SlowGuess/NeurIPS_2023_OCR huggingface dataset:
(a) First-order stochastic dominance  $F\succeq_{(1)}G$
(b) Second-order stochastic dominance  $F\succeq_{(2)}G$
Figure 2: Graphical illustration of stochastic dominance among two exemplary distributions  $F$  and  $G$ .
Note that conditional on  $\widehat{\varphi}$ , the pseudo-outcomes  $(X, \widetilde{Y}_{\varphi})$  in calibration data are drawn from the target distribution, which maintains the exchangeability of conformity scores and addresses covariate shift (challenge (1) in Section 2.4). However, the conformity scores  $V\varphi$  are evaluated on transformed outcomes, which means that  $V_{\varphi}$  and  $V^{*}$  are not exchangeable, even though they are drawn from the same covariate distribution. Consequently, the usual CP guarantees, i.e.,  $\mathbb{P}(\widetilde{Y}_{\varphi} \in \widehat{C}_{\varphi}(X)) \geq 1 - \alpha$ , do not immediately translate to coverage guarantees for the true ITE  $Y(1) - Y(0)$ . In the next section, we show that for certain choices of the pseudo-outcomes, the corresponding pseudo-intervals can provide valid inferences for ITE without requiring the exchangeability of  $V_{\varphi}$  and  $V^{*}$ .
4 Validity of Conformal Meta-learners: A Stochastic Ordering Framework
Under what conditions are pseudo-intervals valid for inference of ITEs? Recall that these intervals are constructed by evaluating the empirical quantile of pseudo-outcome conformity scores. Intuitively, the pseudo-intervals will cover the true ITE if the conformity scores are "stochastically larger" than the oracle scores in Section 2.3, i.e.,  $Q_{\mathcal{V}_{\varphi}}(\alpha) \geq Q_{\mathcal{V}^*}(\alpha)$  in some stochastic sense (Figure 1(b)). Hence, to study the validity of conformal meta-learners, we analyze the stochastic orders of  $V_{\varphi}$  and  $V^*$ , and identify conditions under which pseudo-intervals cover oracle intervals.
Stochastic orders are partial orders of random variables used to compare their location, magnitude, or variability [28, 29]. In our analysis, we utilize two key notions of stochastic order among cumulative distribution functions (CDFs)  $F$  and  $G$ , which we formally define below.
Definition 1 (Stochastic dominance)  $F$  has first-order stochastic dominance (FOSD) on  $G$ ,  $F \succeq_{(1)} G$ , iff  $F(x) \leq G(x), \forall x$ , with strict inequality for some  $x$ .  $F$  has second-order stochastic dominance (SOSD) over  $G$ ,  $F \succeq_{(2)} G$ , iff  $\int_{-\infty}^{x} [G(t) - F(t)] dt \geq 0, \forall x$ , with strict inequality for some  $x$ .
Definition 2 (Convex dominance)  $F$  has monotone convex dominance (MCX) over  $G$ ,  $F \succeq_{mcx} G$ , iff  $\mathbb{E}_{X \sim F}[u(X)] \geq \mathbb{E}_{X \sim G}[u(X)]$  for all non-decreasing convex functions  $u: \mathbb{R} \to \mathbb{R}$ .
Stochastic ordering is useful tool in decision theory and quantitative finance used to analyze the decisions of utility maximizers with varying risk attitudes [30]. A distribution  $F$  has FOSD over  $G$  if it is favored by any decision-maker with a non-decreasing utility function, i.e.,  $F$  is more likely to give higher outcomes than  $G$  because its CDF is strictly lower (Figure 2(a)). If  $F$  has SOSD over  $G$ , then it is favored by risk-averse decision-makers, i.e.,  $f$  has smaller spread than  $g$  and is favored by all decision-makers with a non-decreasing concave utility function [31]. In this case, the CDFs can cross but  $G$  is always lower after the last crossing point (Figure 2(b)).  $F$  has MCX over  $G$  if it is favored by decision-makers with a non-decreasing convex utility—in this case, the CDFs can cross but  $F$  is always lower after the last crossing point (See Appendix A for a detailed analysis).
In the following Theorem, we provide sufficient conditions for the validity of conformal meta-learners in terms of the stochastic orders of their conformity scores.
