SlowGuess/NeurIPS_2023_OCR huggingface dataset:
UC Berkeley and UCSF amalaa@berkeley.edu
Zaid Ahmad
UC Berkeley  
zaidahmad@berkeley.edu
Mark van der Laan
UC Berkeley
laan@stat.berkeley.edu
We investigate the problem of machine learning-based (ML) predictive inference on individual treatment effects (ITEs). Previous work has focused primarily on developing ML-based "meta-learners" that can provide point estimates of the conditional average treatment effect (CATE)—these are model-agnostic approaches for combining intermediate nuisance estimates to produce estimates of CATE. In this paper, we develop conformal meta-learners, a general framework for issuing predictive intervals for ITEs by applying the standard conformal prediction (CP) procedure on top of CATE meta-learners. We focus on a broad class of meta-learners based on two-stage pseudo-outcome regression and develop a stochastic ordering framework to study their validity. We show that inference with conformal meta-learners is marginally valid if their (pseudo-outcome) conformity scores stochastically dominate "oracle" conformity scores evaluated on the unobserved ITEs. Additionally, we prove that commonly used CATE meta-learners, such as the doubly-robust learner, satisfy a model- and distribution-free stochastic (or convex) dominance condition, making their conformal inferences valid for practically-relevant levels of target coverage. Whereas existing procedures conduct inference on nuisance parameters (i.e., potential outcomes) via weighted CP [1], conformal meta-learners enable direct inference on the target parameter (ITE). Numerical experiments show that conformal meta-learners provide valid intervals with competitive efficiency while retaining the favorable point estimation properties of CATE meta-learners.
Code: https://github.com/AlaaLab/conformal-metalearners
Identifying heterogeneity in the effects of interventions across individuals is a central problem in various fields, including medical, political, and social sciences [2, 3, 4]. In recent years, there has been growing interest in developing machine learning (ML) models to estimate heterogeneous treatment effects using observational or experimental data [5, 6, 7, 8, 9]. However, most of these models only provide point estimates of the conditional average treatment effect (CATE), which is a deterministic function that describes the expected treatment effect based on a given individual's covariates. In this paper, we focus on quantifying uncertainty in these estimates, which arises from both errors in the model and the variation of individual treatment effects (ITEs) for individuals with the same covariates. We adopt a predictive inference approach to this problem, with the goal of devising valid procedures to issue predictive intervals that cover ITEs on unseen data with a predetermined probability.
Traditionally, predictive inference on ITEs has been conducted through Bayesian methods such as BART [8] and Gaussian processes [9]. These methods can provide interval-valued predictions of ITEs through their induced posterior distributions (e.g., posterior credible intervals). However, Bayesian methods tend to be model-specific and cannot be straightforwardly generalized to modern ML models, e.g., transformer-based architectures used to model visual and textual covariate spaces [10]. More importantly, Bayesian methods generally do not provide guarantees on the frequentist coverage of
their credible intervals—achieved (finite-sample) coverage depends on the prior [11]. This paper is motivated by the advent of conformal prediction (CP), a frequentist alternative that can be used to conduct model-agnostic, distribution-free valid predictive inference on top of any ML model [12, 13, 14]. Throughout this paper, we will study the validity of CP-based procedures for inference of ITEs.
What makes CP-based inference of ITEs different from its application to the standard regression (supervised) setup? The "fundamental problem of causal inference" is that we never observe counterfactual outcomes [15]. That is, our "label" is the ITE which is a difference between two potential outcomes (treated and untreated) for an individual subject—this label is never observed for any given subject because we only ever observe factual outcomes. This poses two challenges [16]:
(1) How to handle covariate shift? When treatments are assigned to individuals with probabilities that depend on their covariates, then the distributions of covariates in treated and untreated groups differ. Consequently, the distribution of training data differs from that of the target population.
(2) How to incorporate inductive biases? Unlike supervised learning wherein we fit a single function using examples of covariates and observed targets, models of treatment effects cannot be directly fit to the unobserved effects. Thus, estimates of treatment effects comprise intermediate estimates of nuisance parameters. Different approaches for combining nuisance estimates entail different inductive priors on the potential outcomes that affect the accuracy of the resulting ITE estimates.
