and Px. As consequence,
P ( (Tn)(()) 2 (T()) ().))
P (((w) n+1 c
P ((()) n+1 < Quantile
1 a- a; Σ
n
(x ((--1))6 Poo 100 (irr)
1=1
1 p
((()) Quantile
1-a: Σ pp ((Xn+))o + Pao (Xxnz)) (Dus
((w) m+1 s Quantile
1-a; Σ P 8m )o + Dao (xnr)))0 [E(D(U)), Der
EP
1=1
(0)
1 1- a
(18)
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where (i) uses Lemma A.2 and (ti) uses (17 and the definition of Quantile(B;FF) By Lemma
drv (Px PT((n)) e- Px PT(w))x,e-1) drv PP),
and as a consequence.
P (Ta+1 (w)) 2 T( (Xu+1) Deal) - (Tu-1 (w) T((w) (Xx+ Deai) drv (Pxx, Px), 2 (1w) (Xn+1) (X
(19)
which implies that P (w) I((W) (X (Xa+1) (Drr, Deai) Taking expectation over Deal, we have
Deal) drv (Px Ixx)
P (w)) 2 T,(w) (X.= Deal) 2 TP (T n+1 (w) (w)w) (Xn+1) 1 Deal) drv (Px,
21-08 drv (Pxx Px).
Using the integral definition of total-variation distance and
13
1 drv Pxx,
X|W - (x
(x) dP dPpx (x)]
2
[ (F) x w (r) e=
2
Ex Ex-Px
-x( w(X]].
2
Then, taking expectation over Du. we have
P P(Tnri (w) I( (X.. 21- (x
when P Px-Px (w0 < =0)== =1
00) 1. If Px (w(() 00) < 1, it is clear that Ex-Fxi (X) w(X) oo and Theorem 4.1 holds trivially. Thus, assume Px-Pxiwaw.can (w(X) < 00)= 1 in the remainder.
Step I.. Under Px-Py (w
Let x denote the distribution Fx conditional on the event Ens & (x:w(z) < oo); that is, dx) I(x Eoo xx)x(x))
(20)
PX-PX (Exo) x - (2) and [+>)(<< E.x//Px-xx (Em)). Note that ((w) remains the same on E. when w is replaced by w and Px is replaced by Ix', because the weighted-split-COO algorithm is invariant with respect rescaling the covariate shift estimate. Since Px- Px-ry (x) 00) =1, (4) implies that
Further, set w'(x)
((X.(()) Xxx
xx( (T(w) 2 [(w) (Xn-1)) 11--a- a- Ex-7 wwse
It can be reformulated as
P (Tn+i(w)) T[(w) (X xn+1))
1
E Ex Ex-Pxxiwe... (xX))
2Pxxxx (Exo)
On the other hand, when w (Xn+1) = 00, c (Xn-1) = 00, implying that ((w) = 000 As
result,
P(T+i1w) 2 I(m)
=1. =0
Putting the two pieces together, we have that
P (Tam (w) ((w) (X..)) = (Tni 2 ((w) (X (Xn-1) co) P(w(Xxn1) co)
IP (TT+1 (w) 2 I( (X (w o0) ((w(Xnn11) = 00)
(1 a a) Px-Px (E.o) + Px-Px (E%)
E Bx-Pxi-weerr
|0 X) w X)]
2
} Ex.Px
2 1 - a
2
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B PROOF OF DOUBLY ROBUSTNESS
first prove the nonasymptotic result for one side of the double robustness and then present simpler asymptotic result a corollary in Appendix B.2 And then we prove Theorem 4.2 in Appendix B.3
Theorem B.1. In the setting of Theorem 4. further assume that
(1) there exists v, by. b2 > O, such that P(T(() = t|X x) [6,, b2] uniformly over all (x, t) with (w)
Le Lgo (x) -T (x)) +r];
(2) Px-Px (xx(X) 00) = 1, and there exist 6, M > 0 such that (X( 1+6 1 $ M;
(3) there exists k, l 0 such that lim (X) (X)] lim (X)(
(X)] (X)) 0, where
N-550
N
Ex 17,
(w) (z) 93
Then there is constant B1 that only depends on r. b1, b2, 5, M, k such that
(xXT(w)))PxxPPPrx)x) (T(w) 2 [(w) (X)) - -B
(logn) (logn)(188)))))))) (1-86)(())/2(2+))) n 822)))))))
(X) (1/2+11)) (X)))))
+([[ (V(1)))
(21)
where 8' 6i.
Furthermore, for any E (0, 1) there a constant B2 that only depends on r, b1, bz 6, M, k. e such that, with probability at least 3,
P ((x,T(1))- xPrwxxxx.. (T(w) ( - B2
(1/(2+))) (E (X)E (1/2401)
(22)
B. PROOF THEOREM B.I
We start with the following two Rosenthal-tyye -tepe inequalities for sums of independent random vari- ables with finite (1 8)th moments.
Proposition B.2. (Theorem of Rosenthal 1970)). Let {Z:)...... be independent mean-zero ran- dom variables. Then for any 8 2 I, there exists L(d) 0 that only depends on 5 such that
1+5
(1+8)/2
E
Ze
L(6)
Σ EZZZ
Σ 1=1
E|Z,1
Proposition B.3, (Theorem 2 of von Bahr & Esseen
1965)).
Let { (Zi) E1..,. independent mean-
zero random variables. Then for any ò 1),
E
Σ Z
Σ 2 1+8
Proof. For notational convenience, we suppress the subscripts N and in 7(("), 0, F(w)) as well as in (x)) and (z). Note that Assumption (2) implies that wXX) almost surely finite under F and (XX is almost surely finite under I [x] By the same reasoning as in the Section A.
w(X) is almost surely finite under I' ex]W=w,e=1 and (XX) is almost surely finite under Fx.