Theorem 1. If  $(X_{i},W_{i},Y_{i}(0),Y_{i}(1))$ ,  $i = 1,\ldots ,n + 1$  are exchangeable, then  $\exists \alpha^{*}\in (0,1)$  such that the pseudo-interval  $\widehat{C}_{\varphi}(X_{n + 1})$  constructed using the dataset  $\mathcal{D} = \{(X_i,W_i,Y_i)\}_{i = 1}^n$  satisfies
$$
\mathbb {P} \big (Y _ {n + 1} (1) - Y _ {n + 1} (0) \in \widehat {C} _ {\varphi} (X _ {n + 1}) \big) \geq 1 - \alpha , \forall \alpha \in (0, \alpha^ {*}),
$$
if at least one of the following stochastic ordering conditions hold: (i)  $V_{\varphi} \succeq_{(1)} V^{*}$ , (ii)  $V_{\varphi} \preceq_{(2)} V^{*}$ , and (iii)  $V_{\varphi} \succeq_{mcx} V^{*}$ . Under condition (i), we have  $\alpha^{*} = 1$ .
All proofs are provided in Appendix A. Theorem 1 states that if the conformity score  $V_{\varphi}$  of the meta-learner is stochastically larger (FOSD) or has a larger spread (SOSD and MCX) than the oracle conformity score, then the conformal meta-learner is valid for high-probability coverage (Figure 3). (This is the range of target coverage that is of practical relevance, i.e.,  $\alpha$  is typically set to 0.05 or 0.1.)
Figure 3: Validity condition in Theorem 1.
Meta-learner	Conformity score
Absolute residual	Signed distance
X-learner	No stochastic order	No stochastic order
IPW-learner	Vφ ≥mcx V*	Vφ ≧(2)V*
DR-learner	Vφ ≥mcx V*	Vφ ≧(2)V*
Table 2: Stochastic orders of conformity scores for the three meta-learners considered in Table 1.
Because stochastic (or convex) dominance pertain to more variable conformity scores, the predictive intervals of conformal meta-learners will naturally be more conservative than the oracle intervals.
Whether a meta-learner meets conditions (i)-(iii) of Theorem 1 depends on how the pseudo-outcome,  $\widetilde{Y}_{\varphi} = f(Z;\widehat{\varphi})$ , is constructed. The following Theorem provides an answer to the question of which of the meta-learners listed in Table 1 satisfy the stochastic ordering conditions in Theorem 1.
Theorem 2. Let  $V_{\varphi}(\widehat{\tau}) = |\widehat{\tau}(X) - \widetilde{Y}_{\varphi}|$  and assume that the propensity score function  $\pi : \mathcal{X} \to [0,1]$  is known. Then, the following holds: (i) For the  $X$ -learner,  $V_{\varphi}$  and  $V^{*}$  do not admit to a model- and distribution-free stochastic order; (ii) For any distribution  $P(X,W,Y(0),Y(1))$ , CATE estimate  $\widehat{\tau}$ , and nuisance estimate  $\widehat{\varphi}$ , the IPW- and the DR-learners satisfy  $V_{\varphi} \succeq_{mcx} V^{*}$ .
Theorem 2 states that the stochastic ordering of  $V_{\varphi}$  and  $V^{*}$  depends on the specific choice of the conformity score function  $V(X, \widetilde{Y}_{\varphi}; \widehat{\tau})$  as well as the choice of the meta-learner, i.e., the pseudo-outcome generation function  $\widetilde{Y}_{\varphi} = f(Z; \widehat{\varphi})$ . The IPW- and DR-learners ensure that, by construction, the pseudo-outcome is equal to CATE in expectation:  $\mathbb{E}[\widetilde{Y}_{\varphi} | X = x] = \tau(x)$ . This construction enables the IPW- and DR-learners to provide unbiased estimates of average treatment effects (ATE) independent of the data distribution and the ML model used for the nuisance estimates  $\widehat{\mu}_0$  and  $\widehat{\mu}_1$ . By the same logic, IPW- and DR-learners also guarantee stochastic (convex) dominance of their conformity scores irrespective of the data distribution and the ML model choice, hence preserving the model- and distribution-free nature of the CP coverage guarantees. Contrarily, the X-learner does not use the knowledge of  $\pi$  to construct its pseudo-outcomes, hence it does not guarantee a (distribution-free) stochastic order and the achieved coverage depends on the nuisance estimates  $\widehat{\mu}_0$  and  $\widehat{\mu}_1$ . In Table 2, we list the stochastic orders achieved for different choices of meta-learners and conformity scores. (The analysis of stochastic orders for the signed distance score used in [25] and [32] is provided in Appendix A.)