The literature on ML-based CATE estimation focuses on addressing the two questions above. Covariate shift affects the generalization performance of ML models—existing CATE estimation models address this problem using importance weighting [17] or balanced representation learning methods for unsupervised domain adaptation [6, 18, 19]. In [5], the notion of “meta-learners” was coined to describe various model-agnostic approaches to incorporating inductive biases and combining nuisance estimates. In [5, 20], it was shown that the choice of the meta-learner influences the CATE estimation rates. While the impact of (1) and (2) on the generalization performance of CATE estimators has been extensively investigated, their impact on the validity and efficiency of predictive inference methods for ITE is less well-understood. This forms the central focus of our paper.
Contributions. We propose a CP procedure for predictive inference of ITEs that jointly addresses (1) and (2) in an end-to-end fashion. Our proposed inference strategy applies the standard CP procedure on top of a broad class of CATE meta-learners based on two-stage pseudo-outcome regression. These meta-learners operate by first estimating pseudo-outcomes, i.e., transformed targets that depend on observable variables only, and then regressing the pseudo-outcomes on covariates to obtain point estimates of CATE. We then construct intervals for ITEs by computing the empirical quantile of conformity scores evaluated on pseudo-outcomes in a held-out calibration set. Conformal meta-learners address (1) because the distribution of covariates associated with pseudo-outcomes is the same for training and testing data, and they address (2) since the calibration step is decoupled from model architecture, enabling flexible choice of inductive priors and the possibility of re-purposing existing meta-learners and architectures that have been shown to provide accurate estimates of CATE.
Conformal meta-learners inherit the guarantees of CP, i.e., their resulting intervals cover pseud outcomes on test data with high probability. However, the original CP guarantees do not immediately translate to guarantees on coverage of ITEs. To this end, we develop a unified stochastic ordering framework to study the validity of conformal meta-learners for inference on ITEs. We show that inference with conformal meta-learners is valid if their conformity scores satisfy certain stochastic ordering conditions with respect to "oracle" conformity scores evaluated on unobserved ITEs. We prove that some of the commonly used meta-learners, such as the doubly-robust learner [20], satisfy a weaker stochastic (or convex) dominance condition which makes them valid for relevant levels of target coverage. Our numerical experiments show that, with careful choice of the pseudo-outcome transformation, conformal meta-learners inherit both the coverage properties of CP as well as the efficiency and point estimation accuracy of their underlying CATE meta-learners.
2 Predictive Inference of Individual Treatment Effects (ITEs)
We consider the standard potential outcomes (PO) framework with a binary treatment ([21, 22]). Let  $W \in \{0,1\}$  be the treatment indicator,  $X \in \mathcal{X}$  be the covariates, and  $Y \in \mathbb{R}$  be the outcome of interest. For each subject  $i$ , let  $(Y_{i}(0),Y_{i}(1))$  be the pair of potential outcomes under  $W = 0$  and  $W = 1$ ,
respectively. The fundamental problem of causal inference is that we can only observe the factual outcome, i.e., the outcome  $Y_{i} = W_{i}Y_{i}(1) + (1 - W_{i})Y_{i}(0)$  determined by  $W_{i}$ , but we cannot observe the counterfactual  $Y_{i}(1 - W_{i})$ . For  $n$  subjects, we assume that the data generation process
$$
\left(X _ {i}, W _ {i}, Y _ {i} (0), Y _ {i} (1)\right) \stackrel {{i i d}} {{\sim}} P (X, W, Y (0), Y (1)), i = 1, \dots , n, \tag {1}
$$
satisfies the following assumptions: (1) Unconfoundedness:  $(Y(0), Y(1)) \perp W \mid X$ , (2) Consistency:  $Y = Y(W)$ , and (3) Positivity:  $0 < P(W = 1 \mid X = x) < 1, \forall x \in \mathcal{X}$ . These assumptions are necessary for identifying the causal effects of the treatment from a dataset  $\{Z_i = (X_i, W_i, Y_i)\}_{i=1}^n$ . The causal effect of the treatment on individual  $i$ , known as the individual treatment effect (ITE), is defined as the difference between the two potential outcomes, i.e.,  $Y_i(1) - Y_i(0)$ .
Previous modeling efforts (e.g., [5, 6, 7, 8]) have focused primarily on the (deterministic) conditional average treatment effect (CATE), i.e.,  $\tau(x) \triangleq \mathbb{E}[Y(1) - Y(0) \mid X = x]$ . In this paper, we focus on the (random) ITE as the inferential target of interest. That is, our goal is to infer the ITE for a given subject  $j$  given their covariate  $X_{j}$  and the observed sample  $\{Z_{i} = (X_{i}, W_{i}, Y_{i})\}_{i=1}$ .
