Title: Conformal Prediction for Causal Effects of Continuous Treatments
Venue: NeurIPS 2025
Accepted: yes
Keywords: causality

dosage response curves

conformal prediction

uncertainty quantification
Abstract: Uncertainty quantification of causal effects is crucial for safety-critical applications such as personalized medicine. A powerful approach for this is conformal prediction, which has several practical benefits due to model-agnostic finite-sample guarantees. Yet, existing methods for conformal prediction of causal effects are limited to binary/discrete treatments and make highly restrictive assumptions, such as known propensity scores. In this work, we provide a novel conformal prediction method for potential outcomes of continuous treatments. We account for the additional uncertainty introduced through propensity estimation so that our conformal prediction intervals are valid even if the propensity score is unknown. Our contributions are three-fold: (1) We derive finite-sample validity guarantees for prediction intervals of potential outcomes of continuous treatments. (2) We provide an algorithm for calculating the derived intervals. (3) We demonstrate the effectiveness of the conformal prediction intervals in experiments on synthetic and real-world datasets. To the best of our knowledge, we are the first to propose conformal prediction for continuous treatments when the propensity score is unknown and must be estimated from data.

Conformal prediction for causal effects of continuous treatments

Abstract

Uncertainty quantification of causal effects is crucial for safety-critical applications such as personalized medicine. A powerful approach for this is conformal prediction, which has several practical benefits due to model-agnostic finite-sample guarantees. Yet, existing methods for conformal prediction of causal effects are limited to binary/discrete treatments and make highly restrictive assumptions, such as known propensity scores. In this work, we provide a novel conformal prediction method for potential outcomes of continuous treatments. We account for the additional uncertainty introduced through propensity estimation so that our conformal prediction intervals are valid even if the propensity score is unknown. Our contributions are three-fold: (1) We derive finite-sample validity guarantees for prediction intervals of potential outcomes of continuous treatments. (2) We provide an algorithm for calculating the derived intervals. (3) We demonstrate the effectiveness of the conformal prediction intervals in experiments on synthetic and real-world datasets. To the best of our knowledge, we are the first to propose conformal prediction for continuous treatments when the propensity score is unknown and must be estimated from data.

1 Introduction

Machine learning (ML) for estimating causal quantities such as causal effects and the potential outcomes of treatments is nowadays widely used in real-world applications such as personalized medicine [12]. However, existing methods from causal ML typically focus on point estimates [e.g., 38, 40], which means that the uncertainty in the predictions is neglected and hinders the use of causal ML in safety-critical applications [12, 31]. As the following example shows, uncertainty quantification (UQ) of causal quantities is crucial for reliable decision-making.

Motivating example: Let us consider a doctor who seeks to determine the dosage of chemotherapy in cancer care. This requires estimating the tumor size in response to the dosage for a specific patient

profile. A point estimate will predict the average size of the tumor post-treatment, but it will neglect that chemotherapy is ineffective for some patients. In contrast, UQ will give a range of the tumor size that is to be expected post-treatment, so that doctors can assess the probability that the patients will actually benefit from treatment. This helps to understand the risk of a treatment being ineffective and can guide doctors to choose treatments that are effective with large probability .

A powerful method for UQ is conformal prediction (CP) [35, 39, 51]. CP provides modelagnostic and distribution-free, finite-sample validity guarantees for quantifying uncertainty. CP has been widely used for traditional, predictive ML [e.g., 4, 6, 18], where it has been shown to yield reliable prediction intervals in finitesample settings (see Fig. 1). Recently, there have been works that adapt CP for estimating causal quantities (see Fig. 2 for an overview). Yet, existing methods for CP focus on binary or discrete treatments [e.g., 2, 27, 36], but

Uncertainty intervals on

finite data samples

Point estimation

Predicition intervals

No uncertainty covered

A

Y

A

Y

Coverage guarantees:

Conformal interval

UQ                with

asymptotic guarantees

+

Potential

coverage gap

not

Adapting CP to causal quantities is non-trivial for two main reasons. Challenge a : Intervening on the treatment induces a shift in the covariate distribution, specifically in the propensity score. As a result, the so-called exchangeability assumption , which is inherent to CP [51], is violated between the observational and interventional distribution 1 , and because of this, standard CP intervals are not valid. Thus, we must later account for the distribution shift and derive treatment-conditional guarantees . Challenge b : Assessing the aforementioned shift in the distribution requires information about the propensity score; yet, the propensity score is typically unknown. Hence, estimating the propensity score introduces additional uncertainty . However, incorporating the additional uncertainty in the overall CP intervals cannot be done in a simple plug-in manner, and it is highly non-trivial.

Unique to CP for effects of continuous treatments is a third challenge c : data points with the same treatment value are rarely observed. Thus, we later employ smoothing to model the propensity shift.

In this paper, we develop a CP method for causal quantities, such as potential outcomes, of continuous treatments. Our method is designed to account for the additional uncertainty introduced during propensity estimation and is thus applicable to settings where the propensity score is unknown.

Our contributions: 2 (1) We propose a novel method for CP of causal quantities such as potential outcomes or treatment effects of continuous treatments. For this, we mathematically derive finitesample prediction intervals for potential outcomes under known and unknown propensity functions. (2) We provide an algorithm for efficiently calculating the derived intervals. (3) We demonstrate the effectiveness of the derived CP intervals in experiments on multiple datasets.

2 Related Work

UQfor causal effects: Existing methods for UQ of causal quantities are often based on Bayesian methods [e.g., 1, 21, 22, 24]. However, Bayesian methods require the specification of a prior distribution based on domain knowledge and are thus neither robust to model misspecification nor generalizable to model-agnostic machine learning models. A common ad hoc method for computing uncertainty intervals is Monte Carlo (MC) dropout [14]. However, MC dropout yields approximations of the posterior distribution, which are not faithful [34].

Conformal prediction: CP [35, 39, 51] has recently received large attention for finite-sample UQ. For a prediction model ϕ trained on dataset D T = ( X i , Y i ) i =1 ,...,m and a new test sample X k , CP aims to construct a prediction interval C ( X k ) such that P ( Y k ∈ C ( X k )) ≥ 1 -α for some significance level α . We refer to [4] for an in-depth overview. Due to its strong finite-sample validity guarantees, CP is widely used for traditional, predictive ML with widespread applications such as in medical settings [57] or drug discovery [3, 10].

Several extensions have been developed for CP. One literature stream focuses on CP with marginal coverage under distribution shifts between training and test data [e.g., 8, 11, 15, 16, 17, 18, 19, 36, 41, 48, 55]. Our setting later also involves a distribution shift due to the intervention on the treatment but differs from the latter in that the true distribution shift is unknown. Another literature stream constructs intervals conditional on the variables following the shifted distribution. Since, in general, exact conditional coverage has been proven impossible [35, 50], the works in

Causal conformal

prediction

Unknown

propensity

Finite sample

exact guarantees

Wang et al. (2024)

treatment

continuous

Ours

this literature stream have two key limitations: (1) they only guarantee approximate conditional coverage [e.g., 5, 7, 35, 43]; or (2) they are restricted to specific data structures such as binary variables [e.g., 35, 50]. Because of that, none of the existing methods for marginal and conditional coverage can be applied to derive prediction intervals with finite-sample validity guarantees for causal quantities of continuous treatments.

Conformal prediction for causal quantities: Only a few works focus on CP for causal quantities (see Fig. 2). Examples are methods aimed at off-policy learning [47, 58], conformal sensitivity analysis [56], or meta-learners for the conditional average treatment effect (CATE) [2, 27, 36, 52]. However, there are crucial differences to our setting: First, the existing works (a) assume that the propensity is known and thus achieve finite-sample coverage guarantees, or the existing works (b) focus on the easier task of giving asymptotic guarantees but then might suffer from under-coverage because of which the intervals are not faithful. Only Lei and Cand` es [36] provides finite-sample coverage guarantees under estimated propensity scores. However, all existing CP methods are designed for binary or discrete treatments. Applying such methods to discretized continuous treatments leads to ill-defined causal estimands. Therefore, none of the existing methods are applicable to our continuous treatment setting. We offer a detailed discussion in Supplement F.

Research gap: To the best of our knowledge, no work has provided prediction intervals with finite-sample validity guarantees for causal quantities of continuous treatments.

3 Problem formulation

Notation: We denote random variables by capital letters X with realizations x . Let P X be the probability distribution over X . We omit the subscript whenever it is obvious from the context. For discrete X , we denote the probability mass function by P ( x ) = P ( X = x ) and the conditional probability mass functions by P ( y | x ) = P ( Y = y | X = x ) for a discrete random variable Y . For continuous X , p ( x ) is the probability density function w.r.t. the Lebesgue measure.