Key limitations of conformal meta-learners. While conformalized meta-learners can enable valid end-to-end predictive inference of ITEs, they have two key limitations. First, the propensity score  $\pi$  must be known to guarantee model- and distribution-free stochastic ordering of conformity scores. However, we note that this limitation in not unique to our method and is also encountered in methods based on weighted CP [1, 26]. The second limitation is peculiar to our method: exact characterization of  $\alpha^{*}$  is difficult and depends on the data distribution. Devising procedures for inferring  $\alpha^{*}$  based on observable variables or deriving theoretical upper bounds on  $\alpha^{*}$  are interesting directions for future work. Here, we focus on empirical evaluation of  $\alpha^{*}$  in semi-synthetic experiments. A detailed comparison between our method and previous work is provided in Appendix B.
Since the true ITEs are never observed in real-world datasets, we follow the common practice of conducting numerical experiments using synthetic and semi-synthetic datasets [1, 8, 19]. We present a number of representative experiments in this Section and defer further results to Appendix C.
Synthetic datasets. We consider a variant of the data-generation process in Section 3.6 in [1] which was originally proposed in [7]. We create synthetic datasets by sampling covariates  $X \sim U([0,1]^d)$  and treatments  $W|X = x \sim \mathrm{Bern}(\pi(x))$  with  $\pi(x) = (1 + I_x(2,4)) / 4$ , where  $I_x(2,4)$  is the regularized incomplete beta function (i.e., CDF of a Beta distribution with shape parameters 2 and 4). Outcomes are modeled as  $\mu_1(x) = \zeta(x_1) \cdot \zeta(x_2)$  and  $\mu_0(x) = \gamma \zeta(x_1) \cdot \zeta(x_2)$ , where  $\gamma \in [0,1]$  is a parameter that controls the treatment effect, and  $\zeta$  is a function given by  $\zeta(x) = 1 / (1 + \exp(-12(x - 0.5)))$ .
(a) Empirical assessment of stochastic orders
(b) Coverage, efficiency and RMSE for Setup A (top) and Setup B (bottom)
(c) Performance at different levels of target coverage
Figure 4: Performance of all baseline in the synthetic setup described in Section 5.1. In (b), red vertical lines correspond to target coverage  $(1 - \alpha = 0.9)$ , and blue vertical lines correspond to optimal interval width. In (c), baseline methods are color-coded as follows: CM-DR, CM-IPW, CM-X, WCP-Naive, WCP-Exact, and WCP-Inexact. Here, WCP stands for weighted CP and CM stands for conformal meta-learners.
We assume that POs are sampled from  $Y(w)|X = x \sim \mathcal{N}(\mu_w(x), \sigma^2(x))$ ,  $w \in \{0,1\}$  and consider a heteroscedastic noise model  $\sigma^2(x) = -\log(x_1)$ . We define two setups within this model: Setup A where the treatment has not effect  $(\zeta = 1)$ , and Setup B where the effects are heterogeneous  $(\zeta = 0)$ .
Semi-synthetic datasets. We also consider two well-known semi-synthetic datasets that involve real covariates and simulated outcomes. The first is the National Study of Learning Mindsets (NLSM) [3], and the second is the IHDP benchmark originally developed in [8]. Details on the data generation process for NLSM can be founded in Section 2 in [33]. Details on the IHDP benchmark can be found in [6, 8, 16, 19]. Appendix C provides detailed description of both datasets for completeness.
Baselines. We consider baseline models that provide valid predictive intervals for ITEs. Specifically, we consider state-of-the-art methods based on weighted conformal prediction (WCP) proposed in [1]. These methods apply weighted CP to construct intervals for the two POs or plug-in estimates of ITEs. We consider the three variants of WCP in [1]: (1) Naïve WCP which combines the PO intervals using Bonferroni correction, (2) Exact Nested WCP which applies WCP to plug-in estimates of ITEs in treatment and control groups followed by a secondary CP procedure, and (3) Inexact Nested WCP which follows the same steps of the exact version but replaces the secondary CP with conditional quantile regression. (Note that Inexact Nested WCP does not provide coverage guarantees.) For all baselines, we use the same model (Gradient Boosting) for nuisance and pseudo-outcome modeling, and we use the conformal quantile regression method in [25] to construct predictive intervals.