Let r/2 and (X, ?(()) denote generic random vector drawn from Px x (P which is
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independent of the data.Then Then
P(T(w) 2 {())
la w) (())
(x) -
2 (1
(((" (f()) -tt_ < <c(
(0) -0)
- - - <<()):
(23)
above, (i) uses the condition that c <<r/2, Assumption (1) and the definitions ga that PPT((w) (1)) (x))=1- a.
2
qa
Next, we derive an upper bound on (Cc (w) (X) < < Let G denote the cumulative distribution function of the random distribution Σ (X))v, Pao (X (8)0 Again, G implicitly depends on N, n, and X. Then (w) implies G(-e) 2 at, and thus,
P
s (G(-c)21-a(X) a.s.
Let (-c) denote the expectation of G(-c) conditional on (Dir. (X;) X), namely,
( " -E(D).
For any s > 0, the triangle inequality implies that
P( PFC
a...
(24)
To bound the first term, we note that
G(-)--G(--) -(-)-G(-)--x(XXXIY ( --c)- - -cV)s
a
t=1
Conditional on G(-c) G (-c)
Lab-Gaussian with parameter
n
0² = Σ (
For any s 0,
82
P(G(-c) G (-c) S sexp
27
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Published as a conference paper at ICLR 2026
Let "/n be any fixed sequence with 7/n O(1). Taking expectation over DX we obtain that
P
P(G(-0)
G" (-c)
2 s/X)
8
S
exp
X
2
82
$ Sexp
+ P(G 2 X)
20
2
Σi (
=exp
+
(Eh-) 2 X2)) + +(())) 271x
20
8²
E (X.)²
S Sexp
+ +
27118
2//n/
EE 2 (X.))
(1)
Σ ((x))
exp
+P IP
2 7n
2m
(ER,0 @(X.))
n²mn (X.)²² n²
eexp
P
Σ (X,)) <
+P Σ (xx)) 2
2//n
4
22
(0)
á exp
+
p 1/ 2
2//n
4
1=11
where (i) uses the fact that X is independent of ( Note that this bound holds uniformly with X. Throughout the rest of the proof, we write O1n v a2n if there exists constant B that only depends on r, b1, b2, 6, M, k, l such that ain Bagn for all n. We consider two cases:
(w) If 5 2 1. then by Markov's inequality,
4E E (xx)
1
P
Σ
(Xx))
4
2
70
71
Since E
[ (X or
1, we have B[o(X.)] By Markov's inequality and Proposition
B.3
Σ / (X;) 1/2 72
PP
21+8
1+8
E
Σ (X.) -[x
n
{nBlo(X) (X,) B[ (200)) [(((())
1-8
(
1
(nE
1+5 (1+5)/2 (1+51 ({X
ni+
1
(25)
(1166/22
where (i) follows from Hölder's inequality which gives
(X,) [[W(X,)] 1+8 2 (I)O [[0(x))]
22 (x)o))))))
Piecing things together yields
1
32
P (G(-c)-G's (-c) 2 exp
2%m
(1+6)/22
717m
1 exp
2
where the last step follows from the fact that 8 and 7n 0(1)
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(2) If < 1, then by Markov's inequality,
n²mn
E [E (2))
E (E: 22)
IP Σ (xx))²
(n²2m (1+6)/2
(n² (1=6)/2
where the last step follows from the simple fact that ||l|p s |||||] for 1, with 1/(1+ (11+0) () and n, (Xx)) By Markov's inequality and Proposition B.3
n P
n 2 21-8
1-8
Σ i-11
(X,))
1/2
n (X,,) - [[x((X)))
2
2x,)) [
(26)
n2 1+8
72
Piecing things together yields
82
1
P(G(-))-G((-e22d))) exp
2//
n
n
(1+6)/2
s²
1
exp
2
+
n
(1-5)/2*
where the last step follows from 7n- 0(1.
In all cases,
52
P(G(-t)-C"(t) 2 s|X) 1 exp
+
n nm (1481)22 8' = =min(0,1 1}.
(27)
2//n
Next, we almost surely bound the term P(G' 1- s/X). By the triangle inequality and defi- nition of E())
v(w) 2 Th (X.) -x)))))))(())))))) =((w)= -E(X,). -(()))((x))))
By Assumption (1) and the definitions of da that P(T 2 (w) (x)) 1-a, (w)) (x)))1-a,P()((") sDD) = 1-a. Conditional on D. {Xx)) is deterministric. Since r/2 2r.
G (-(-)) s Σ (x) (1( 8 9) + [[(w)= +P(((W)> $ 2 |11)]
n
Σ xx) ( 9) +P }
s ODD)
cb
2
Σx(x) (1 (((x)))) 2 5) + +1-a- thi
2
1=1
eb
+ Σ ((X,) 2 g
"
< - a
(28)
2
i=l
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Published as a conference paper at ICLR 2026
where the last step follows from the fact that Ei-1 (i) 1. If eb./4, 28 implies that
P(G" (-c) 2 a- s(x)
p (X) (((X)) 2 ( bb9
<P
s|X
E (X,) 2 2012)
cb
2 s|X
EL (X.) (X)
((X))
ebj s
SP
2i10((1)
2
(
E ((x)) 2 2/2
eh
P
2
(XX,)
2
(X.) 2 +P Σ ((x)) (E(x)) ;/) n(ebi -2s)
Σ
+11
4
(10) $ P
Σ D (X.)