Setting: Let the data ( X i , A i , Y i ) i =1 ,...,n consisting of observed confounders X ∈ X , a continuous treatment A ∈ A , and an outcome Y ∈ Y be drawn exchangeably from the joint distribution P . Additionally, let a new sample of confounders X n +1 be drawn independently from the marginal distribution P X . Throughout our work, we split the dataset into a proper training dataset D T = ( X i , A i , Y i ) i =1 ,...,m , and a calibration dataset D C = ( X i , A i , Y i ) i = m +1 ,...,n . Furthermore, let π ( a | x ) define the generalized propensity score for treatment A = a given X = x .

Throughout this work, we build upon the potential outcomes framework [44]. We denote the potential outcomes of a hard intervention a ∗ by Y ( a ∗ ) and of a soft intervention A ∗ ( x ) ∼ ˜ π ( a | x ) = P A ∗ | X = x by Y ( A ∗ ( x )) . 3 We make three standard identifiability assumptions for causal effect estimation: positivity, consistency, and unconfoundedness [e.g., 2, 27]. Finally, we consider an arbitrary machine learning model ϕ to predict the potential outcomes. Hence, we define the outcome prediction function as ϕ : X × A → R , ϕ ( X,A ) ↦→ Y . We assume the dose-response curve to be sufficiently H¨ older-smooth. This is common in settings with continuous treatments [e.g., 40, 45].

Our objective: In this work, we aim to derive conformal prediction intervals C ( X n +1 , ♢ ) for the prediction of a potential outcome Y n +1 ( ♢ ) of a new data point under either hard, ♢ = a ∗ , or soft intervention, ♢ = A ∗ ( X n +1 ) ∼ ˜ π ( a | X n +1 ) . The derived intervals are called valid for any new exchangeable sample X n +1 with non-exchangeable intervention ♢ , i.e., for ♢ ∈ { a ∗ , A ∗ ( X n +1 ) } and significance level α ∈ (0 , 1)

Of note, our CP method can be used with an arbitrary ML model ϕ to predict the potential outcomes.

In CP, the interval C is constructed based on so-called non-conformity scores [51], which capture the performance of the prediction model ϕ . For example, a common choice for the non-conformity score is the residual of the fitted model s ( X,A,Y ) = | Y -ϕ ( X,A ) | , which we will use throughout our work. For ease of notation, we define S i := s ( X i , A i , Y i ) .

Whyis CP for causal quantities non-trivial? There are two main reasons. First, coverage guarantees of CP intervals essentially rely on the exchangeability of the non-conformity scores. However, intervening on treatment A shifts the propensity function and, therefore, induces a shift in the covariates ( X,A ) ( → Challenge a ). Formally, we have a propensity shift in which the intervention ♢ shifts the propensity function π ( a | x ) to either a Dirac-delta distribution of the hard intervention, δ a ∗ ( a ) , or to the distribution of the soft intervention, ˜ π ( a | x ) , without affecting the outcome function ϕ ( x, a ) . As a result, the test data sample under ♢ does not follow the same distribution as the train and calibration data, i.e., the exchangeability assumption is violated.

Second, the propensity score π is commonly unknown in observational data and, therefore, must be estimated, which introduces additional uncertainty that one must account for when constructing CP intervals ( → Challenge b ). Crucially, existing coverage guarantees [e.g., 50, 48] do not hold in our setting. Instead, we must derive new intervals with valid coverage under propensity shift .

In the following, we address the above propensity shift by performing a calibration conditional on the shift induced by the intervention, which allows us then to yield valid prediction intervals with significance level (1 -α ) for potential outcomes of a specific hard or soft intervention. We emphasize that the extension to intervals for causal effects is straightforward, in that one combines the intervals for each potential outcome under a certain treatment and without treatment, so that eventually arrives at CP intervals for the individual treatment effect (ITE). Details are in Supplement A.

4 CP intervals for potential outcomes of continuous treatments

Recall that intervening on test data breaks the necessary exchangeability assumption, i.e., the guaranteed coverage of at least (1 -α ) . Therefore, we now construct CP intervals where we account for a (potentially unknown) propensity shift in the test data induced by the intervention.

Scenarios: In our derivation, we distinguish two different scenarios(see Fig. 3):

1 Known propensity score (see Section 4.1): If the propensity score in the observational data is known, it means that the treatment policy is known. Then, we aim to update the policy by increasing/decreasing the treatment by a value ∆ A , i.e., A ∗ ( X ) = A +∆ A . Example: A doctor prescribes a medication to a new patient. Instead of prescribing the same dosage as he would have prescribed to a similar patient in the past, the doctor is interested in the potential health outcome when increasing (or decreasing) the original dosage by an amount ∆ A .

=

Different

physician

Same

physician

=

1

2

Known

propensity score

Challenge:

Unknown

propensity score

Challenge:

+

a

b

a

2 Unknown propensity score (see Section 4.2): In observational data, the propensity score is unknown. Therefore, we usually assess the effect of hard interventions, i.e., a ∗ . Here, we face

additional uncertainty from propensity score estimation ( → Challenge b ). 4 Example: In our running example, a patient sees a doctor who has never prescribed the respective medication and thus will base the decision on observational data (electronic health records), which was collected under a different, unknown treatment policy (e.g., from another physician). Thus, the prescribed intervention (dosage) cannot be expressed in terms of the policy in the observational data.

In our derivations, we make use of the following two mathematical tools. First, we define the propensity shift . Formally, it is the shift between the observational and interventional distributions, P and ˜ P , in terms of the tilting of the propensity function by a non-negative function f . Hence, we have

for some f with E P [ f ( X,A )] > 0 and a ∈ A , x ∈ X .

Second, our CP method will build upon ideas from so-called split conformal prediction [39, 51], yet with crucial differences. In our methods, the calibration step differs from the standard procedure in that we conditionally calibrate the non-conformity scores depending on the tilting function f to achieve marginal coverage for the interventional - and thus shifted - data.

High-level outline: Our derivation in Sections 4.1 and 4.2 proceed as follows. Following [18, 42], we reformulate split conformal prediction as an augmented quantile regression. Let S i represent the non-conformity score of the sample ( X i , A i , Y i ) for i = m +1 , . . . , n of the calibration dataset and S n +1 = S an imputed value for the unknown score of the new sample. We define

where

Of note, ˆ θ S is an estimator of the (1 -α ) -quantile of the non-conformity scores [32, 46]. Using ˆ θ S , we construct the CP interval with the desired coverage (1 -α ) . However, the interval is only valid for exchangeable data. Quantile regression might yield non-unique solutions that can depend on the indices of the scores [18], so we later restrict the analysis to solvers invariant to the data ordering. 5

4.1 Scenario 1: Known propensity score

We first consider scenario 1 with known propensity scores. Here, existing CP intervals are not directly applicable due to the shift from old to new propensity ( → Challenge a ). For our derivation, we need the following lemma building upon and generalizing the intuition presented above.

Lemma 4.1 ([18]) . Let F define a finite-dimensional function class that includes the function f characterizing the shift in the (potentially unknown) propensity function π (see Eq. 2). Define the distribution-shift-calibrated (1 -α ) -quantile of the non-conformity scores as

for an imputed guess S of the ( n +1) -th non-conformity score S n +1 . The prediction interval

for the true S n +1 given a realization of Y n +1 = y satisfies the desired coverage guarantee under all distribution shifts f ∈ F , i.e.,

Building upon Lemma 4.1, we derive our first main result in Theorem 4.2. We define the finitedimensional function class of interest as F := { θ π ( a +∆ A | x ) π ( a | x ) | θ ∈ R + } . It is easy to verify that all f ∈ F represent the desired propensity shift to ˜ π ( a | x ) = π ( a +∆ A | x ) as defined in Eq. 2.

However, note that the optimization problem in Eq. 5 requires knowledge about the true or optimal imputed scores S n +1 . Directly solving the problem in this form would require running it for all possible imputed values S ∈ R , i.e., an infinite amount of times. As a remedy, we exploit the strong duality property and present our results in terms of a dual problem formulation.

Theorem 4.2 (Conformal prediction intervals for known baseline policy) . Consider a new datapoint with X n +1 = x n +1 , A n +1 = a n +1 , and A ∗ ( X n +1 ) = a ∗ = a n +1 + ∆ A . Let η S = { η S m +1 , . . . , η S n +1 } ∈ R n -m be the optimal solution to

(8) for an imputed unknown S n +1 = S . Furthermore, let S ∗ be defined as the maximum S s.t. η S n +1 < 1 -α . Then, the prediction interval

satisfies the desired coverage guarantee

Proof. We provide a full proof in Supplement D.2. Here, we briefly outline the underlying idea of the proof. First, we show that the function class F indeed satisfies Eq. equation 2 for the intervention A ∗ ( X ) = A +∆ A , and we then rewrite Eq. equation 5 as a convex optimization problem. Next, we exploit the strong duality property. We optimize over the corresponding dual problem to receive a dual prediction set with equal coverage probability. Finally, we derive S ∗ from the dual prediction set to construct C n +1 and prove the overall coverage guarantee.