5.2 Results and discussion
Our experimental findings yield the following key takeaways: Firstly, the  $IPW$ - and  $DR$ -learners demonstrate a robust FOSD (i.e.,  $\alpha^{*} = 1$ ) in the majority of experiments, surpassing the MCX conditions outlined in Theorem 2. Secondly, the  $DR$ -learner exhibits superior point estimation accuracy and interval efficiency in most experiment compared to all other baselines that ensure valid inference. Thirdly, the effectiveness of conformal meta-learners depends on the discrepancy between the CDFs of conformity scores and oracle scores—pseudo-outcome transformations that induce thicker tails in the resulting conformity scores can cause conformal meta-learners to under-perform.
Empirical assessment of stochastic orders. Figure 4(a) depicts the empirical CDF of the conformity scores  $V_{\varphi}$  and oracle scores  $V^{*}$  for the three meta-learners under study (DR-, IPW- and X-learners). These CDFs are averaged over 100 runs of Setups A and B of the synthetic generation process outlined in Section 5.1. (The shaded regions represent the lowest and highest bounds on the empirical
IHDP	NLSM
Coverage	Avg. len.	RMSE	Coverage	Avg. len.	RMSE
Naive	0.89 (0.02)	18.9 (4.04)	4.73 (1.00)	0.99 (0.00)	4.82 (0.02)	0.15 (0.00)
Exact	0.99 (0.00)	29.8 (7.60)	4.50 (0.97)	0.99 (0.00)	4.92 (0.07)	0.19 (0.01)
Inexact	0.61 (0.04)	8.49 (1.36)	4.61 (0.99)	0.96 (0.00)	2.38 (0.01)	0.18 (0.00)
X	0.65 (0.04)	11.0 (3.04)	3.34 (0.56)	0.27 (0.00)	0.38 (0.00)	0.14 (0.00)
IPW	0.99 (0.00)	112 (23.0)	19.9 (3.44)	0.99 (0.00)	6.48 (0.07)	0.61 (0.02)
DR	0.96 (0.01)	16.7 (3.30)	3.32 (0.53)	0.99 (0.00)	6.24 (0.07)	0.37 (0.01)
Table 3: Performance of all baselines in semi-synthetic datasets. Bold numbers correspond to best performance.
Figure 5: Stochastic orders in IHDP/NLSM.
CDFs evaluated across all runs.) In both setups, the conformity scores for the DR- and IPW-learners demonstrate FOSD over the oracle scores with respect to the average CDFs and in almost all realizations. This aligns with the result of Theorem 2, and shows that the stochastic dominance condition achieved in practice is even stronger than our theoretical guarantee since FOSD  $(V_{\varphi} \succeq_{(1)} V^{*})$  implies the weaker conditions of  $V_{\varphi} \preceq_{(2)} V^{*}$  and  $V_{\varphi} \succeq_{mcx} V^{*}$ . On the contrary, the conformity scores of the X-learner are dominated by oracle scores in the FOSD sense. This is not surprising in light of Theorem 2, which indicates that X-learners do not guarantee a distribution-free stochastic order. These observations are also replicated in the semi-synthetic datasets as shown in Figure 5.
Based on Theorem 1, the empirical stochastic orders observed in Figures 4(a) and 5 predict that the IPW- and DR-learners will cover ITEs, whereas the X-learner will not achieve coverage. This is confirmed by the results in Figures 4(b), 4(c) and Table 3. The fact that the IPW- and DR-learners satisfy a stronger FOSD condition is promising because it indicates that the range of validity for these models spans all levels of coverage ( $\alpha^{*} = 1$  in Figure 3). It also means that a stronger version of Theorem 2 outlining the conditions under which IPW- and DR-learners achieve FOSD could be possible.
Coverage, efficiency and point estimation accuracy. The performance of a predictive inference procedure can be characterized in terms of three metrics: achieved coverage for true ITEs, expected length of predictive intervals, and root-mean-square error (RMSE) in CATE estimates. In most experiments, we find that the DR-learner strikes a balance between these metrics (See Appendix C for further experiments). In Figure 4(b), we can see that the DR-learner outperforms the valid (naive and exact) WCP procedures in terms of RMSE and interval length, while achieving the target coverage of  $90\%$ . The X-learner outperforms all baselines in terms of RMSE, but as expected, it under-covers ITEs in all experiments. The inexact WCP baseline offers competitive efficiency and calibration, however, in addition to not offering coverage guarantees it also lacks consistency in RMSE performance under different inductive biases (i.e., no treatment effects in Setup A and heterogeneous effects in Setup B). These performance trends hold true across all levels of target coverage as shown in Figure 4(c).