The distribution of the observed variable  $Z = (X, W, Y)$  is indexed by the covariate distribution  $P_{X}$ , as well as the nuisance functions  $\pi(x)$ ,  $\mu_0(x)$  and  $\mu_1(x)$  defined as follows:
$$
\pi (x) = \mathbb {P} (W = 1 \mid X = x),
$$
$$
\mu_ {w} (x) = \mathbb {E} [ Y \mid X = x, W = w ], w \in \{0, 1 \}. \tag {2}
$$
The function  $\pi(x)$  is known as the propensity score and it captures the treatment mechanism underlying the data generation process. Throughout this paper, we assume that  $\pi(x)$  is known, i.e., data is drawn from an experimental study or the treatment assignment mechanism is known.
Predictive Inference of ITEs. Given the sample  $\{Z_{i} = (X_{i},W_{i},Y_{i})\}_{i = 1}^{n}$ , our goal is to infer the ITE for a new individual  $n + 1$  with covariate  $X_{n + 1}$ . In particular, we would like to construct a predictive band  $\widehat{C} (x)$  that covers the true ITE for new test points with high probability, i.e.,
$$
\mathbb {P} \left(Y _ {n + 1} (1) - Y _ {n + 1} (0) \in \widehat {C} \left(X _ {n + 1}\right)\right) \geq 1 - \alpha , \tag {3}
$$
for a predetermined target coverage of  $1 - \alpha$ , with  $\alpha \in (0, 1)$ , where the probability in (3) accounts for the randomness of the training data  $\{Z_i\}_i$  and the test point  $(X_{n+1}, Y_{n+1}(1) - Y_{n+1}(0))$ . Predictive intervals that satisfy the coverage condition in (3) are said to be marginally valid.
2.2 Conformal prediction
Conformal prediction (CP) is a model- and distribution-free framework for predictive inference that provides (finite-sample) marginal coverage guarantees. In what follows, we describe a variant of CP, known as split (or inductive) CP [12, 13, 14], for the standard regression setup. Given a training dataset  $\mathcal{D} = \{(X_i,Y_i)\}_i$ , the CP procedure starts by splitting  $\mathcal{D}$  into two disjoint subsets: a proper training set  $\{(X_j,Y_j):j\in \mathcal{D}_t\}$ , and a calibration set  $\{(X_k,Y_k):k\in \mathcal{D}_c\}$ . Then, an ML model  $\widehat{\mu} (x)$  is fit using the samples in  $\mathcal{D}_t$  and a conformity score  $V(.)$  is evaluated for all samples in  $\mathcal{D}_c$  as follows:
$$
V _ {k} (\widehat {\mu}) \triangleq V \left(X _ {k}, Y _ {k}; \widehat {\mu}\right), \forall k \in \mathcal {D} _ {c}. \tag {4}
$$
The conformity score measures how unusual the prediction looks relative to previous examples. A common choice of  $V(.)$  is the absolute residual, i.e.,  $V(x,y;\widehat{\mu})\triangleq |\widehat{\mu} (x) - y|$ . For a target coverage level of  $1 - \alpha$ , we then compute a quantile of the empirical distribution of conformity scores, i.e.,
$$
Q _ {\mathcal {V}} (1 - \alpha) \triangleq (1 - \alpha) \left(1 + 1 / \left| \mathcal {D} _ {c} \right|\right) - \text {t h q u a n t i l e} \mathcal {V} (\widehat {\mu}), \tag {5}
$$
where  $\mathcal{V}(\widehat{\mu}) = \{V_k(\widehat{\mu}): k \in \mathcal{D}_c\}$ . Finally, the predictive interval at a new point  $X_{n+1} = x$  is
$$
\hat {C} (x) = \left[ \hat {\mu} (x) - Q _ {\mathcal {V}} (1 - \alpha), \hat {\mu} (x) + Q _ {\mathcal {V}} (1 - \alpha) \right]. \tag {6}
$$