1/2 5 Σ E [(X:) 2
neb
8
where (i) follows from the independence between X and {D {X}} and (ii) follows from the fact that s S ebi/4. /4. By 25 and (26). we have that
Σ (xx)) 2) "(6+0///2*
1
where 8' = min(8, 1}. By Markov's inequality,
n Σ (xx))
(((x)) [((x)) 2 9 nebi
} [ (((,)) {
P
cl+b
where the last step uses the simple fact that T(E(X,) 2 c/2) s (2/c) (x))) Therefore, for any /b, we obtain an almost sure bound of the form
(X))
P(G (-t)211-----3 a = 16
(29)
cl+b
Combining
24
27
and
29
together and setting s eb,/4, 4, we obtain that for any sequence
7n O())
b2 2
E
P (0) (X) -e) s exp
32 7m
+
8 Tn
(1+8)/22
((8+8)/2
el+b
1/ 2
([((X)E((X))
exp
(30)
32 7n
n
(1+80)/2
I+h
7m
Substitute with c, and assume en $ r/2 (recall the beginning of proof). Set
DJ 2
(31)
32 log
Clearly, 7n o(1). Then the first term of (30h is 1/n, and thus,
(logn) (logn)(140)) (1+51)/2
E[O(X)E
P (o (X) -en. (x)
n c 1+0"
cl
Equivalently, there exists constant B that only depends r, by, M, k. l, such that
(log () ())g))(+='))) (1+01)/2
E (XX))*(
PP (41 (X) -c,/ --,,1))) s B
n
ch
1+8
a.s..
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Published as a conference paper at ICLR 2026
Together with 23 it implies that
((T(w) 2 (x))x) (x)
(10gn)
(X)]
2 1 a- b2(en (()))()((((()))) -B
n 210
1+
n
almost surely. Assume B 2b2 without loss of generality, Then
(logn) (logn)(++8)) (1+8
P 2 (1)x)
<P(E()) en).
clk
(32)
For any B e (0, 1), let
(log n) (1+8)/2((28))
(XI(())((1)))
+ (EIO(X)E* 1/(2+))
(2+()))))
B1
Then
(logn) (1+81)/2
(X)(
Ens
(33)
n=" cl-8'
cl
and by Markov's incquality and (13)
[X){ c
[X)( $B.
P(E(X)> c..)
c
Furthermore, Assumption (3) implies that en s r/2 when N N(r and 2 n(r) for some con- stants N(r),n(r) that only depend on r. Replacing by 3B, we obtain that, for N 2 N(r), n n(r).
PP (P T (X)18) s a- BBn)
-BBen) s 3.
We can further enlarge B so that Be, 2 - a when N < <N)) or n < n(r), in which case 22 holds trivially.
To prove the unconditional result, note that 32 implies
P(f(w) 2I(")
(logn)(1+0)) (1+80/2
(
2 1 a - B B cn nz' c0'1
(1-((()))> xw))
clak lak
Let
6, (log n) (1+6))2(225)
(X) ((+0)) (I])( 1(140)
82+(()))))
Then
33 remains to hold. By Markov's inequality and 13
EX)(
([ (X)]
((E(() en)
c
e
Ex.
Furthermore, Assumption (3) implies that e, r/2 when N and are sufficiently large, in which case,
((((w) 2 (w)) (x)) 2 a- 31.) (1..))
(3B 1)cn-
(34)
Similar to 22 we can enlarge the constant to make 21 hold when or n is not sufficiently large.
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B.2 ASYMPTOTIC RESULT
Theorem 4.1 and Theorem B. together imply the following asymptotic result, which is general-
ization Theorem 4.2 in Section 4.2
Corollary B.4. With the same notation as in Theorem 4.1 assume that either Bl or B2 (or both) is
satisfied:
B/ lim - - 0;
N-000
B2 the assumptions ( (1)-3) and the definition of Th in Theorem B. hold.
Then
lim ((XIT())) (T(w) 2 [(w) N,
N.n (X)) 2 1-a
(35)
Furthermore, under B2, for any e 0.
lim Px-Tx (P(T(W)2 T. (x) s 11 -1- No
(36)
B.3 PROOF OF THEOREM 4.2
Proof. Since (X) <
and [8]1/(X)] co, we can here set
KW- w. (z)
1//x(x))
@x(x)= E[p(W w, 1)/ (X) [uu (X)
dP, x (x)
(W = w 1)/7(x)
1/7(x)
w(x) dIP x x=== (x) Elp(W w,e= ())r
[[]]+(
B (ONN(X) (X) = [x)]
Thus, the assumption BI of Corollary B.4 reduce to
1/)x(w) (X)
1/(xx) =0,
lim
N N00
[[1/((X]]
and the assumptions (2) and (3) in B2 of Corollary
B.4 redree to
([) (E]L/7(X)))4+ co, lim No00
E[EN (X))TN = lim No
E[En(X)/1(X)) =0.
lim sup
B[1/5N
E E[I/
N N-
Clearly, A2 implies B2 since (N)) [0,1]. 1].
Now we prove that AI implies B1. fact,
(x)
1/7(x)
lim E
N N0
E[1/TA
(X) Duel
[[1//(X]]
1
limsup N
E
E 111/TN (X)
(X) x(X)
1
+ limsup E
IE
N-
(XX)
([1/
(i)
S lim E
+ lim sumsupli E No
EE E
DDr
IE
(x)
2(X)
(x))
(X)
(x))
1
1
1
1
1
lim Simsupli E N-
lim sup N-+50
E
E
1Drr
TeN (X)
7(X)
(XX)
(x)
7(X)
(8)
I 1+B
limsup N
7(X)
7(X)
(i) above uses the fact that ((x),x((*) E [0, and (ii) uses Assumption A and the condition that B[1/>(X)] c0.
26

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Table 2: Parameters utilized in the six settings of synthetic datasets.
Setting
p
m(x))
(xx)
C(z)
1
5
//2
Exp(2)
2
5
(3 - 1-5. 112 2 [122 2}
2
Exp(3)
3
10
VTa 11 2 (2 /08) [(r2 2]
1.5
Exp 2/3
+0.5
4
10
2.5- 122 2 + 13 [1 2)
1.5
lognormal
3 2-z2 50
5
20
0.1 (I2 T64e))
2
Exp
Tolis 50
+0.6
6
50
0.1 22 + 2406
Pa+2 2
Exp
Pap
Table 3: The censored and treated rate of synthetic datasets, with W e {0, 1}.