The semi-synthetic experiments on IHDP and NLSM datasets shed light on when meta-learners may perform poorly. The DR-learner outperforms all baselines on the IHDP dataset in terms of RMSE, interval efficiency, while achieving the desired coverage of  $90\%$ . However, we observe that empirical performance depends on how closely the CDF of conformity scores matches the oracle CDF. The DR-learner performance deteriorates when conformity scores have "very strong" dominance over oracle scores, as observed in the NLSM dataset (Figure 5, bottom). Conversely, when the CDF of conformity scores is a closer lower bound on the oracle CDF, the DR-learner performance is competitive (Figure 4 and Figure 5, top). This is intuitive because if the pseudo-outcome transformation induces significant variability in regression targets, it will result in a lower CDF, poorer accuracy of pseudo-outcome regression, and longer predictive intervals. This is why the DR-learner consistently outperforms the IPW-learner, as it provides a closer approximation of the oracle CDF. Future work could focus on analyzing the gap between pseudo-outcome and oracle score CDFs and designing pseudo-outcome transformations that optimize efficiency while preserving stochastic orders.
Estimation and inference of treatment effects is challenging because causal effects are not directly observable. In this paper, we developed a general framework for inference of treatment effects, dubbed conformal meta-learners, that is compatible with any machine learning model. Our framework inherits the model- and distribution-free validity of conformal prediction as well as the estimation accuracy of model-agnostic meta-learners of treatment effects. Additionally, we introduce a new theoretical framework based on stochastic ordering to assess the validity of our method, which can guide the development of new models optimized for both accurate estimation and valid inference.
[42] Muhammad Faaiz Taufiq, Jean-Francois Ton, Rob Cornish, Yee Whye Teh, and Arnaud Doucet. Conformal off-policy prediction in contextual bandits. arXiv preprint arXiv:2206.04405, 2022.
[43] Isaac Gibbs and Emmanuel Candes. Adaptive conformal inference under distribution shift. Advances in Neural Information Processing Systems, 34:1660-1672, 2021.
[44] Rina Foygel Barber, Emmanuel J Candes, Aaditya Ramdas, and Ryan J Tibshirani. Conformal prediction beyond exchangeability. arXiv preprint arXiv:2202.13415, 2022.
[45] Hongxiang Qiu, Edgar Dobriban, and Eric Tchetgen Tchetgen. Distribution-free prediction sets adaptive to unknown covariate shift. arXiv preprint arXiv:2203.06126, 2022.
[46] Jeanne Brooks-Gunn, Fong-ruey Liaw, and Pamela Kato Klebanov. Effects of early intervention on cognitive function of low birth weight preterm infants. The Journal of pediatrics, 120(3):350-359, 1992.
[47] Vincent Dorie, Jennifer Hill, Uri Shalit, Marc Scott, and Dan Cervone. Automated versus do-it-yourself methods for causal inference: Lessons learned from a data analysis competition. Statistical Science, 2019.
Appendix A: Technical Proofs
A.1. Equivalent Definitions for Stochastic Orders
The following section introduces alternative definitions for stochastic dominance between two cumulative distribution functions (CDFs), denoted as  $F$  and  $G$ . These definitions will be used in the proof for Theorem 2 (Section A.4.). Note that these definitions are equivalent to Definition 1 provided in the main text but are expressed in decision-theoretic terms.
Definition A1 (First-order stochastic dominance)  $F$  has first-order stochastic dominance (FOSD) over  $G$ ,  $F \succeq_{(1)} G$ , if and only if  $\mathbb{E}_{X \sim F}[u(X)] \geq \mathbb{E}_{X \sim G}[u(X)]$  for all non-decreasing (utility) functions  $u: \mathbb{R} \to \mathbb{R}$  (Theorem 1.A.3 in [28] proves the equivalence of Definitions 1 and A1).
Definition A2 (Second-order stochastic dominance)  $F$  has second-order stochastic dominance (SOSD) over  $G$ ,  $F \succeq_{(2)} G$ , if and only if  $\mathbb{E}_{X \sim F}[u(X)] \geq \mathbb{E}_{X \sim G}[u(X)]$  for all non-decreasing concave functions  $u: \mathbb{R} \to \mathbb{R}$  (Refer to Theorem 4.A.1 and Eq. (4.A.7) in [28]).