4.2 Scenario 2: Unknown treatment policy

If the underlying treatment policy is unknown, the only possible intervention is a hard intervention a ∗ . As described above, measuring the induced propensity shift is non-trivial due to two reasons: (i) The propensity model needs to be estimated, which introduces additional uncertainty affecting the validity of the intervals ( → Challenge b ). (ii) The density function corresponding to a hard intervention is given by the Dirac delta function

̸

which hinders a direct adaptation of Theorem 4.2 due to the inherent discontinuity of the improper function. Hence, we make the following assumption on the propensity estimator.

Assumption 1. The estimation error of the propensity function ˆ π ( a | x ) is bounded in the sense that, for all i = 1 , . . . , n +1 , there exists M > 0 such that

Under Assumption 1, the distribution shift induced by the intervention is then defined as

for a suitable function f . We further formulate δ a ∗ ( a ) in terms of a Gaussian function as

This motivates the following lemma. Therein, we specify the class F of tilting functions f that represent the distribution shift induced by the hard intervention a ∗ .

Lemma 4.3. For σ > 0 , we define

with E P [ f ( A,X )] = 1 . Furthermore, we define the finite-dimensional function class F

Then, f ( a, x ) ∈ F for all c a ∈ [ 1 M , M ] and σ → 0 . As a result, the distribution shift

can be represented in terms of Eq. equation 2 through functions f ∈ F .

Following the motivation in scenario 1 , we thus aim to estimate the (1 -α ) -quantile of the nonconformity scores under the distribution shift in Lemma 4.1. We can reformulate this problem as

for the imputed score S n +1 = S . As the score is unknown, computing the CP interval would require solving equation 18 for all S ∈ R , yet which is computationally infeasible. Before, we exploited properties of the dual optimization problem and the Lagrange multipliers of the convex problem in Theorem 4.2 to efficiently compute the CP intervals. However, the present non-convex problem does not automatically allow for the same simplifications. Instead, we now present a remedy for efficient computation of the CP intervals in the following lemma. We prove Lemma 4.4 in Supplement D.

Lemma 4.4. The problem equation 18 is Type-I invex and satisfies the linear independence constraint qualification (LICQ).

Lemma 4.4 allows us to derive properties of the present non-convex optimization problem in terms of the Karush-Kuhn-Tucker (KKT) conditions. For this, we note that the fulfillment of the LICQ serves as a sufficient regularity condition for the KKT to hold at any (local) optimum of equation 18. Combined with the Type-I invexity of the objective function and the constraints, the KKT conditions are not only necessary but also sufficient for a global optimum. As a result, we can employ the KKT conditions at the optimal values 6 σ ∗ and c ∗ a to derive coverage guarantees of our CP interval in a similar fashion as in Theorem 4.2. We thus arrive at the following theorem to provide CP intervals for the scenario with unknown propensity scores.

for an imputed unknown S n +1 = S . Let S ∗ be defined as the maximum S s.t. v S n +1 > 0 . Then,

satisfies the desired coverage guarantee

Proof. See Supplement D.3.

In certain applications, it might be beneficial to fix σ to a small value σ 0 to approximate δ a ∗ ( a ) though a soft intervention and only construct the CP interval through optimizing over c a . We present an alternative theorem for this case in Supplement A.

We now use Thm. 4.5 to present an algorithm for computing CP intervals of potential outcomes from continuous treatment variables under unknown propensities in Alg. 1. We present a similar algorithm for scenario 1 with known propensities and discuss the computational complexity in Supplement B.

5 Experiments

Baselines: As we have discussed above, there are no baselines that directly compute prediction intervals with finite-sample guarantees for potential outcomes of continuous treatments. Therefore, we compare our method against MC dropout [14] and deep ensemble methods [33]. Yet, we again emphasize that MC dropout is an ad hoc method with poor approximations of the posterior, which is known to give unfaithful intervals [34]. Furthermore, we report the empirical coverage achieved by intervals from a Gaussian process regression (GP). By doing so, we consider a method that assesses the underlying aleatoric uncertainty. Additionally, we compare our method against the naive vanilla CP and the method by [36] for binary treatments in Supplements C and G.

Implementation: All methods are implemented with ϕ as a multi-layer perceptron (MLP) and an MCdropout regularization of rate 0.1. Crucially, we use the identical MLP for both our CP method and MC dropout. Hence, all performance gains must be attributed to the coverage guarantees of our conformal method. In the MC dropout baseline, the uncertainty intervals are computed via Monte Carlo sampling. In scenario 2 , we perform conditional density estimation by conditional normalizing flows [49]. Implementation and training details are in Supplement G.

Performance metrics: We evaluate the methods in terms of whether the prediction intervals are faithful [e.g., 21]. That is, we compute whether the empirical coverage of the prediction intervals surpasses the threshold of 1 -α for different significance levels α ∈ { 0 . 05 , 0 . 1 , 0 . 2 } . Additionally, we report the width of the resulting intervals in Supplement H.

5.1 Datasets

Synthetic datasets: We follow common practice and evaluate our methods using synthetic datasets [e.g., 2, 25]. Due to the fundamental problem of causal inference, counterfactual outcomes are never observable in real-world datasets. Synthetic datasets enable us to access counterfactual outcomes and thus to benchmark methods in terms of whether the computed intervals are faithful. Additionally, we perform experiments on the semi-synthic TCGA dataset in Supplement C. We hereby show the applicability of our method to high-dimensional real-world data in a controlled environment.

Medical dataset: We demonstrate the applicability of our CP method to medical datasets on the MIMIC-III dataset [26]. MIMIC-III contains de-identified health records from patients admitted to critical care units at large hospitals. Our goal is to predict patient outcomes in terms of blood pressure when treated with a different duration of mechanical ventilation. We use 8 confounders from medical practice (e.g., respiratory rate, hematocrit). Overall, we consider 14,719 patients, split into train (60%), validation (10%), calibration (20%), and test (10%) sets. Details are in Supplement G.

5.2 Results for synthetic datasets

We consider two synthetic datasets with different propensity scores and outcome functions. Dataset 1 uses a step-wise propensity function and a concave outcome function. Dataset 2 is more complex and uses a Gaussian propensity function and oscillating outcome functions. Both datasets contain a single discrete confounder, a continuous treatment, and a continuous outcome.

By choosing low-dimensional datasets, we later render it possible to plot the treatment-response curves so that one can inspect the prediction intervals visually. (We later also show that our method scales to highdimensional settings as part of the real-world dataset.) Details about the data generation are in Supplement G.





































a











































a







We evaluate the faithfulness of our CP intervals. On each dataset, we analyze the performance of the intervals in the presence of various soft interventions ∆ a ∈ { 1 , 5 , 10 } and hard interventions a ∗ ∈ { 7 x, 10 x } for each X = x . We average the empirical coverage across 50 runs with random seeds. The results are in Fig. 4 and Fig. 5. Additionally, we report the empirical coverage of the GP.

We make the following observations. First, our CP intervals comply with the targeted significance level α and are therefore faithful. Second, both MC dropout and the deep ensemble method have a considerably lower coverage, implying that the intervals are not faithful. This is in line with the literature, where MC dropout is found to produce poor approximations of the posterior [34]. In particular, the ensemble method is highly unfaithful. Thus, we will not consider this baseline in all the following experi-

ments. Third, our method has only a small variability in terms of empirical coverage, whereas the empirical coverage of MC dropout varies greatly. This corroborates the robustness of our method. Fourth, the results are consistent for both datasets. Fifth, the GP is only able to capture the true potential outcome in the prediction intervals for small distribution shifts ( ∆ = 1 )

on dataset 2. However, the empirical coverage is extremely low: α = 0 . 05 : 0.125; α = 0 . 1 : 0.125; α = 0 . 2 : 0.0833. This aligns with our expectations, as the aleatoric uncertainty in our experiments is low. Therefore, the GP intervals are small (average width of 0.1293) and barely valid after the intervention. In sum, this demonstrates our CP method's effectiveness.

Table 1 presents the empirical coverage of the intervals from our CP method vs. MC dropout across different α and hard interventions a ∗ . We observe that our CP intervals are effective and achieve the intended coverage. In contrast, MC dropout does not provide faithful intervals. Our findings are again in line with the literature, where MC dropout is found to produce poor













































































approximations of the posterior and thus might provide poor coverage [34]. We present further results in Supplement H.