Setting
1
3
4
5
6
Censored Rate (%)
37.3
18.6
37.8
46.5
36.4
23.5
Treated Rate (%)
75.0
73.0
60.0
52.0
43.0
71.0
pWW e 1) (%)
47.3
60.6
38.0
32.8
38.9
55.3
C DATASET DETAILS
C.I SYNTHETIC DATASETS
In the six synthetic datasets used in Section 5.1 covariates X are sampled from a uniform distribu- tion X ~ UT0, 4] while the treatment variable is set as binary {0, 1) and determined by sampling from a binomial distribution B(N,b), b), which indicates whether the treatment is administered. The base conditional survival time T|X:
T X ~ (xp(N(
(37)
where (x)) ((((X))(((X)) is the normal distribution with mean ((X) and standard deviation (XX) described in Table 2 along with dimension of X and censoring time Cl X to mimic real-world cen- sorship scenarios commonly encountered in clinical trials Zhou et al. 22020); Wilson et al. (202T). Furthermore, to simulate patients with random early mortality, we choose 5% of subjects, and reas- sign them the parameters μ = In 0.1 and a 0.2 instead, as utilized in Davidov et al. (2025). For the remaining 95%, the event time is defined as T Tsurv.
our work, under the SUTVA assumption, we aim to explore the differences in potential outcomes across different treatments. Therefore, while simulating the true censoring mechanism, the propor- tion of uncensored data also crucial for our method, as shown Table
C.2 REAL-WORLDD CLINICAL DATASET
The dataset comes from a retrospective, single-institution cohort, comprising patients treated be- tween January 2015 and June 2023. The institutional ethics committee has approved the usage of the dataset. The dataset includes 541 patients diagnosed with unresectable locally advanced non- small cell lung cancer (LA-NSCLC) who received radically intended radiotherapy (RT), with or without concurrent chemotherapy. Specific inclusion and exclusion criteria detailed as follows:
Inclusion criteria: 1) pathologically confirmed diagnosis of NSCLC; 2) age at least 18 years old: 3) KPS score at least 70; and 4) diagnosed as inoperable locally advanced NSCLC (AJCC 8th edition) before RT, with definitive RT.
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Table 4: The ratio of different radiochemotherapy regimens in the clinical dataset analyzed in Sec- tion 5.2 For radiotherapy techniques, we have 0 :IIMT,, =VMA,, and for other chemotherapies =
Fase. True.
Radiochemo Therapy
Radiotherapy Techniques
Concurrent Chemotherapy
Induction
Consolidation Chemotherapy
Chemotherapy
Rate
P W 0) p 1)
0.495
0.12
0.665
0.645
0.505
0.88
0.335
0.354
Table 5: The value of six covariates in the clinical dataset analyzed in Section
5.2 where Xowooeer {X; : X; & Quantile }.)) X.-upper {X; X, > Quantile (0.5, F.)} and F. is the distribution
of the corresponding covariate. The values of these six covariates described below.
Covariate
Max3I)
Voxel
Stage
T-stage
N-stage
KPS
Value
Diameter
Volume
0
IIB,
1,2
0,1
70,80
Lowwer
VV-leer
IBI.IIIC
3,4
2,3
90,100
MaD-uppper
Xvp-uppe
Exclusion criteria: 1) history of lung surgery before RT; 2) prior chest RT; 3) poor quality of planning CT images; 4) unacceptable dose deviations; 5) treated with immunotherapy; and 6) follow-up shorter than six months.
The overall survival (OS) is defined the time from the start of RT any cause of death the last follow-up. A comprehensive set of variables was collected, including 17 clinical baseline charac- teristies (e.g., age, gender, Karnofsky Performance Status, etc), tumor information (e.g., pathology, staging, tumor location, etc), comorbidities, and treatment details. List of the 17 clinical baseline characteristics in the dataset
Age (21,...,88)
Gender (Female, Male)
Karnofsky Performance Status (KPS) {70, 80,90, 100)
Pathology (Squamous cell carcinoma, Non-squamous cell carcinoma]
Staging (AJCC 8th) (IB, III, IIB,III))
T-stage {1, 2,
N-stage {0, 1,
Tumor Location (Central, Peripherl))
Weight Loss (NO, YES)
Hypertension (NO, YES)
Diabetes (NO, YES]
Smoking status (NO, YES)
Radiation Dose (2 60Gy) e (NO, YES)
Radiation Technique e (IMRT, VMAT)
Induction Chemotherapy (NO, YES)
Concurrent Chemotherapy e (NO, YES)
Consolidation Chemotherapy (NO, YES)
Additionally, the dataset includes 107 quantitative radiomic features of the tumor, for which the
extraction method followed prior work Luo et al. (2025). These features comprised 14 shape features, 18 first-order features, and 75 textural features. List of the 107 radiomic features extracted in the dataset:
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Table 6: Configurations of the number of hidden layers of the MLP regressor implemented for the synthetic and real datasets.
Setting
Hidden layers
Synthetic dataset
[8]
Clinical dataset
[128, 64, 16]
Table 7: Average runtime (seconds) based on the linear quantile regressor for each run comparison of different methods on simulated datasets, with 10 independent trials.
Setting
4
5
6
Method
Uncab
0.024
0.029
0.048
0.043
0.064
0).220
Naive
0.416
0.483
0.991
0.832
1.26
6.76
Focus
0.212
0.234
0.329
0.291
0.397
1.16
Fused
10.89
12.81
21.1
13.09
25.07
58.6
Ours
1.32
1.42
1.52
1.51
1.72
3.13
Elongation, Flatness, LeastAxisl.ength, MajorAxisL Maxi- mum2DDiameterColumm. Maximun2DDiamoterRww Maxinum2DDiamettsSlic.. Maxi- mum3DDiameter, Volume, MinorAxisLength. Sphericity, SurfaceArca, Surface Vol- umeRatio, VoleIVolume.
