SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \mathbb {E} _ {V} [ u (V) ] = P (W = 1) \cdot \mathbb {E} _ {V} [ u (V) \mid W = 1 ] + P (W = 0) \cdot \mathbb {E} _ {V} [ u (V) \mid W = 0 ] \\ = \pi \cdot \mathbb {E} [ u (| X |) ] + (1 - \pi) \cdot \mathbb {E} [ u (| Y |) ] \\ \stackrel {(a)} {\geq} \mathbb {E} [ u (\pi | X | + (1 - \pi) | Y |) ] = \mathbb {E} [ u (| \pi X | + | (1 - \pi) Y |) ] \\ \geq \mathbb {E} [ u (| \pi X + (1 - \pi) Y |) ] = \mathbb {E} [ u (Z) ] \Rightarrow V \succeq_ {m c x} Z, \tag {A.5} \\ \end{array}
$$
where the last two inequalities follow from (a) the convexity of  $u$ , and (b) the monotonicity of  $u$  along with the application of the triangle inequality  $|\pi X| + |(1 - \pi)Y| \geq |\pi X + (1 - \pi)Y|$ .
Lemma A3. (Stochastic dominance and sign changes in CDFs) Let  $X$  and  $Y$  be two non-negative random variables with distribution functions  $F$  and  $G$  and with finite means such that  $\mathbb{E}[X] \leq \mathbb{E}[Y]$ . Let  $S^{-}(f)$  be the number of sign changes of a function  $f$  defined as
$$
S ^ {-} (f) = \sup  S ^ {-} [ f (x _ {1}), f (x _ {2}), \dots , f (x _ {m}) ], \tag {A.6}
$$
where  $S^{-}[a_{1}, a_{2}, \ldots, a_{m}]$  is the number of sign changes of the sequence  $[a_{1}, a_{2}, \ldots, a_{m}]$ , with the zero terms being discarded, and the supremum in (A.6) is taken over all sets  $x_{1} < x_{2} < \ldots < x_{m}$  such that  $m < \infty$ . Then,  $F \succeq_{(2)} [\succeq_{mcx}] G$  if and only if there exist random variables  $Z_{1}, Z_{2}, \ldots$ , with distribution functions  $G_{1}, G_{2}, \ldots$ , such that  $Z_{1} = _{st} X$ ,  $\mathbb{E}[Z_j] \leq \mathbb{E}[Y]$ ,  $j = 1, 2, \ldots, Z_j \to_{st} Y$  and  $\mathbb{E}[Z_j] \to \mathbb{E}[Y]$  as  $j \to \infty$  and  $S^{-}(G_{j+1} - G_j) = 1$  and the sign sequence is  $\{+, -\}[\{-, +\}]$ . Here,  $=_{st}$  denotes equality in law and  $\to_{st}$  denotes convergence in distribution.
Proof. Refer to Theorems 4.A.22 and 4.A.23 in [28] for a full proof.
Corollary A4. (CDF crossings under SOSD and MCX) Let  $X$  and  $Y$  be non-negative random variables with CDFs  $F$  and  $G$  with finite means such that  $\mathbb{E}[X] \leq \mathbb{E}[Y]$ . Then, if  $F \succeq_{(2)} [\succeq_{mcx}] G$ , we have  $S^{-}(F - G) \geq 1$  and the corresponding sign sequence is  $\{\ldots, -, +\}[\{\ldots, +, -\}]$ . Equivalently, there exists  $\alpha^{*} \in (0,1)$  such that  $F^{-1}(\alpha^{*}) = G^{-1}(\alpha^{*}) = v^{*}$  and  $F(v) \geq [\leq] G(v), \forall v \geq v^{*}$ .
Proof. From Lemma A3, we know that if  $F \succeq_{(2)} [\succeq_{mcx}] G$  then there exists a sequence of random variables  $Z_1, Z_2, \ldots$ , with distributions  $G_1, G_2, \ldots$ , such that  $Z_1 =_{st} X$ ,  $Z_j \to_{st} Y$ , and the CDFs satisfy  $S^{-}(G_{j+1} - G_j) = 1$  with a sign sequence  $\{-, +\}[\{+, -\}]$ .