Insights: We plot the intervals across different levels α and covariates X (Fig. 6), allowing us to inspect the intervals visually. We observe that the intervals behave as expected: they become sharper with increasing significance level α . We further see that our CP intervals are slightly wider (see details in Supplement H), yet this is intended as it ensures that the intervals are faithful. Our CP intervals (blue) generally include the true outcome. In contrast, the intervals from MC dropout (orange) often do not include the true outcome (e.g., see the bottom row Fig. 6) and are thus not faithful.

5.3 Results for the MIMIC dataset

Recall that numerically evaluating causal inference methods on real-world data is not possible due to the fundamental problem of causal inference. Therefore, we provide insights on the MIMIC dataset in Fig. 7. We compare the CP intervals of two male and two female patients of differing ages when treated with increasing duration of mechanical ventilation. Our intervals show higher uncertainty in treatment regions rarely included in the training data (high medium to high treatments). The intervals given by MC-dropout are narrower, which suggests lower coverage, confirming the effectiveness of our proposed method. This finding aligns with our observation from the synthetic datasets.

6 Discussion

Limitations: As with any other method, our UQ method has limitations that offer opportunities for future research. Our method relies on the quality of the propensity estimator. Although we incorporate estimation errors when constructing intervals, poorly estimated propensities could potentially lead to wide prediction intervals. We acknowledge that our intervals are conservative for point interventions and for segments of the output space with limited calibration data, implying that a representative calibration dataset is essential for the performance of our method. As for all CP methods, the use of sample splitting may reduce data efficiency. Furthermore, we note that the optimization procedure can be computationally expensive for large CATE vectors.

Broader impact: Our method makes a significant step toward UQ for potential outcomes and, thus, toward reliable decision-making. We provided strong theoretical and empirical evidence that our prediction intervals are valid. To this end, our method fills an important demand for using causal ML in medical practice and other safety-critical applications with limited data.

Considerations for practical application: First, our method is designed for single continuous treatments with a univariate outcome, which is common in dosing (e.g., determining the dosage of chemotherapy/insulin/hypertension drugs). In randomized controlled trials (when the propensity score is known), we can directly apply Theorem 4.2 on top of the trained outcome prediction model ϕ to construct our CP intervals at target coverage level α . In observational studies, when the propensity score is unknown, we require the practitioner to have sufficient prior knowledge to set a bound M on the propensity estimation error. The practitioner should choose this bound with care and rather in a conservative way to prevent undercoverage. Then, the CP intervals can be derived based on Theorem 4.5. We emphasize again that our intervals do not suffer from undercoverage due to limited data, as CP guarantees are valid for any sample size and our method is kept fully model agnostic , enabling the practitioner to make reliable judgements based on limited data and an ML model of choice.

Conclusion: We presented a novel conformal prediction method for potential outcomes of continuous treatments with finite-sample guarantees. Our method extends naturally to treatment effects. A key strength of our method is that the intervals are valid under distribution shifts introduced by the treatment assignment, even if the propensity score is unknown and has to be estimated.





















































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A Additional theoretical results

A.1 Calculating prediction intervals for further causal quantities and differences

We presented a method for calculating conformal prediction intervals for potential outcomes of continuous treatments. In the following, we show how the intervals can be combined to yield valid prediction intervals for further causal quantities, such as the individual treatment effect (ITE) γ i of treatment a :

Here, we consider the setting in which the non-conformity score is chosen to be the absolute residual.

Lemma A.1. Let S ∗ a and S ∗ 0 denote the optimal imputed non-conformity scores S n +1 for treatment a and no treatment at significance level 1 -α 2 for α ∈ (0 , 1) , respectively. Furthermore, let

Then the interval C γ ( X i , a ) := [ C -, C + ] contains the ITE γ i with probability 1 -α .

Proof. Let ε i ( a ) be the estimation error of the potential outcome, i.e.

We can rewrite the coverage guarantee of the original conformal prediction intervals for the potential outcome Y ( a ) as

Now observe that

Thus, it follows directly that

A.2 Alternative scenario 2: Fixing an approximation of δ a ∗ ( a )

In Section 4.2, we formulated the unknown propensity shift in terms of

and minimized over σ and c a in Theorem 4.5 to construct the CP intervals. However, in certain applications, it might be beneficial to control the spread of the approximation of δ a ∗ ( a ) through fixing

σ to a small value σ 0 and performing the soft intervention ˜ π ( a | x ) = c a √ 2 πσ 0 exp ( -( a -a ∗ ) 2 σ 2 0 ) ˆ π ( a | x ) . In this case, the resulting optimization problem is a convex problem similar to Theorem 4.2. We present the alternative optimization problem below.

Theorem A.2 (Alternative for Theorem 4.5: Conformal prediction intervals for unknown propensity scores) . Let a new datapoint be given with X n +1 = x n +1 and A n +1 = a n +1 . Let η S = { η S 1 , . . . , η S n +1 } ∈ R n +1 be the optimal solution to

for an imputed unknown S n +1 = S . Furthermore, let S ∗ be defined as the maximum S s.t. η S n +1 < 1 -α . Then, the prediction interval

satisfies the desired coverage guarantee

where with a slight abuse of notation Y ( a ∗ ) denotes the potential outcome under the soft intervention ˜ π above.

Proof. The statement follows from Theorem 4.5.

A.3 Soft-interventions on estimated propensities

In the main paper, we presented algorithms for constructing prediction intervals for soft interventions if the propensity function is known and hard interventions if it is unknown. These are arguably the most common scenarios in practice. However, in some cases, one might also be interested in the effect of soft interventions on estimated propensity scores [e.g., ? ]. Therefore, we present an alternative theorem for calculating conformal prediction intervals under soft interventions with estimated propensity scores below.

Theorem A.3 (Conformal prediction intervals for soft interventions with unknown propensity scores) . Let a new datapoint be given with X n +1 = x n +1 and A n +1 = a n +1 . Furthermore, let ˆ π denote the estimated propensity score with estimation error bounded by [ 1 M , M ] , M > 0 . The soft intervention is represented by the shift given through ∆ ∈ R . Let η S = { η S 1 , . . . , η S n +1 } ∈ R n +1 be the optimal solution to

(37) for an imputed unknown S n +1 = S . Furthermore, let S ∗ be defined as the maximum S s.t. η S n +1 < 1 -α . Then, the prediction interval

satisfies the desired coverage guarantee

where with a slight abuse of notation Y ( a ∗ ) denotes the potential outcome under the soft intervention represented by ∆ .

Proof. The statement follows from Theorem 4.2 and Theorem 4.5.

B Algorithm

We now use Thm. 4.5 to present an algorithm for computing CP intervals of potential outcomes from continuous treatment variables under unknown propensities in Alg. 1. We present a similar algorithm for scenario 1 with known propensities and discuss the computational complexity in below.

We make the following comments: In our algorithm, an optimization solver is used to calculate v n +1 according to Theorem 4.5. The specific choice of the solver is left to the user. In our experiments in Section 5, we perform the optimization via interior point methods. Further, the overall goal of our algorithm is to find the optimal imputed non-conformity score S ∗ such that the coverage guarantees hold. It can be implemented through suitable iterative search algorithms.

Algorithm explanation: Theorem 4.5 requires knowledge of the optimal imputed non-conformity score S ∗ . Since it is unknown beforehand, we implement Algorithm 1 as a binary search algorithm. We find suitable upper and lower bounds for S ∗ (until line 8). To validate that the bounds are indeed smaller/larger than the optimal non-conformity score, we observe the corresponding dual value ν n +1 (Theorem 4.5). After finding valid bounds, we start searching for S ∗ via standard binary search, continuously increasing the lower and decreasing the upper bound. When the difference between the bounds is less than a specified error tolerance ε , we stop and take S ∗ to be the mean of the interval. In every iteration (when calling 'solver'), we make use of the optimization in Theorem 4.5 and check whether the dual value fulfills the requirement in the Theorem.

Below, we state a second algorithm that is applicable if the propensity score is known. In this case, a convex optimization solver can be used.

Computational complexity: The complexity of running our algorithms depends heavily on the employed optimization solver with complexity σ s ( n c ) (e.g., polynomial complexity for suitable 7 convex solvers) and the size of the calibration dataset n c . This might become costly for large-scale calibration datasets in practice. The outer algorithm has a time complexity of at most O (log( S up -S low ε ) + 1) .