Shape features:
First-order features: 10Percentile, 90Percentile, Energy, Entropy. InterquartileRange, Kur- tosis, Maximum, MeanAbsolluteDeviatton Mean, Median, Minimum, Range, Robust- McanAbsoluteDDviiatin,, RootMeanSquared, Skewness, TotalEnergy. Uniformity, Vari- ance.
Textural features:
GLCM: Autocorrelation. ClusterProminence. ClusterShade, ClusterTendency. Con- trast, Correlation, DifferenccAverage, DifferennceEntropy. Difference DifferenceVVrrince,, Id, Idm, Idmn, Idn, Ime1, Imc2, everseVariancee JointAverage, JointEnergy, JointEn- tropy, MCC, MaximumProbabilliy, SumAverage, SumEntropy, -SumSquares.
GLDM: DependenceEnntropy. DependdeneeNonnnnfffrmii,, DependenceNon Unifor- Varniinnee.nnnyLeeeolvvonoiiiirt GrayLevel Vari- GrayLeevelEmphasis,, JargeDependennceefmmaass, LargeDDependenceHfigh LowGrayleevelia phasis, SmallDepeendeneeEEppaaas.., ShanlDopppoeenneeffggGGoe.eeeees SnallDDppedddneeeoooooagggeeeeep
ance, eveeippaas,ee
GLRLM: GrayLeeve/NonUnnffrmii,, GueyLLeee/NaantiioooiiyXXXAmaii
GrayLevvll Variance,
/rayyLee//Vriince, HighGrayyLeveiiuuEEnhasss LongRunEmphasis, Lon- KaunHighhrayLLeeeEmppaaa
www.aunnLooraaLLeeeeeee LowGrayLevel-
RunEmphasis, RunEntropy, monlengthhonUniffrmiy MunLeengthNonUnifoon Uniformi- tyNormalized, RunPercentage, Run Variance, ShorrRunEmphasis, unnrtRunHigh- GrayLevelEmphasis,
GLSZM: GrayLevaNNonnnffomiit
ized, GrayLevelVariance, HigghGaayLLvllZonEEEEmpaaaas LargeAreaEmpha-
tyNormalized, SmallArreaEmhhasis, htwp:A/eeraiiiiininieeeeeeeeepaa SmallAre- sis. Low.GrayLeve/ZooeEEmphssii SizzZZoneNomnniformityy CreeZonNNonnnifomm
ZoneEntropy, ZonePercentage, Zone Variance.
NGTDM: Busyness, Coarseness, Complexity, Contrast, Strength.
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Switing
3.22
3.0
-01
1300
2.33 2030 1553 5003
4500
Figure 6: Performance of the naive and our methods as function of sample size in Setting 3. Top: empirical coverage rate, with a red dashed line indicating the nominal 90% level. Bottom: LPB. A higher LPB is better. The performance metrics are evaluated 10 independent trials, each consisting of newly sampled train, validation, calibration, and test sets ratios 50%, 10%, 30%, and 10% of the synthetic dataset with different sample sizes, respectively.
D EXPERIMENT SETUP
The data is split into four folds in all experiments: 50% for training, 30% for calibration, 10% for validation (used for early stopping), and 10% for testing to evaluate performance. As described in Table 2 the synthetic data is generated through distribution simulations. The collection and inclusion of the clinical dataset are described in Appendix C.2
In all experiments, we estimate the counterfactual quantile regression function by using an MLP neural network implemented in Py'Torch based on CQRNN Pearce et al. (2022). The MLP regressor consists three hidden layers with ReLU activation, early stopping (triggered after epochs without improvement), and training cycle of 400 epochs. The SGD optimizer is used to optimize the model with parameters learning rate = 1e - 3. weight decay= 1e 4, and momentum= 0.9, a batch size of 256, dropout layers with rate of p 0.1, and varying configurations of hidden layers, detailed in Table 6 Besides, we employ scikit-learn Pedregosa et al. (201I) to train a Random Forest Classifiers with maximum depths of 10 and 4, to estimate the weight function 7(x) for real data and synthetic ones, respectively. The lower max depth in the synthetic experiment for reducing potential overfitting.
D.1 DETAILS MULTI-TREATMEN
SETTING 4.
We randomly divide the data into three parts with the same ratio. Based the data generation process of Setting 4, we add Gaussian perturbations of (V.55, 0.2) and (1100, 0.2) to the data under Treatment and Treatment 2, respectively. The experiment aims simulate the differential effects of various treatments on survival time. The results show that our method can distinguish the distinct impacts from different treatments and provide informative LPB.
D.2 DETAILS OF SIMULATION ON ABOUT EXTREME CASES.
The extreme cases (outliers) are added into setting 4 shown in Figure to simulate real-world scenarios where conditions such as interstitial lung abnormalities (ILA) Kashihara et al. (2023) and untreated chronic obstructive pulmonary disease (COPD) Jo al. (2022) lead drastically reduced patients' survival times during treatment. In practical terms, we introduce perturbations to the survival times for 10% of the data randomly selected from the dataset generated under Setting
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De Lyaite
Ormerion1
2003
20 UP
5
Dreerio15
Dineriand
Figure 7: LPB as function of different covariates in the synthetic datasets of Setting I and Setting 2, with a total sample size of 3000, with 10% of the data for testing.
4. by subtracting Gaussian noise sampled from (11, ((,,,),(((O)) (10, and (20, 2), respectively. The perturbed values are then truncated at zero by taking the maximum with zero
D.3 MACHINE SPECIFICATION
The hardware and operating system used for the experiments are as follows.