Now observe that for any three integers  $k < m < n$ , if  $S^{-}(G_{m} - G_{k}) = 1$  with a sign sequence of  $\{-, +\}[\{+, -\}]$ , and  $S^{-}(G_{n} - G_{m}) = 1$  with a sign sequence  $\{-, +\}[\{+, -\}]$ , then by monotonicity of CDFs it follows that in the last point of crossing,  $G_{n}$  crosses  $G_{k}$  from below [above], i.e.,  $G_{n} - G_{k}$  has at least one sign change with  $\{\ldots, -, +\}[\{\ldots, +, -\}]$ . Thus, for any  $j$ , the three random variables  $Z_{1} =_{st} X$ ,  $Z_{j}$  and  $Y$  satisfy  $S^{-}(G_{j} - F) = 1$  and  $S^{-}(G - G_{j}) = 1$  with a sign sequence  $\{\ldots, -, +\}[\{\ldots, +, -\}]$ , which implies that in the last point of crossing,  $F$  crosses  $G$  from below [above] and concludes the statement of the corollary.
A.3. Proof of Theorem 1
$$
\mathbb {P} \left(Y _ {n + 1} (1) - Y _ {n + 1} (0) \in \widehat {C} _ {\varphi} \left(X _ {n + 1}\right)\right) \geq 1 - \alpha , \forall \alpha \in [ 0, \alpha^ {*} ],
$$
Proof. Without loss of generality, assume that the conformity scores are sorted,  $V_{\varphi,1} < \ldots < V_{\varphi,n_c}$ , where  $n_c = |\mathcal{D}_c|$ . Recall from (10) that the pseudo-interval is constructed as:
$$
\widehat {C} _ {\varphi} (x) = \left[ \widehat {\tau} (x) - Q _ {\mathcal {V} _ {\varphi}} (1 - \alpha), \widehat {\tau} (x) + Q _ {\mathcal {V} _ {\varphi}} (1 - \alpha) \right], \tag {A.7}
$$
where  $Q_{\mathcal{V}_{\varphi}}(1 - \alpha)$  is the empirical quantile defined as
$$
Q _ {\mathcal {V} _ {\varphi}} (1 - \alpha) = \left\{ \begin{array}{l l} V _ {\varphi , \lceil (n _ {c} + 1) (1 - \alpha) \rceil}, & \alpha \geq \frac {1}{n _ {c} + 1}, \\ \infty , & o. w. \end{array} \right. \tag {A.8}
$$
Combining (A.7) and (A.8), we notice that the following two events are equivalent
$$
\left\{Y _ {n + 1} (1) - Y _ {n + 1} (0) \in \widehat {C} _ {\varphi} \left(X _ {n + 1}\right) \right\} \Longleftrightarrow \left\{V _ {n + 1} ^ {*} \leq Q v _ {\varphi} (1 - \alpha) \right\}, \tag {A.9}
$$
$$
\left\{Y _ {n + 1} (1) - Y _ {n + 1} (0) \in \widehat {C} _ {\varphi} \left(X _ {n + 1}\right) \right\} \Longleftrightarrow \left\{V _ {n + 1} ^ {*} \leq V _ {\varphi , \lceil (n _ {c} + 1) (1 - \alpha) \rceil} \right\}. \tag {A.10}
$$