Overall, our algorithm has a fixed complexity of O (log( S up -S low ε ) σ s ( n c )) . The complexity of deriving intervals through MC-dropout or ensemble methods depends, however, on the number of MC samples or models, respectively. The latter thus scales with the precision of the intervals, which might be difficult to control. Furthermore, we emphasize that MC intervals are generally not faithful and therefore not directly comparable. While the theoretical runtime of CP may exhibit more complex scaling behavior (e.g., cubic or non-linear), our empirical results demonstrate that CP scales well in practice. In our work, we use optimization as a tool to provide conformal prediction intervals. Future research should focus on developing more efficient optimization algorithms for this task.

Note on the stability of the optimization algorithm: In Scenario 1, the only source of potential instability can be a very low propensity score in low-overlap regions of the covariate space. This, however, is only a problem if π ( a | x ) << π ( a +∆ | x ) . We can thus consider this unlikely in practice. For example, consider a patient who would be treated with a dosage of 10mg of some medication, which is prescribed in the range from 0 to 50mg. A practitioner is likely to be interested in the effect of an increase of the dosage to 15mg (i.e., ∆ = 5 ) in contrast to an increase to 50mg. Furthermore, it is reasonable to assume the propensity function to be locally smooth. Therefore, the instability of π ( a | x ) << π ( a +∆ | x ) is unlikely to occur.

In Scenario 2, we could additionally face an underflow of exp( -( a i -a ) 2 2 σ 2 ) or a blow-up of the prefactor ( σ ˆ π ( x i )) -1 . As a remedy, we can reparameterize the problem to work in the log domain and only exponentiate after subtracting a stable (maximum) constant across all datapoints. This additionally prevents the Jacobian/Hessian of the constraints from becoming nearly singular or wildly varying, causing Newton-type steps to blow up or stall.

C Semi-synthetic experiments

To underline the effectiveness of our method, we perform additional experiments on the semi-synthetic TCGA dataset. The Cancer Genome Atlas (TCGA) dataset [54] consists of a comprehensive and diverse collection of gene expression data. The data was collected from patients with different cancer types. In our experiment, we consider the gene expression measurements of the 4,000 genes with the highest variability, which we employ as our features X . The study cohort consisted of a total of 9659 patients. We model a continuous treatment based on the sum of the 10 covariates with the highest variance and assign a treatment effect that is constant in the sum of the covariates. Specifically, we model the treatment to follow a normal distribution centered at 100*sum of the 10 covariate values, and the outcome to follow a normal distribution centered at the sum of the treatment and the covariate sum times 100.

As in the main paper, we construct CP intervals for different interventions and confidence levels α . We state the empirical coverage of our method in Table 2 as well as the coverage of the intervals returned by the ensemble method and MC dropout below. The prediction performance of the trained model on the hold-out test dataset is reported. We find that our method is highly effective.

We observe that our method consistently achieves the desired coverage. To evaluate the usefulness of our intervals, we also report the interval width in Table 3 below. The range of the outcomes was 2.0.

Comparison to the CP methods for binary treatments [36]:

To investigate whether simple weighted CP methods for causal tasks on binary treatments sufficiently address the non-exchangeability due to the distribution shift induced by the intervention, we compare our method with the method by Lei and Cand` es [36]. Of note, both methods fulfill the coverage guarantees. However, when comparing the interval width, we see that our method is superior: Our intervals have an average width of 0.6255 (sd = 0.1714). In contrast, the intervals on the binarized treatment obtained through the method by Lei and Cand` es [36] have an average width of 3.2876 (sd = 0.5587). Hence, the intervals by our method are by far more informative.

D Proofs

D.1 Proofs of the supporting lemmas

In the following, we prove Lemma 4.3 and Lemma 4.4 from our main paper.

Proof of Lemma 4.3 Recall the definition of the hard intervention

Under Assumption 1, we have for some M > 0 and all a, x . Then

Therefore, the distribution shift induced by the hard intervention can be represented as

Proof of Lemma 4.4 We first prove that the problem equation 18 fulfills the linear independence constraint qualifications. For all i = m + 1 , . . . , n + 1 , we denote the constraints of problem equation 18 as

The gradient of h i is given by

Therefore, with ∇ h := ( ∇ h m +1 , . . . , ∇ h n +1 ) and λ ∈ R n +1 , we obtain

where

As a result, the constraints are linearly independent. This property suffices for the KKT conditions to hold at any (local) optimum of equation 18.To furthermore show that the KKT conditions are also sufficient for a global optimum, we show that equation 18 is Type-I invex. An optimization problem with objective function f ( x ) and constraints g ( x ) < = 0 with x ∈ R n +1 is Type-I invex at x 0 , if there exists ν ( x, x 0 ) ∈ R n +1 , such that

[20]. In problem equation 18, the gradients of the objective function and of each constraint h i for all i, j = m +1 , . . . , n +1 at x 0 are given by

∀ i = m +1 , . . . , n +1 and

and

For

the definition of Type-I invexity holds for equation 18. Thus, the KKT conditions are also sufficient for a global optimum.

D.2 Proof of Theorem 4.2

We prove Theorem 4.2 in three steps: (i) We show that function class F := { θ π ( a +∆ A | x ) π ( a | x ) | θ ∈ R + } indeed satisfies Eq. equation 2 for the intervention a ∗ = a + ∆ A and rewrite Eq. equation 5 as a convex optimization problem. (ii) We retrieve the corresponding dual problem, derive a dual prediction set, and show the equality of the coverage guarantee of the dual and the primal prediction sets. (iii) We derive S ∗ from the dual prediction set to construct C n +1 and prove the overall coverage guarantee. For further theoretical background on the idea of the proof, we refer to Gibbs et al. [18].

Justification of the distribution shift Observe that E [ f ( A,X )] = θ for all f ∈ F := { θ π ( a +∆ A | x ) π ( a | x ) | θ ∈ R + } . Therefore, Eq. equation 2 simplifies to

Thus, F satisfies the propensity shift from Eq. equation 2 for the soft intervention a ∗ = a +∆ A . Following Lemma 4.1, we thus aim to find

Dual problem formulation First, we rewrite the primal problem as

For a reference, see [18]. The Lagrangian of the primal problem states

Setting derivative of L w.r.t. u i and v i to 0 results in

Since γ 1 i , γ 2 i ≥ 0 ∀ i , it follows for all i = m +1 , . . . , n +1 that

Therefore, the dual problem is formulated as

Coverage guarantee Recall from Lemma 4.1 that, for

we can construct C n +1 = { y | S n +1 ( y ) ≤ ˆ q S n +1 ( y ) } to achieve the desired coverage guarantee

It is infeasible to calculate ˆ q S n +1 ( y ) directly. Therefore, we optimize the dual problem to receive

with S ∗ the maximum S , s.t. for η S n +1 maximizing the dual problem, η S n +1 < 1 -α . Hence, it is left to show that replacing ˆ q S n +1 ( y ) by S ∗ in C does not change to coverage guarantee.

To do so, we fix some θ > 0 to obtain a specific f ( a, x ) := θ π ( a ∗ | x ) π ( a | x ) . Let ˆ g ( a, x ) ∈ F denote the primal optimal solution. Recall the Lagrangian

Deriving wrt. f yields the stationarity condition

The complementary slackness Karush-Kuhn-Tucker conditions yield

Therefore, we can rewrite the equation from above as

Before deriving the coverage guarantee from the stationarity condition, we state the following lemma to underline the definition of S ∗ .

Lemma D.1 (Gibbs et al. [18]) . The mapping S ↦→ η S n +1 is non-decreasing in S for all η S n +1 maximizing

for non-negative function classes F .

To prove the final coverage guarantee, we observe that

✶ With the definition of S ∗ as the maximum optimizer η S n +1 with η S n +1 < 1 -α and Lemma D.1, it follows that and, by exchangeability of ( f ( a i , x i ) , ˆ q S ( a i .x i ) , S i ) , that

Since f is positive and η i ∈ [ -α, 1 -α ] , it follows and thus

D.3 Proof of Theorem 4.5

We follow the same outline as in the proof of Theorem 4.2 in Section D.2. In Lemma 4.3, we motivated the functional class of distribution shifts. Therefore, it is left to prove the coverage guarantee of C n +1 .

Key to our proof is the following lemma.

Lemma D.2. The mapping S ↦→ v S n +1 is non-increasing in S for all g S ( x, a ) minimizing

for non-negative function classes F and imputed S n +1 = S stemming from a non-negative nonconformity score function (e.g., the residual of the prediction).

Proof. Assume for contradiction that there exists ˜ S > S such that v ˜ S n +1 > v S n +1 . Then

We observe that

Reformulating the equation above yields

This is equivalent to

which contradicts the assumption that v ˜ S n +1 > v S n +1

.