CPU: Inten())Xeon(R) Platinum 8369B CPU @ 2.90GHz
GPU: NVIDIA A A100-80G-SXM
System: Ubuntu 22.04
D.4 COMPUTATIONAL EFFICIENCY
In addressing the covariate shift from p to Px, it requires the computation of the non- conformality score v.(1) = g(()) (X:) - T, within Deat and finding the weighted (1-ar)-th quantile, resulting in computational complexity of O(Pcail) for the calibration process. The remaining computational overhead lies in the training procedures of the counterfactual regression function
(w) (x: and the weight function (xx). The runtime comparison is shown in Table 7 the runtime 4
increases with the increase in dimension. The runtime of the fused method the longest due to its implementation, which results in three predictions from the weight function for each calibration data point. Our method demonstrates a computationally efficient property of runtime: (comparable to the naive method, and faster than the fused method) while providing strong theoretical guarantees and robust performance challenging scenarios (c.g., multi-treatmmntt extreme cases). The "naive cali- bration process" does not significantly increase runtime, demonstrating that our method is practical for real-world applications- can deliver accurate and robust results without compromising speed.
E ADDITIONAL EXPERIMENT
In addition to comparisons with other baseline methods, we conduct several additional experiments on synthetic datasets to demonstrate the property of our conformalized survival counterfactuals pre- diction procedure between the naive method and ours. Section E.I we explore the impact of sample size on performance. And we examine the adaptiveness of our method in Section E.2 Then, in Section E.3 we investigate whether the method's performance is affected by p(W = w = 1) within reasonable range in the dataset. The experiment of different regression algorithms for our method is provided in Section E.4 Finally, we compare the baselines with coverage guarantee to method in real data in Section E.6
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Setting
Ratre Nean
Parge GurNean
Caverage Rate
0.92
0.93
0.52
0.303
LPO
0.250
0.200
0.200
0.20
0.25 0.33
6.35
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Figure 8: Performance of the naive and our methods as function of p(W = 1, 1) in Setting Top: empirical coverage rate, with red dashed line indicating the nominal 90% level. Bottom: LPB. A higher LPB is better. The performance metrics are evaluated consisting of newly sampled train, validation, calibration, and test sets with ratios of 50%, 10%, 30%, and 10% of the synthetic datasets, with total sample size of 3000, respectively.
10 independent trials, each
E. EFFECT FROM THE SIZE DATA
In this section, we examine both the coverage rate and LPB with varying total sample sizes of the dataset, increasing from 200 to 500, then to 4500 in intervals of 500. The results shown in Figure 6 indicate that our method, based on the SUTVA and strong ignorability assumptions, utilizes part of training and calibration data in practice, making more sensitive to changes data volume compared to the naive method when the data volume is small. When the total sample size is below 1000, our calibration strategy demonstrates less robustness to the effect from sample size compared the naive method, while the interval of LPB shows marked reduction when the sample size exceeds 500. Both methods stabilize after the total data volume exceeds 2000, and our method achieves a tighter LPB with valid coverage rate.
E.2 ADAPTIVENESS ANALYSIS
In this section, we explore the adaptiveness of our method across different covariates. As shown in Figure 7 the LPB of our method demonstrates significant variations corresponding to changes in æ2. and in both x2 and F3, Settings and 2, respectively. Similarly, adaptiveness of the method on real clinical data, with the results presented in Section 5.22 The results show that our method exhibits data adaptability both in synthetic and clinical data for providing reliable LPB for survival counterfactuals prediction.
also investigate the
E.3 EFFECT FROM THE RATIO OF p(W = 1)
Based on the derivation in (3) the theoretical performance of our method remains invariant to changes in p(W w. 1). In this section, we investigate the changes in empirical coverage rates and LPB of the naive method and our method under Setting ( (W e (0, 1)) when p(W 1, 1) varies within a reasonable range. In the simulation experiment, we maintain the censoring rate Setting 3 and vary the treatment ratio p(W = =1) from 0.2 to 0.5 in increments of 0.05 by adjusting the parameter of the binomial distribution. The results, shown in Figure & demonstrate that both
methods achieve valid coverage rates across the reasonable range of p( W = 1,e = 1) variations, while method produces less conservative L.P.
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Published as a conference paper at ICLR 2026
Setting 4
1.00
Coverage Rate
0.95
0.90
0.80
1.1
1.0
LPB
0.9
0.8
0.7
Linear
Forest
Figure 9: Different quantile regressors, linear regression (Linear), neural network (NN), random forest regressor (Forest), and Cox model (Cox), are utilized our method on setting with 10 independent trials.
E.4 THE CHOICE OF DIFFERENT REGRESSION ALGORITHMS.
Based on setting 4, apply four different quantile regressors, namely linear regression (Linear), neural network (NN), random forest regressor (Forest), and Cox model (Cox), with the weight func- tion being random forest, as shown in Figure 9 The results indicate that our method achieves the coverage guarantee and exhibits low sensitivity across the four different regression algorithms evaluated.
E.5 THE CHOICE OF DIFFERENT WEIGHT FUNCTIONS.
We explore four different classifiers for the weight function, namely random forest (Forest), logistic regression (Logistic), neural network (NN), and support vector machine (SVM), with the regres- being neural network, shown in Figure [2] The results indicate that method achieves robustness with different weight functions, both for the coverage rate and LPB.
E.6 COMPARISON OF BASELINES ON REAL DATA.
To evaluate the performance different methods on real-world data, we conducted a four-group experiment comparing Radiotherapy Techniques (IMRT vs. VMAT) and Consolidation Chemother- apy (True vs. False), The methods achieving the desired coverage guarantee are compared and illus- trated in Figure TO The results demonstrate that each method can effectively distinguish the impact of IMRT vs. VMAT and the presence or absence of Consolidation Chemotherapy on survival time. Notably, method not only satisfies these conditions but also provides the most informative LPB.
F
THE USE OF LARGE LANGUAGE MODELS (LLMs)
We declare that in this work. LLMs were used solely for grammatical correction in writing.
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IMRT
N/O Consolication
Consolication
0.7
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Dura
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Figure 10: LPB comparison from the baselines with coverage guarantee of a 0.1 for different treatment regimes, with 10 independent trials.
3.50
3.5
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Ame
1.0
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2:0
Myd
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Figure 11: The variation of the LPB provided by our method with different 7 of quantile regression under various settings.