Hence, the achieved coverage probability for the pseudo-interval is given by
$$
\mathbb {P} \left(V _ {n + 1} ^ {*} \leq V _ {\varphi , \lceil (n _ {c} + 1) (1 - \alpha) \rceil}\right). \tag {A.11}
$$
By exchangeability of the variables  $(X_{1},Y_{1}(1) - Y_{1}(0)),\ldots ,(X_{n + 1},Y_{n + 1}(1) - Y_{n + 1}(0))$  , we have
$$
\mathbb {P} \left(V _ {\varphi , n + 1} \leq V _ {\varphi , k}\right) = \frac {k}{n _ {c} + 1}, \tag {A.12}
$$
for any integer  $k$ . (A.12) holds because ranks are uniformly distributed under exchangeability, i.e.,  $V_{\varphi ,n + 1}$  is equally likely to fall in anywhere between the calibration points  $V_{\varphi ,1},\ldots ,V_{\varphi ,n_c}$ . From (A.12), it follows that for  $\lceil (n_c + 1)(1 - \alpha)\rceil$  we have:
$$
\mathbb {P} \left(V _ {\varphi , n + 1} \leq V _ {\varphi , \lceil (n _ {c} + 1) (1 - \alpha) \rceil}\right) = \frac {\lceil (n _ {c} + 1) (1 - \alpha) \rceil}{n _ {c} + 1} \geq 1 - \alpha . \tag {A.13}
$$
Now, we examine the three conditions: (i)  $V_{\varphi} \succeq_{(1)} V^{*}$ , (ii)  $V_{\varphi} \preceq_{(2)} V^{*}$ , and (iii)  $V_{\varphi} \succeq_{mcx} V^{*}$ .
(i) FOSD  $V_{\varphi} \succeq_{(1)} V^{*}$ :
If  $V_{\varphi} \succeq_{(1)} V^{*}$ , then from Definition 1:  $F_{V_{\varphi}}(v) \leq F_{V^{*}}(v), \forall v$ . Equivalently, FOSD can be written as:
$$
V _ {\varphi} \succeq_ {(1)} V ^ {*} \iff \mathbb {P} \left(V _ {\varphi} \leq v\right) \leq \mathbb {P} \left(V ^ {*} \leq v\right), \forall v. \tag {A.14}
$$
Since (A.14) applies for any  $v$ , then the following holds:
$$
V _ {\varphi , n + 1} \succeq_ {(1)} V _ {n + 1} ^ {*} \Longleftrightarrow \mathbb {P} \left(V _ {\varphi , n + 1} \leq V _ {\varphi , \lceil (n _ {c} + 1) (1 - \alpha) \rceil}\right) \leq \mathbb {P} \left(V _ {n + 1} ^ {*} \leq V _ {\varphi , \lceil (n _ {c} + 1) (1 - \alpha) \rceil}\right). \tag {A.15}
$$
Combining (A.13) and (A.15) we have
$$
V _ {\varphi , n + 1} \succeq_ {(1)} V _ {n + 1} ^ {*} \Rightarrow \mathbb {P} \left(V _ {n + 1} ^ {*} \leq V _ {\varphi , \lceil (n _ {c} + 1) (1 - \alpha) \rceil}\right) \geq \mathbb {P} \left(V _ {\varphi , n + 1} \leq V _ {\varphi , \lceil (n _ {c} + 1) (1 - \alpha) \rceil}\right) \geq 1 - \alpha .
$$
which holds for any  $\alpha$ , hence  $\alpha^{*} = 1$ . This concludes statement (i).