In addition to the alternative definitions of FOSD and SOSD above, we also state the definitions for two additional notions of stochastic ordering that we will use in the proofs.
Definition A3 (Statewise stochastic dominance) A random variable  $X$  has statewise stochastic dominance (SWD) over a random variable  $Y$ ,  $X \succeq_{\mathrm{SWD}} Y$ , if and only if  $X \geq Y$  for every draw of the bivariate random variable  $(X, Y) \sim P_{X,Y}$ . Note that SWD implies FOSD.
Definition A4 (Convex dominance)  $F$  has convex dominance (CX) over  $G$ ,  $F \succeq_{\mathrm{cx}} G$ , if and only if  $\mathbb{E}_{X \sim F}[u(X)] \geq \mathbb{E}_{X \sim G}[u(X)]$  for all convex functions  $u: \mathbb{R} \to \mathbb{R}$ .
A.2. Useful Lemmas
Lemma A1. (Pointwise and marginal stochastic orders) For random variables  $X$ ,  $Y$ , and  $Z$ , the following conditions hold for any marginal distribution  $P_X$ :
(i)  $Y|X = x\succeq_{(l)}Z|X = x,\forall x\in \mathcal{X}\Rightarrow Y\succeq_{(l)}Z,$
(ii)  $Y\mid X = x\succeq_{(2)}Z\mid X = x,\forall x\in \mathcal{X}\Rightarrow Y\succeq_{(2)}Z,$
(iii)  $Y\mid X = x\succeq_{mcx}Z\mid X = x,\forall x\in \mathcal{X}\Rightarrow Y\succeq_{mcx}Z.$
Proof. Recall from the definition of FOSD (Definition A1) that if  $Y \mid X = x \succeq_{(1)} Z \mid X = x$ , then:
$$
\mathbb {E} _ {V \sim P _ {Y | X = x}} [ u (V) ] \geq \mathbb {E} _ {W \sim P _ {Z | X = x}} [ u (W) ], \forall x \in \mathcal {X}, \tag {A.1}
$$
for all non-decreasing functions  $u:\mathbb{R}\to \mathbb{R}$ . Since expectation is a positive linear operator, marginalizing both sides with respect to the density  $P_{X}$  preserves the inequality in (A.1):
$$
\mathbb {E} _ {X \sim P _ {X}} \left[ \mathbb {E} _ {V \sim P _ {Y | X}} [ u (V) ] \right] \geq \mathbb {E} _ {X \sim P _ {X}} \left[ \mathbb {E} _ {W \sim P _ {Z | X}} [ u (W) ] \right], \tag {A.2}
$$
which recovers the definition of FOSD between random variables  $Y$  and  $Z$ , i.e.,
$$
\mathbb {E} _ {Y \sim P _ {Y}} [ u (Y) ] \geq \mathbb {E} _ {Z \sim P _ {Z}} [ u (Z) ], \tag {A.3}
$$
and concludes the proof for (i). Note that (A.3) holds for all non-decreasing functions  $u: \mathbb{R} \to \mathbb{R}$ , which also includes all non-decreasing concave and convex functions that define SOSD and MCX. Hence, combining (A.1)-(A.3) with Definitions 2 and A2 concludes the proof for (ii) and (iii).
Lemma A2. (Convex dominance of mixture distributions over convex combinations of random variables) Let  $X \in \mathbb{R}$  and  $Y \in \mathbb{R}$  be two real-valued random variables with finite means, and let  $\pi \in [0,1]$  be a constant. If  $Z$  and  $V$  are two random variables constructed as:
-  $Z$  is the absolute value of the convex combination of  $X$  and  $Y$ , i.e.,  $Z = |\pi X + (1 - \pi)Y|$ ,
-  $V$  is a mixture of  $|X|$  and  $|Y|$ , i.e.,  $V = W|X| + (1 - W)|Y|$ ,  $W \sim \text{Bernoulli}(\pi)$ ,
then we have  $V\succeq_{mcx}Z$  for any  $\pi \in [0,1]$  and any distribution  $P(X,Y)$ .
Proof. Recall from Definition 2 that  $V \succeq_{mcx} Z$  if and only if
$$
\mathbb {E} _ {V} [ u (V) ] \geq \mathbb {E} _ {Z} [ u (Z) ], \tag {A.4}
$$
for all non-decreasing convex functions  $u:\mathbb{R}\to \mathbb{R}$ . Given the definition of  $V$  above, we have