Setting
1.00
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0.90
0.85
0.80
1.1
1.0
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0.9
0.7
Forest
Logistic
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Figure 12: Different weight functions (classifier), random forest (Forest), logistic regression (Logis-
tic), neural network (NN), and support vector machine (SVM), are applied in our method on setting 4, with 10 independent trials.
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Reviews:
[{"review_id": "Hep3upQQnz", "reviewer": "Reviewer_2PvE", "summary": "The paper proposes a new method to provide lower prediction bounds for counterfactual outcomes of binary treatments in right-censored survival data based on conformal prediction. In contrast to the former work, which only provides approximate guarantees, the proposed method provides exact marginal coverage guarantees. The core idea of the method is based on a transformation of the coverage probability into a reweighted expectation.", "questions": "- Lines 141 following: what is $\\tau$? The notation has not been introduced.\n- Counterfactual quantile regression (lines 168 following): What exactly is the relation between $\\tau$ and $\\alpha$?\n- Double robustness provides asymptotic guarantees in contrast to the finite-sample CP guarantees. Why does it make sense to assess the double robustness of the CP interval? What does it mean in practice? Which value does Theorem 4.2 provide for the CP intervals?\n- Experiments: how does the variance of the coverage and the lower bound compare to other methods? This is not observable from the current evaluation. How does the method perform over different levels of alpha? In comparison to the fused method, one can observe, on average, a larger variance in coverage and a lower bound. This does not support the effectiveness of the proposed method.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- The paper addresses an interesting and important topic: Finite sample uncertainty quantification in counterfactual prediction in time-to-event data.\n- The paper provides rigorous mathematical guarantees and derivations.", "weaknesses": "- The paper lacks a proper related work discussion on conformal prediction for causal effects.\n- In the motivation, the paper states that it provides bounds for \"different treatments\". However, later on, the treatment is considered binary, rendering the former an overstatement.\n- The paper combines existing work in a straightforward manner. I do not see much novelty but rather a direct application of Lei and Candès (2021) and Candès et al (2023). \n- The work contains many wrong statements (potentially due to a last-minute submission). For example, line 188 states that the upper bound exactly equals alpha. This is not true. In line 74, the paper (presumably) confuses upper and lower bounds.\n- The paper requires the weight function to be well estimated. If it is not, the coverage guarantee could even be 0. This is not useful in practice.\n- The experimental evaluation is insufficient to assess the benefit of the proposed method over, e.g., the fused method by Davidov et al. (2025). Statements made on the comparison are not necessarily true. A significantly larger lower bound is not observable in the stated settings 3,4,5. \n- The paper is not yet ready for publication. Besides an often unstructured presentation with confusing mistakes and unexplained mathematical notation, this can be seen in the incorrect specification of the references, e.g., journal name \"find this article online\" or incomplete conference proceedings \"proceedings of the... conference.\"", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "9rTtLWEzeZ", "reviewer": "Reviewer_rUA7", "summary": "The proposed CSC method develops a weighted conformal calibration procedure under the potential outcome framework, producing exact marginal coverage and a doubly robust property against model misspecification. Theoretically, the authors derive formal coverage guarantees and empirically validate the approach on synthetic simulations and a real lung cancer dataset.", "questions": "1. Could the authors provide runtime comparisons with existing conformal survival approaches?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The authors provide formal proofs for marginal validity and robustness.\n2. Introducing conformal calibration for counterfactual survival with exact coverage is a meaningful advance.\n3. Enhances credibility and robustness under imperfect model estimation.", "weaknesses": "1. The experiments, though well organized, are limited to one real-world dataset (a single lung cancer cohort). This constrains the generalizability of the findings. Additional datasets from other clinical domains or public survival benchmarks (e.g., SUPPORT, METABRIC, MIMIC, SEER) would strengthen the empirical evidence.\nMoreover, ablation studies isolating the contributions of key components—such as the choice of quantile regression model (MLP vs. CQR forest), weighting scheme, and calibration size—are missing. Such ablations would help quantify where performance gains arise.\n\n2. The proposed method introduces several estimation layers: quantile regression, density ratio estimation ($\\hat{\\gamma}(x)$), and weighted conformal calibration. These can be computationally intensive and may suffer from instability when sample sizes are small or censoring rates are high. The paper does not analyze runtime complexity, convergence stability, or the effect of hyperparameter choices. In large-scale or high-dimensional settings, density ratio estimation can be error-prone and potentially undermine the theoretical guarantees if $\\hat{\\omega} (x)$ is poorly estimated.\n\n3. The paper mainly compares against a few conformal survival calibration baselines (uncalibrated, naive, focused, fused). However, it omits other relevant conformal-based approaches that are published in recent years. Inclusion of such baselines would clarify whether CSC’s advantages stem from conformalization or from model choice.\nAdditionally, it would be useful to report not only coverage and LPB but also metrics like mean absolute deviation, average interval width, and computational cost for a fuller comparison.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "pPkcIvLFvo", "reviewer": "Reviewer_Mprf", "summary": "This paper aims to give lower prediction bounds (LPBs) for counterfactual survival time under different treatments with general right-censoring. While prior conformal survival methods gave PAC-type guarantees; this work targets marginal population coverage for counterfactual survival times, under the strong ignorability assumption.\n\nThe method adopts the weighted conformal prediction framework. Experiments on synthetic and a 541-patient NSCLC lung-cancer data show near-nominal coverage and informative bounds. \nTheory gives a finite-sample lower bound on marginal coverage that degrades with density-ratio error, and a doubly robust asymptotic result if either the weight estimator or the quantile estimator is consistent.", "questions": "1. Could you please provide some results for for other values of $\\alpha$ to show consistency of the performance?\n\n2. Compare to previous conformal survival works, is it true that one reason the theory successfully avoids the PAC-style guarantee is because of the survival counterfactuals setting. Could you please provide some intuition for why this joint setting enables that?