The interval in (6) is guaranteed to satisfy marginal coverage, i.e.,  $\mathbb{P}(Y_{n + 1}\in \widehat{C} (X_{n + 1}))\geq 1 - \alpha$  . The only assumption needed for this condition to hold is the exchangeability between calibration and test data [12, 23, 24]. Note that the interval in (6) has a fixed length of  $2Q_{\mathcal{V}}(1 - \alpha)$  that is independent of  $x$  To enable adaptive intervals, [25] proposed a variant of the CP procedure where the base model is a quantile regression with interval-valued predictions  $[\widehat{\mu}_{\alpha /2}(x),\widehat{\mu}_{1 - \alpha /2}(x)]$  , and the conformity score is defined as the signed distance  $V_{k}(\widehat{\mu})\triangleq \max \{\widehat{\mu}_{\alpha /2}(X_{k}) - Y_{k},Y_{k} - \widehat{\mu}_{1 - \alpha /2}(X_{k})\}$
2.3 Oracle conformal prediction of ITEs
How can we adapt the CP framework for predictive inference of ITEs? In a hypothetical world where we have access to counterfactual outcomes, we can apply the standard CP in Section 2.2 to a dataset of covariates and ITE tuples,  $\mathcal{D}^{*} = \{(X_{i},Y_{i}(1) - Y_{i}(0))\}_{i}$ , and compute conformity scores as:
$$
V _ {k} ^ {*} (\widehat {\tau}) \triangleq V \left(X _ {k}, Y _ {k} (1) - Y _ {k} (0); \widehat {\tau}\right), \forall k \in \mathcal {D} _ {c} ^ {*}, \tag {7}
$$
where  $\widehat{\tau}$  is an ML model fit to estimate the CATE function  $\tau(x)$  using  $\mathcal{D}_t^*$ , and  $\mathcal{D}^* = \mathcal{D}_t^* \cup \mathcal{D}_c^*$ . We will refer to this procedure as "oracle" conformal prediction and to  $V_k^*(\widehat{\tau})$  as the oracle conformity scores. Since the oracle problem is a standard regression, the oracle procedure is marginally valid—i.e., it satisfies the guarantee in (3),  $\mathbb{P}(Y(1) - Y(0) \in \widehat{C}^*(X)) \geq 1 - \alpha$ . However, oracle CP is infeasible since we can only observe one of the potential outcomes (colored in red and blue in (7)), hence we need an alternative procedure that operates only on the observed variable  $Z = (X, W, Y)$ .
2.4 The two challenges of predictive inference on ITEs
A naive approach to inference of ITEs is to split the observed sample  $\{Z_{i} = (X_{i},W_{i},Y_{i})\}_{i}$  by the treatment group and create two datasets:  $\mathcal{D}_0 = \{(X_i,Y_i):W_i = 0\} _i$ $\mathcal{D}_1 = \{(X_i,Y_i):W_i = 1\} _i$  , then generate two sets of conformity scores for the nuisance estimates  $\widehat{\mu}_0$  and  $\widehat{\mu}_1$  as follows:
$$
V _ {k} ^ {(0)} \left(\widehat {\mu} _ {0}\right) \triangleq V \left(X _ {k}, Y _ {k} (0); \widehat {\mu} _ {0}\right), \forall k \in \mathcal {D} _ {c, 0}, \quad V _ {k} ^ {(1)} \left(\widehat {\mu} _ {1}\right) \triangleq V \left(X _ {k}, Y _ {k} (1); \widehat {\mu} _ {1}\right), \forall k \in \mathcal {D} _ {c, 1}, \tag {8}
$$
where  $\mathcal{D}_{c,0}$  and  $\mathcal{D}_{c,1}$  are calibration subsets of  $\mathcal{D}_0$  and  $\mathcal{D}_1$ . In order to construct valid predictive intervals for ITE using the conformity scores in (8), we need to reconsider how the two distinct characteristics of CATE estimation, previously discussed in Section 1, interact with the CP procedure:
(1) Covariate shift
The distributions of covariates for treated and untreated subjects differ from that of the target population:  $P_{X|W = 0} \neq P_{X|W = 1} \neq P_{X}$ , i.e., the following holds for the conformity scores in (8):
$$
P _ {X, V ^ {(0)} \mid W = 0} \neq P _ {X, V ^ {(0)}} \quad P _ {X, V ^ {(1)} \mid W = 1} \neq P _ {X, V ^ {(1)}}
$$
(2) Inductive biases
Different choices for the joint model of  $\mu_0$  and  $\mu_{1}$  encode different inductive biases that impose different forms of regularization on the implied CATE function, i.e.,  $\widehat{\tau} (x) = \widehat{\mu}_1(x) - \widehat{\mu}_0(x)$  [5]. These biases influence the induced distributions  $P_{V^{(0)}}$  and  $P_{V^{(1)}}$  of the conformity scores in (8).