Coverage guarantees. As in D.2, we fix some σ > 0 and c a ∈ [ M , M ] to obtain a specific f ( a, x ) := c a √ 2 πσ exp ( -( a i -a ∗ ) 2 2 σ 2 ) ˆ π ( a i | x i ) . We further denote ˆ g ( a, x ) ∈ F the optimal solution given by the optimal values ˆ σ and ˆ c a .

1

With the definition of S ∗ as the minimum S such that v S ∗ n +1 = 0 and Lemma D.2, we now can state

and, by exchangeability of ( f ( a i , x i ) , ˆ q S ( a i .x i ) , S i ) , that

Deriving the Lagrangian above wrt. f yields the stationarity condition

The complementary slackness Karush-Kuhn-Tucker conditions yield

Therefore, we receive

Since f is positive and η i ∈ [ -α, 1 -α ] , it follows and thus

E Additional background

E.1 Extended literature review

Uncertainty quantification for causal quantities

There exist various methods for uncertainty quantification of causal quantities. These are often based on Bayesian methods [e.g., 1, 21, 22, 24]. However, Bayesian methods require the specification of a prior distribution based on domain knowledge and are thus neither robust to model misspecification nor generalizable to model-agnostic machine learning models. Other methods only provide asymptotic guarantees [e.g., 25, 27]. The strength of conformal prediction, however, is to provide finite-sample uncertainty guarantees.

In the following, we present related work on CP for causal quantities in more detail.

Recently, Alaa et al. [2] provided predictive intervals for CATE meta-learners under the assumption of full knowledge of the propensity score. As an extension, Jonkers et al. [27] proposed a MonteCarlo sampling approach to receive less conservative intervals. Chen et al. [9] provide prediction intervals for counterfactual outcomes. However, the proposed method requires access to additional interventional data and is thus not applicable to real-world applications on observational data. All methods are restricted to binary treatments.

Other works focus on prediction intervals for off-policy prediction [47, 58] and conformal sensitivity analysis [56], thus neglecting estimation errors arising from propensity or weight estimation or for randomized control trials [30]. Wang et al. [52] constructed intervals with treatment-conditional coverage of discrete treatments. Aiming for group-conditional coverage, Wang et al. [52] adapted CP to cluster randomized trials. Nevertheless, the method only applies to a finite number of treatments and thus is not applicable to continuous treatments.

Lei and Cand` es [36] consider the estimated propensity by incorporating the estimation error as a TV-distance term in the coverage guarantees. However, for large TV-distances (close to 1), the proposed method can only construct intervals with a very limited coverage α ∈ (0 , 1 -TV ) . Hence, the method is not suitable for applications in medical practice. Our method, however, can also construct intervals with high coverage guarantees for high estimation errors. An increased error will widen the prediction intervals instead of reducing the coverage guarantee. We consider our approach more suitable for medical practice, as one can visually inspect the intervals and decide on the suitability of the task at hand.

Overall, no method can provide exact intervals for continuous treatments. Especially, no method considers the error arising from propensity estimation in the analysis.

Conformal prediction under covariate shift Multiple works on CP with marginal coverage under distribution shifts between training and test data have been introduced in the literature [e.g., 8, 11, 15, 16, 17, 18, 19, 36, 41, 48, 55]. Our setting also involves a distribution shift due to the intervention on the treatment but differs from the latter in that the true distribution shift is unknown.

Gibbs et al. [18] introduced an approach to derive CP intervals under unknown distribution shifts. It proves valid finite-sample prediction intervals for all distribution shifts in a finite-dimensional function class. However, the approach does not directly apply to causal inference settings. Nevertheless, our framework builds upon the work by Gibbs et al. [18] in that we re-frame the proposed approach to apply to the distribution shift induced through the intervention in causal effect estimation. In this setting, the distribution shift is captured by the shift of the propensity function. Adapting Gibbs et al. [18] to a causal inference setting requires carefully addressing the underlying challenges that come from computing CP intervals in a causal inference setting (e.g., propensity score estimation, hard/soft interventions), which we regard as our main novelty and which is of immediate practical relevance (e.g., in personalized medicine).

E.2 The need for exchangeability in CP

Coverage guarantees of existing CP intervals essentially rely on the exchangeability of the nonconformity scores. Exchangeability assures that the nonconformity score of the test point n +1 is equally likely to fall anywhere among the calibration scores, its rank is uniform, and that uniformity

is exactly what yields the distribution-free coverage guarantee. Without exchangeability, the rank is not guaranteed to be uniform, and the marginal coverage bound can fail.

However, intervening on treatment A shifts the propensity function and, therefore, induces a shift in the covariates between calibration and test data, specifically in treatment A . Therefore, exchangeability is not fulfilled, and the coverage guarantees might fail.

As a remedy, we present a novel and powerful remedy in our work: The overall distribution of the confounders X is assumed to stay constant between train, calibration, and test data (as standard in ML problems). This is completely orthogonal to constructing intervals for different (e.g., young or old) patients. Note that CP intervals are constructed for only one sample/patient at a time. This means that different intervals are constructed for patients with different features X . In other words, the intervals are constructed conditionally on X , but the coverage guarantee is marginal across the complete population of X . Overall, the shift from one patient to another does not pose any challenges for CP methods.

F Extended discussion

F.1 Discussion on the tightness of our CP intervals:

Our method builds upon the idea of CP to provide finite-sample coverage guarantees. Notably, standard CP does not provide intervals that are proven to be sharp. To our knowledge, there is no method that provides sharp/the tightest possible intervals for potential outcomes. Exploring the tightness of our CP intervals is an interesting and important direction for future research.

In practice, it is possible to observe the width, and thus the informativeness for decision-making, of the intervals. However, coverage guarantees cannot be observed. Therefore, we help the decision-maker by providing valid intervals. The decision-maker has to decide, case by case, if the returned intervals are beneficial for the problem at hand. Note that this aspect of the informativeness of intervals holds true for any uncertainty quantification method (including those to be proven to be sharp).

F.2 A note on challenges and difficulties in CP for causal effects of continuous treatments

Existing works on conformal prediction for binary or (low-dimensional) discrete treatment are commonly based on (a) weighted conformal prediction [48] or (b) conformal prediction local coverage guarantees [35]. The first approach provides marginal coverage under a distribution shift through reweighting. It requires computing the weights based on the probability of treatment A = a . However, for continuous treatments, this is always zero. Although applicable to binary or low-dimensional discrete treatments [e.g., 36], this weighting approach cannot be extended similarly to continuous treatments. Furthermore, the propensity of a continuous treatment given by the Dirac delta function δ a would require us to restrict the calibration to data samples of the specific treatment, which are extremely rare or even might be missing . Therefore, the calibration step cannot be employed in our setting. The second approach provides treatment group-conditional coverage. Although again possible for binary or low-dimensional treatments, this approach does not apply to continuous treatment s as no treatment groups can be defined. Instead, we propose a novel method for conformal predictions that circumvents the above problems and is carefully tailored to continuous treatments.

F.3 Causal effects of continuous treatments & kernel smoothing

Causal inference becomes challenging with continuous treatments primarily due to the infinite number of potential outcomes per sample, from which only one outcome is observed. Continuous treatments thus result in causal effects that are generally represented by curves (called dose-response curves) [29]. This is unlike binary treatment, where the causal effects are represented by a single discrete value.

For continuous treatments, the dose-response curves are typically assumed to fulfill some smoothness criterion [e.g., 40, 45]. Hence, when estimating treatment effects, interpolation and kernel smoothing of the outcome function are commonly employed [e.g., 29, 37].

Underlying causal estimation with continuous treatments is the generalized propensity score [23]. It is defined as the conditional probability of receiving treatment a ∗ given the covariates X under the following regularity conditions: (i) For each i , Y i ( a ) , x i , A i are defined on a common probability space; (ii) A i is continuously distributed with respect to the Lebesgue measure; and (iii) Y i = Y i ( A i ) is a well-defined random variable.

Approximating the density δ a ∗ ( a ) of the hard intervention a ∗ through a Gaussian kernel follows directly from the definition of δ a ∗ ( a ) as the limit of such kernel. This is also common in the literature [e.g., 28]. Importantly, we note that we do not directly approximate the potential outcome Y ( a ∗ ) (but only the propensity scores). Thus, we do not have a bias-variance trade-off of the estimated outcome. Due to the smoothness of the dose-response curve, it is now valid to employ observed samples within a treatment region of a ∗ defined by σ to construct the intervals. We note that the importance of the samples is weighted by the inverse distance of the sample to a ∗ in treatment space. We give further intuition on the relationship between σ , the importance of observational samples, and the prediction interval width in the following.