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Important problem: The paper tackles a meaningful and practical challenge—making counterfactual survival time prediction reliable when data are right-censored. This is essential for treatment-effect estimation in medicine, where censoring is unavoidable.\n\n2. Solid theoretical foundation: The authors derive finite-sample marginal coverage guarantees and show that coverage error depends only on the estimation error of a single density ratio. \n\n3. Consistent empirical performance: The method performs reliably on six diverse synthetic setups and on a real high-dimensional clinical dataset, showing both strong coverage and clinically plausible counterfactual predictions.", "weaknesses": "1. Coverage depends on ratio accuracy: The claim of “exact” coverage only holds when the density ratio is estimated perfectly. Theorem 4.1 actually provides a lower bound that subtracts an error term based on the ratio estimation error, so the language precision may be worth reviewing\n\n2. Limiting assumptions: The method assumes strong ignorability, meaning the potential outcomes are independent of both treatment and censoring once covariates are conditioned on. This can be a demanding assumption in practice.\n\n3. Narrow empirical validation: Experiments only test a single coverage level ($\\alpha=0.1$). Broader evaluation would make the empirical evidence more convincing.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "mLupXNjtnr", "reviewer": "Reviewer_xp44", "summary": "This paper proposes a method to estimate the lower prediction bound for the survival time of different treatments under general right-censored data. For that, the authors leverage ideas from weighted conformal prediction in the survival setting under standard assumptions of the potential outcomes framework. The authors provide theory for coverage guarantees and a double robustness property of their framework. Lastly, they perform experiments on synthetic data to compare their method against baselines, and provide results on a real-world medical dataset to show the applicability of their method in practice.", "questions": "- The authors mention “Candes et al. (2023) also provided survival counterfactuals prediction, but only used on cases with censoring time exceeding a specific threshold.” Could the authors elaborate on the difference to that and what the implications of this shortcoming are? That would help to stress their contribution more clearly.\n- For the real-world experiments, how can the empirical coverage given the different treatment strategies be evaluated without access to ground-truth counterfactuals/survival times?\n- For the real-world experiments, how do the results and findings of the authors’ method deviate or coincide with the baselines?", "limitations": "", "rating": 4, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- Uncertainty quantification for counterfactual prediction of survival data is an important problem for real-world medical treatment settings and using conformal prediction seems to be an useful research direction here. \n- The authors provide results on real world data sets and give insights and evaluation of the plausibility of their estimates.", "weaknesses": "Some parts of the theoretical contribution seem unclear to me and the comparison to prior related work could be improved (however, I do not have further background knowledge on conformal prediction for survival data).\n- Especially the contribution over Candes (2023) is not clear to me (see also Questions).\n- E.g., in ll. 182, the authors state, unlike previous methods, their method would achieve exact marginal coverage. However, Theorem 4.1 shows that the method only achieves marginal coverage when $\\hat{w}(x) = w(x)$ if I understand it correctly? Could the authors elaborate on this? \n\nThe experimental results section could be improved especially with respect to clarity and presentation.\n- At least a short summary of the used datasets and a bit more details should be included in the main paper as this is relevant to check the robustness of their method (simple vs complex setup) and practical relevance (real-world applicability, and realistic setup?).\n- the baselines should be cited and described shortly again in the experiments to give intuition about their potential shortcomings.\n- The used metrics (empirical coverage, relative LPB) should be at least once defined formally. \n- The presentation of the results could be improved. In Figure 1, especially the lower row is not easy to interpret. The authors state in the caption of Fig. 1, a higher LPB would be better but write in the text uncab (with highest LPB) would produce a conservative LPB estimate, this seems inconsistent to me. I guess, overall, the authors would like to display a tradeoff between valid empirical coverage and less conservative/tighter LBP estimates? One possible way to show this would be to display both in one graph with coverage and LPB on the axis and show this for over different values for $\\alpha$ to show that their method dominates the others?\nNo code is provided for reproducibility.\n\nMinor:\n- citation style needs to be adjusted (direct vs indirect citations)\n- In the introduction the double robustness property is mentioned. It should be summarized once in the beginning what this means and double robustness with respect to which nuisance functions is meant.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "hliGl9zIPa", "reviewer": "Reviewer_dYEk", "summary": "The paper introduces a novel approach to do conformalized survival counterfactuals prediction, which mainly has two advantages: 1) provide valid LPB for counterfactual survival outcomes for general right-censored data; 2) achieve the marginal coverage with the double robustness property instead of the PAC-type guarantee, which should be more reliable when rare and extreme cases are present.\n\nThe author proposes a new non-conformity score based on the quantile regression and use the probability $P(V(X, \\tilde{T})\\leq c_{1-\\alpha}(\\tau))$ with the help of weight function $w(X)$ to learn the cut-off value, which allows for inference on general censored data with exact coverage guarantee compared to the use of empirical version for $\\alpha(\\tau)$ .", "questions": "None", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. transform the problem into a weighted conformal inference problem with exact coverage for the entire population.\n2. LPB optimization to get the highest LPB with the coverage guarantee.\n3. doubly robust with theoretical guarantee if the weight function is estimated rather well.", "weaknesses": "* The definition of $c_{1-\\alpha}^{(w)}(\\tau)$ appears to be incorrect, as the right-hand side should also depend on $w$. For example, it should be a function of $V^{(w)}(X, T\\leq c)$.\n\n* It would be beneficial to provide the selected $\\tau$ in the LPB optimization. I suspect that the optimal $\\tau$ should be close to $0.5$, as this would allow the majority of the data to be leveraged for a more accurate estimation of $q_{\\tau}(x)$.\n\n* If the extreme cases occur with very small probability, achieving exact marginal coverage should intuitively inflate the conformalized interval (i.e., result in a lower LPB). I did not observe this effect in Figure 1. Is this because the data generation process for the synthetic datasets does not produce too extreme values, or is it an artifact of the LPB optimization step? I believe an ablation study on the LPB optimization would be a valuable addition to the paper, as it would experimentally highlight the importance of achieving exact coverage.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)