Covariate shift breaks the exchangeability assumption necessary for the validity of CP. Current methods have primarily focused on (1) with  $Y(0)$  and  $Y(1)$  as inference targets, and developed approaches for handling covariate shift by reweighting conformity scores [1, 26]. The resulting intervals for POs are then combined to produce intervals for ITEs. However, these method tie the CP procedure to model architecture, requiring inference on nuisance parameters, and hence lose the desirable post-hoc nature of CP. Furthermore, inference on POs is likely to provide conservative ITE intervals, and limits the inductive priors that can be assumed since not all CATE models provide explicit PO estimates.
3 Conformal Meta-learners
In [5], a taxonomy of "meta-learners" was introduced to categorize different inductive priors that can be incorporated into CATE estimators by structuring the regression models for  $\mu_0$  and  $\mu_{1}$ . For example, the  $T$ -learner estimates  $\widehat{\mu}_0$  and  $\widehat{\mu}_{1}$  independently using  $\mathcal{D}_0$  and  $\mathcal{D}_1$ , while the  $S$ -learner models the treatment variable  $W$  as an additional covariate in a joint regression model  $\widehat{\mu} (X,W)$  and estimates CATE as  $\widehat{\tau} (x) = \widehat{\mu} (x,1) - \widehat{\mu} (x,0)$ . In this Section, we propose an end-to-end solution to (1) and (2) by applying CP on top of CATE meta-learners in a post-hoc fashion, thereby decoupling the CP procedure from the CATE model and allowing direct inference on ITEs. In the next Section, we develop a unified framework for analyzing the validity of this broad class of procedures.
3.1 Pseudo-outcome regression for CATE estimation
We focus on a broad subclass of CATE meta-learners based on two-stage pseudo-outcome regression. These models replace the (unobserved) oracle ITEs with "proximal" targets that are estimated from observed variables only, and then train an ML model to predict the estimated targets from covariates. The two stages of this general pseudo-outcome regression framework can be described as follows:
(a) Pseudo-outcome regression for CATE
Figure 1: Pictorial depiction of conformal meta-learners.
(b) Conformal pseudo-intervals for ITE
Algorithm 1: Conformal Meta-learner
Input :  $\mathcal{D} = \{(X_i,W_i,Y_i)\}_{i = 1}^n$ $X_{n + 1}$
Output:  $\hat{C}_{\varphi}(X_{n + 1})$
1 Split  $\mathcal{D}$  into  $\mathcal{D}_{\varphi}, \mathcal{D}_t$  and  $\mathcal{D}_c$ ;
2 Pseudo-outcome regression:
3 (1) Estimate  $\widehat{\varphi} = (\pi, \widehat{\mu}_0, \widehat{\mu}_1)$  using  $\mathcal{D}_{\varphi}$ ;
4 (2) Regress  $\widetilde{Y}_{\varphi} = f(Z, \widehat{\varphi})$  on  $X$  using  $\mathcal{D}$ ;
5 Conformal pseudo-intervals:
6 Evaluate conformity scores  $\nu_{\varphi}$  using  $\mathcal{D}_{c}$ ;
7 Return  $\hat{C}_{\varphi} = [\widehat{\tau}(X_{n+1}) \pm Q_{\mathcal{V}_{\varphi}}(1 - \alpha)]$
Stage 1. We obtain a plug-in estimate  $\widehat{\varphi}$  of the nuisance parameters  $\varphi = (\pi, \mu_0, \mu_1)$ . Note that since we assume that the propensity score is known, we only need to estimate  $\mu_0$  and  $\mu_1$  using  $\mathcal{D}_0$  and  $\mathcal{D}_1$ .
Stage 2. We use the nuisance estimates obtained in Stage 1 to create pseudo-outcomes  $\widetilde{Y}_{\varphi}$  that depend only on  $\widehat{\varphi}$  and the observable variables  $Z = (X,W,Y)$ , i.e.,  $\widetilde{Y}_{\varphi} = f(Z,\widehat{\varphi})$  for some function  $f$ . The CATE estimate is then obtained by regressing the pseudo-outcome  $\widetilde{Y}_{\varphi}$  on the covariate  $X$ . This is typically conducted using a different dataset than the one used to obtain the nuisance estimate  $\widehat{\varphi}$ .
The general framework described above captures various models in previous literature. We study 3 examples of such meta-learners: X-learner, Inverse propensity weighted (IPW) learner and doubly-robust (DR) learner.