F.4 Interpretation of optimal parameters

To obtain CP intervals under an unknown distribution shift, we approximate the Dirac-delta distribution representing the hard intervention by a Gaussian function as

In Theorem 4.5, we thus optimize over σ > 0 and c a ∈ [ 1 M , M ] to obtain the (1 -α ) -quantile of the distribution shift-calibrated non-conformity scores. The optimal parameter σ ∗ represents a trade-off between the uncertainty in the prediction and the uncertainty in the interval construction: A small σ ∗ resembles the propensity of the hard intervention best. Thus, with sufficient or even infinite data close to a ∗ , we could construct the narrowest CP interval. However, the smaller σ , the less data close to a ∗ will be available to calculate the prediction interval in practice. As a result, many calibration data samples will be strongly perturbed during the calculation, which increases the uncertainty and, thus, the interval size.

The parameter c a allows us to incorporate the estimation error in the propensity score. It represents a weighting of the propensity shift such that the (1 -α ) -quantile of the non-conformity scores is increased with higher estimation error.

F.5 Interpretation of parameter M

Our optimization requires the specification of a parameter M , denoting a bound on the propensity estimation error. One can view the parameter M as a type of sensitivity parameter. Therefore, we follow former work in causal inference and propose to incorporate domain knowledge to specify the parameter M [e.g., 13 ? ]. Another way of making use of the parameter M is to observe how the intervals change for varying M . This indicates how much effect the propensity misspecification has on the prediction interval and can help in making reliable decisions. A third option to calibrate M is to employ measures for epistemic uncertainty on top of the propensity estimate when there is no domain knowledge for specifying M .

F.6 A note on the stability of our method

In our experiments, one can observe some instability for certain privacy budgets and intervention combinations. This is likely due to the fact that the CP coverage guarantees are only marginal . Therefore, we might experience under- or over-coverage. However, we note that across all runs, our method, on average, achieves the desired coverage. These instabilities only occur in single settings. Furthermore, the variance in coverage of our method is much lower than the coverage variance of the baselines. A more in-depth analysis of the stability of the proposed method is left for future work.

G Experimentation details

G.1 Synthetic dataset generation

We consider two different propensity and outcome functions. In each setting, we assign two types of interventions: a known propensity shift of ∆ = 1 , 5 , 10 , i.e., three soft interventions a ∗ = a +∆ , and the point interventions a ∗ ∈ { 1 x, 5 x, 10 , } given the confounder X = x .

Wegenerate synthetic datasets for each setting. Specifically, we draw each 2000 train, 1000 calibration, and each 1000 test samples per intervention from the following structural equations. Dataset 1 is given by

and dataset 2 by

G.2 Medical dataset

We use the MIMIC-III dataset [26], which includes electronic health records (EHRs) from patients admitted to intensive care units. From this dataset, we extract 8 confounders (heart rate, sodium, blood pressure, glucose, hematocrit, respiratory rate, age, gender) and a continuous treatment (mechanical ventilation) using an open-source preprocessing pipeline [53]. From each patient trajectory in the EHRs, we sample random time points and average the value of each variable over the ten hours before the sampled time point. We define the variable blood pressure after treatment as the outcome, for which we additionally apply a transformation to be more dependent on the treatment and less on the blood pressure before treatment. We remove all patients (samples) with missing values and outliers from the dataset. Outliers are defined as samples with values smaller than the 0.1th percentile or larger than the 99.9th percentile of the corresponding variable. The final dataset contains 14719 samples, which we split into train (60%), val (10%), calibration (20%), and test (10%) sets.

G.3 Implementation details

Our experiments are implemented in PyTorch Lightning. We provide our code in our GitHub repository. All experiments were run on an AMD Ryzen 7 PRO 6850U 2.70 GHz CPU with eight cores and 32GB RAM.

We limited the experiments to standard multi-layer perception (MLP) regression models, consisting of three layers of width 16 with ReLu activation function and MC dropout at a rate of 0.1, optimized via Adam. We did not perform hyperparameter optimization, as our method aimed to provide an agnostic prediction interval applicable to any prediction model. All models were trained for 300 epochs with batch size 32.

Our algorithm requires solving (non-convex) optimization problems through mathematical optimization. We chose to employ two interior-point solvers in our experiments: For the experiments with soft interventions that pose convex optimization problems, we use the solver MOSEK. For the hard interventions, which included non-convex problems, we used the solver IPOPT. Both solvers were run with default parameters.

G.4 Selection of the interventions in our experiments

The treatment (in the complete dataset) is modeled to lie in the range [0,40]. Therefore, the treatments/interventions can also only fall into this range. All samples that would have achieved a treatment outside this range through the interventions were neglected in our analysis. To guarantee

that a sufficient number of samples were included in our experiments, we chose the maximal soft treatment as an increase of 10.

We sampled the covariates X in the range from 1 to 4. To again perform interventions that fall inside the range of [0,40], we decided to set the intervention as 7 X and 10 X . Other choices of interventions would also have been possible. Reassuringly, there was no systematic selection of interventions in our experiments besides the considerations above.

For the soft interventions, we do not see different effects of the interventions on the two datasets. For the hard interventions, however, we can observe a difference. Recall that in dataset 1, the treatment was sampled uniformly from [0,40] (with a dependence on X ), whereas in dataset 2, it was sampled from a normal distribution with a mean of 5 X . Therefore, the intervention 10 X is far in the tail of the distribution. As a result, we observe a slightly lower coverage for this intervention on dataset 2 compared to dataset 1.

H Further results

We present further results from our experiments in Section 5. Specifically, we state the prediction performance of the underlying models ϕ , discuss the scalability of our approach, and show the prediction intervals per covariate for various significance levels α and soft interventions ∆ of our synthetic experiments on dataset 1 and dataset 2.

Performance: We first report the performance of the underlying prediction models ϕ for the synthetic datasets across 50 runs in Table 4. The prediction model on the real-world dataset achieved a mean squared error loss of 1.2373.

Width of the prediction intervals: We further report the width of the prediction intervals in our synthetic experiments in Table 5. The width is important to assess the usefulness of the resulting prediction intervals. As the performance of the ensemble method is not comparable with the coverage of MC-Dropout and our CP method, we only compare the latter two methods with regard to the interval width.

Comparison to the vanilla CP baseline: We compare our method to the naive vanilla CP (V-CP), i.e., a CP method that does not account for the distribution shift. We observe that V-CP does not achieve any valid prediction interval across all distribution shifts and confidence levels. This can be explained by the good prediction performance of the underlying model. Thus, V-CP intervals are extremely small (average width of 0.0003) and can never cover the true potential outcome after the intervention. Overall, the results confirm the importance of accounting for the distribution shift induced by the intervention.    

Scalability: Calculating the prediction intervals requires an iterative search for an optimal value S ∗ . Therefore, the underlying optimization problem must be fitted multiple times throughout the algorithm, potentially posing scalability problems. In our empirical studies, however, we did not encounter scalability issues. Importantly, we found that the average runtime of our algorithm on a standard desktop CPU is only 16.43 seconds. On the MIMIC dataset, computing CP intervals takes roughly 10 times longer than computing MC intervals. However, we emphasize that MC intervals are generally not faithful and therefore not directly comparable.

Prediction intervals: In Figures 8, 9, and 10, we present the prediction bands given by our





































































method and MC dropout on dataset 1. In particular, for confounder X = 1 , our method shows a large increase in the uncertainty in the potential outcomes of treatments affected by the intervention. MC dropout does not capture this uncertainty.

























































































































































In Figures 11 and 12, we present the prediction bands given by our method and MC dropout on dataset 2 for the soft interventions ∆ = 1 and ∆ = 10 (the results for ∆ = 5 were presented in the main paper). We observe that the prediction intervals for ∆ = 10 become extremely wide for high treatments. This aligns with our expectation, as data for high treatments in combination with low confounders is rare or even absent in the dataset. Thus, the expected uncertainty is very high.















































































































































Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a conformal prediction method for the estimation of potential outcomes under continuous treatments and known/unknown propensity scores. The authors provide finite-sample validity guarantees and validate their method empirically on synthetic and real-world data.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**: \n- *Novelty*: First to provide conformal prediction intervals for continuous treatments with unknown propensities.\n- *Structured*: The methodological contributions are well-structured (e.g., known propensity->soft intervention, unknown propensity-> hard intervention) and carefully motivated (e.g., challenges a,b,c discussions).\n\n**Weaknesses**:\n- *Weak Empirical Evidence*: \n    - Despite the consistent results on the Synthetic Dataset 1-2, such DGP with a single covariate is over-simplistic to relate with real-world applications. \n    - I really appreciate the attempt to complement such experiments with semi-real and real-world data. However, the experiments on TGCA dataset (semi-synthetic) are poorly documented (unclear outcome of interest, treatment assignment, evaluation procedure, ground truth) and lack comparison with other approaches. \n    - Similarly, the experiment on MIMIC dataset (real-data) looks incomplete: Figure 7 only shows the CP-estimated intervals (compared with MC dropout), which a reader can only evaluate by their qualitative width but not their coverage (how to establish that yours is out-performing MC-dropout?). Also unclear if the counterfactual estimation on this dataset isn't just trivial linear interpolation  (also what are the dotted-line in Figure 7?). Finally, I don't understand why showing the intervals per individual if the conditional (over the covariates) coverage is never discussed.\n- *Section 4*: Despite the great high-level structure, I struggled to both follow and interpret the two theoretical results (also not very familiar with the related works though). The math sounds correct (despite I may have missed some details), but I found difficult to interpret both variables (f, l, g, ...) and formulations. In practice, missing interpretations makes difficult to understand the limitations and practical validity of the method. Are the propensity score estimation and calibration data size challenges the only challenges of this approach (as reported in Section 6) to get exact finte sample guarantees?\n\nQUESTIONS:\nCan you elaborate on the numerical stability of your method, particularly in Scenario 2 where the optimization is non-convex and involves small propensity estimates and Gaussian tilting?\n\nLIMITATIONS:\nWeak (practically relevant) empirical validation and uncritical/difficult to interpret theoretical motivation. If I was a physician I wish the potential challenges potentially invalidating the method were better stressed through the paper: e.g., I have limited data, I have a prior on how to bound M, (when) can I use your method?\n\nSee weaknesses and questions.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-19"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a CP framework for constructing finite-sample valid prediction intervals for counterfactual outcomes, under both hard and soft interventions on a continuous treatment. It addresses a key limitation of standard conformal prediction—the violation of exchangeability between calibration and test points due to distribution shifts induced by interventions. To restore valid coverage, the authors introduce a tilting function based on the ratio of interventional to observational treatment densities (i.e., propensity scores). The method is empirically validated on both synthetic data and the MIMIC-III ICU dataset.\n\nSTRENGTHS AND WEAKNESSES:\n1. Strong motivation as the authors clearly identify three challenges and address them under different scenarios. They also provide strong theoretical results with finite-sample validity in sections 4.1 and 4.2.\n\n2. I quite like the idea of causal conformal prediction—the use of a tilting function based on the treatment distribution is clever. However, this also leads to my another concern: the method in this paper does not guarantee causal identifiability in the traditional sense. Instead,  it is more like assumes the predictive model (and, in some cases, the propensity model) is already \"good enough\" and focuses on valid uncertainty quantification under distribution shifts induced by hard or soft interventions. It lies at the intersection of predictive modeling, distribution shift correction / importance weighting, and finite-sample uncertainty quantification, and touches causal inference only to the extent that it predicts under interventions, but does not address identification or estimation of causal effects directly.\n\n3. The authors provide extensive experimental results to illustrate the superior performance of their method. However, the baseline methods are limited to MC dropout and deep ensembles. Later, they find that ensembles are highly unfaithful and remove them from comparison. This raises a concern about fairness: MC dropout and ensembles are not designed to handle counterfactual or distributional shift elements (they rely on the exchangeability assumption), so comparing them to the proposed method is like comparing a bicycle to a jet engine. This might explain why the figure 4 and 5 look too good to be true. It would be more convincing to compare the proposed method against baselines that incorporate causal elements, such as Bayesian causal forests or causal GPs, which are widely used and provide credible intervals for potential outcomes or treatment effects.\n\n4. As mentioned above, the proposed methods heavily rely on the correctness of the predictive model. It would be helpful to include some ablation analysis to show how performance depends on the accuracy of propensity score estimation, model misspecification, or unmeasured confounding.\n\nQUESTIONS:\n1. No comparisons with causal UQ baselines. Additionally, it would be helpful to include a related work section discussing recent progress on causal CP and causal UQ methods.\n2. The method assumes that outcome exchangeability is violated. Is this violation primarily due to treatment shift (e.g., a patient receiving a different treatment than observed), or due to covariate distribution shift (e.g. a young patient and an old patient)? How does the proposed approach handle these two types of shift, especially when they co-occur?\n3. This may be slightly tangential, but in line 276 you mention aleatoric uncertainty. Could you elaborate on how aleatoric and epistemic uncertainties are defined in this context, and whether (or how) the proposed method addresses each type?\n\nLIMITATIONS:\nSee above.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-24"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper develops a conformal prediction framework for estimating uncertainty intervals of potential outcomes under continuous treatment settings in causal inference, which inherently break exchangeability. In this paper, authors propose CP method to  remain valid in terms of exchangeable assumption under known or unknown propensity scores, when interventions shift the treatment assignment mechanism. Synthetic and real-world experiments confirm reliable coverage and adaptive interval widths under various intervention scenarios and significance level.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n(1) The paper addresses a clear gap in the conformal prediction literature: providing finite-sample valid prediction intervals for counterfactual outcomes under continuous treatments\n\n(2) The paper provides finite-sample marginal coverage guarantees for the constructed intervals in both known and unknown propensity settings, which is rare in causal CP work\n\n(3) The method is carefully grounded in the potential outcomes framework, and the authors precisely handle the violation of exchangeability under interventions.\n\nWeakness:\n(1) In the unknown propensity case, the method uses estimated scores but doesn't model or quantify uncertainty in their estimation. This may degrade validity if the estimation is poor.\n\n(2) The experiments do not include baseline methods for conformal prediction under causal settings (e.g., DR-CP, CPC for binary treatments). Though not directly comparable, some kind of ablation or prior method could serve as a sanity check.\n\n(3) Some strong assumptions about the variables and function form, such as $\\delta$ in terms of Gaussian function in Eq. (14).\n\nQUESTIONS:\n(1) The motivation of uncertainty quantification is not clear. As claim in the paper, the point estimate will neglect that certain treatment is ineffective for some patients profile. however, some causal estimands, e.g., CATE, can also fill this gap.\n\n(2) one of the challenges of conformal prediction for causal quantities lies on the non-exchangeability due to the intervention. But why the challenge is non-trivial? The manuscript is unclear for thin point. Why CP needs exchangeability assumption?\n\n(3) in eq. (3) and (4), what does the function $l_{\\alpha}$ represent? this part lacks important details about the quantile regression.\n\n(4) in line 208, the vector \\eta^S has n+1 elements? it is not correct.\n\n(5) it's very confusing that why one needs to estimate the weight $\\eta$ in eq.(8), instead of directly using $(1-\\alpha)$ and $\\alpha$? what's the difference between eq,(9) and (3), (4)?\n\n(6) and it also lacks of specific explanation why use max $\\eta$ in the max-min objective in eq.(8).\n\n(7) how does the Eq.(Ps) is reformulated? what does u_i and v_i represent?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper focuses on the task of conformal prediction of causal effects from continuous-valued treatments. The novelty in this paper stems from handling the cause of continuous-valued treatment, while previous literature focuses on binary or discrete-valued treatments.\n\nThe paper starts by tackling the causal effect estimation setting, making the standard assumptions of positivity, consistency, and unconfoundedness. They also assume access to a ML model to predict potential outcomes. The paper tackles producing CP intervals under either discretized hard or soft interventions, which cause the resulting distribution to violate standard exchangeability assumptions.\nTo address this shift in distribution, the authors perform calibration (conditioned on the propensity shift). They define this shift in Equation (2) and use split conformal prediction with a calibration step based on the shifting function $f$. \n\nThe authors empirically compare their approach with MC-dropout (with the same underlying MLP) and deep ensembles, given that there are no baselines that handle the setting of continuous treatments. They provide results on synthetic data, finding that the proposed approach has much higher coverage.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n- Tackles a new (and well-motivated) setting of CP for causal effect estimation in the setting of continuous-valued treatments\n- Technical novelty in deriving a new theoretically sound approach (i.e., with proper coverage guarantees), adapting the approach from split CP\n- Develops technical methodology to handle both cases where propensity scores are known and unknown\n\nWeaknesses\n- No comparison to a naive CP implementation as a baseline – this would be another valid and interesting baseline to understand the importance of the proposed approach\n- The method seems computationally intensive; it requires calls to the optimization solver during each iteration of a binary search. If I’m not mistaken, shouldn’t the algorithm’s fixed complexity be $O(\\log(\\frac{S_{up} - S_{low}}{\\epsilon} * \\sigma_s(n_c)) (lines 601-602)? \n- Minor formatting issues (e.g., the wrapped figures on Page 8)\n\nQUESTIONS:\nEmpirically, how does the runtime compare to the alternatives on the MIMIC dataset?\n\nAlso, see other weaknesses above.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}