Pseudo-outcome
$$
I P W - l e a r n e r [ 2 7 ] \widetilde {Y} _ {\varphi} = \frac {W - \pi (X)}{\pi (X) (1 - \pi (X))} Y
$$
$$
X \text {- l e a r n e r} [ 5 ] \widetilde {Y} _ {\varphi} = W (Y - \widehat {\mu} _ {0} (X)) + (1 - W) (\widehat {\mu} _ {1} (X) - Y)
$$
$$
D R \text {- l e a r n e r} [ 2 0 ] \widetilde {Y} _ {\varphi} = \frac {W - \pi (X)}{\pi (X) (1 - \pi (X))} \left(Y - \widehat {\mu} _ {W} (X)\right) + \widehat {\mu} _ {1} (X) - \widehat {\mu} _ {0} (X)
$$
Table 1: Existing meta-learners as instantiations of pseudo-outcome regression.
Table 1 lists the pseudo-outcomes  $\widetilde{Y}_{\varphi}$  for the three meta-learners: IPW-learner reweights factual outcomes using propensity scores to match CATE, i.e.,  $\mathbb{E}[\widetilde{Y}_{\varphi} \mid X = x] = \tau(x)$ ; X-learner uses regression adjustment to impute counterfactuals; DR-learner combines both approaches. DR- and X-learners<sup>1</sup>, coupled with specific architectures for joint modeling of  $\widehat{\mu}_0$  and  $\widehat{\mu}_1$ , have shown competitive performance for CATE estimation in previous studies [5, 16, 20]. The conformal meta-learner framework decouples the CP procedure (Section 3.2) from the inductive priors encoded by these meta-learners, hence it inherits their favorable CATE estimation properties and enables a potentially more efficient direct inference on ITEs as opposed to inference on POs. This addresses challenge (2) in Section 2.4.
3.2 Conformal pseudo-intervals for ITEs
Pseudo-outcome regression is based on the notion that accurate proxies for treatment effects can produce reliable CATE point estimates. This concept can be extended to predictive inference: using CP to calibrate meta-learners via held-out pseudo-outcomes can yield accurate "pseudo-intervals" for ITEs.
Given a dataset  $\mathcal{D} = \{Z_i = (X_i, W_i, Y_i)\}_{i}$ , we create three mutually-exclusive subsets:  $\mathcal{D}_{\varphi}$ ,  $\mathcal{D}_t$  and  $\mathcal{D}_c$ .  $\mathcal{D}_{\varphi}$  is used to estimate the nuisance parameters  $\varphi$ . Next, the estimates  $\widehat{\varphi} = (\pi, \widehat{\mu}_0, \widehat{\mu}_1)$  are used to transform  $\{Z_i = (X_i, W_i, Y_i): i \in \mathcal{D}_t\}$  into covariate/pseudo-outcome pairs  $\{(X_i, \widetilde{Y}_{\varphi,i}): i \in \mathcal{D}_t\}$  which are used to train a CATE model  $\widehat{\tau}$ . Finally, we compute conformity scores for  $\widehat{\tau}$  on pseudo-outcomes, i.e.,
$$
V _ {\varphi , k} (\widehat {\tau}) \triangleq V \left(X _ {k}, \widetilde {Y} _ {\varphi , k}; \widehat {\tau}\right), \forall k \in \mathcal {D} _ {c}. \tag {9}
$$
For a target coverage of  $1 - \alpha$ , we construct a predictive interval at a new point  $X_{n + 1} = x$  as follows:
$$
\widehat {C} _ {\varphi} (x) = \left[ \widehat {\tau} (x) - Q _ {\mathcal {V} _ {\varphi}} (1 - \alpha), \widehat {\tau} (x) + Q _ {\mathcal {V} _ {\varphi}} (1 - \alpha) \right], \tag {10}
$$
where  $\mathcal{V}_{\varphi} = \{V_{\varphi ,k}(\widehat{\tau}):k\in \mathcal{D}_c\}$ . We call  $\widehat{C}_{\varphi}(x)$  a pseudo-interval. The conformal meta-learner approach is depicted in Figure 1 and a summary of the procedure is given in Algorithm 1.
<sup>1</sup>Here, we consider a special case of the X-learner in [5] which involves a weighted sum of two regression adjusted models  $\widehat{\tau}_0$  and  $\widehat{\tau}_1$  trained separately on the treated and control datasets  $\mathcal{D}_0$  and  $\mathcal{D}_1$ .