(ii)SOSD  $V_{\varphi}\preceq_{(2)}V^{*}$
If  $V_{\varphi} \preceq_{(2)} V^{*}$ , then from Corollary A4 we know that  $\exists \alpha^{*} \in (0,1)$  where  $F_{V_{\varphi}}^{-1}(\alpha^{*}) = F_{V^{*}}^{-1}(\alpha^{*}) = v^{*}$  and  $F_{V^{*}}(v) \geq F_{V_{\varphi}}(v), \forall v \geq v^{*}$ . Equivalently, SOSD can be written as:
$$
V _ {\varphi} \preceq_ {(2)} V ^ {*} \Rightarrow \exists v ^ {*}, s. t. \mathbb {P} \left(V _ {\varphi} \leq v\right) \leq \mathbb {P} \left(V ^ {*} \leq v\right), \forall v \geq v ^ {*}. \tag {A.16}
$$
Hence, the following holds:
$$
\begin{array}{l} V _ {\varphi , n + 1} \preceq_ {(2)} V _ {n + 1} ^ {*} \Rightarrow \mathbb {P} \left(V _ {\varphi , n + 1} \leq v\right) \leq \mathbb {P} \left(V _ {n + 1} ^ {*} \leq v\right), \forall v \geq v ^ {*}, \\ \Rightarrow \mathbb {P} \left(V _ {\varphi , n + 1} \leq V _ {\varphi , \lceil (n _ {c} + 1) (1 - \alpha) \rceil}\right) \leq \mathbb {P} \left(V _ {n + 1} ^ {*} \leq V _ {\varphi , \lceil (n _ {c} + 1) (1 - \alpha) \rceil}\right), \tag {A.17} \\ \end{array}
$$
for all  $\alpha \leq \alpha^{*}$ . Combining (A.13) and (A.17) we have
$$
V _ {\varphi , n + 1} \preceq_ {(2)} V _ {n + 1} ^ {*} \Rightarrow \mathbb {P} (V _ {n + 1} ^ {*} \leq V _ {\varphi , \lceil (n _ {c} + 1) (1 - \alpha) \rceil}) \geq \mathbb {P} (V _ {\varphi , n + 1} \leq V _ {\varphi , \lceil (n _ {c} + 1) (1 - \alpha) \rceil}) \geq 1 - \alpha ,,
$$
for all  $0 \leq \alpha \leq \alpha^{*}$ . This concludes statement (ii).
(iii)  $\mathbf{MCX}V_{\varphi}\succeq_{mcx}V^{*}$
Since Corollary A4 holds for SOSD and MCX, the proof is identical to the proof for (ii).
A.4. Proof of Theorem 2
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  $V_{\varphi} \succeq_{mcx} V^{*}$ .
Proof. For a CATE estimate  $\widehat{\tau}$ , the oracle scores are
$$
V ^ {*} (\widehat {\tau}) = | \widehat {\tau} (X) - (Y (1) - Y (0)) |. \tag {A.18}
$$
In what follows, we use the notation  $V_{\varphi}(x)$  to denote the conditional random variable  $V_{\varphi}|X = x$ , i.e., the conformity score evaluated at a given feature point. Similarly, we use  $V^{*}(x)$  to denote the conditional random variable  $V^{*}|X = x$  (oracle score evaluated at a given feature point).
Proof of statement (i):
Recall from Table 1 that the pseudo-outcome for the X-learner is given by:
$$
\widetilde {Y} _ {\varphi} = W \left(Y - \widehat {\mu} _ {0} (X)\right) + (1 - W) \left(\widehat {\mu} _ {1} (X) - Y\right). \tag {A.19}
$$
With  $V_{\varphi}(\widehat{\tau}) = |\widehat{\tau}(X) - \widetilde{Y}_{\varphi}|$ , the conformity score for the X-learner can be written as:
$$
V _ {\varphi} (\widehat {\tau}) = | \widehat {\tau} (X) - W (Y - \widehat {\mu} _ {0} (X)) - (1 - W) (\widehat {\mu} _ {1} (X) - Y) |. \tag {A.20}
$$
To prove statement (i), it suffices to find counter-examples for estimators  $\widehat{\tau}$ ,  $\widehat{\mu}_0$  and  $\widehat{\mu}_1$ , or data distributions that cause violations of FOSD, SOSD and MCX between  $V_{\varphi}$  and  $V^{*}$ . Let  $u: \mathbb{R} \to \mathbb{R}$  be a non-decreasing function. The expected value of  $u(V_{\varphi}(x))$  can be evaluated as:
$$
\begin{array}{l} \mathbb {E} [ u (V _ {\varphi} (x)) ] = \mathbb {E} [ u (| \widehat {\tau} (x) - W (Y - \widehat {\mu} _ {0} (x)) - (1 - W) (\widehat {\mu} _ {1} (x) - Y) |) ] \\ = \pi (x) \cdot \mathbb {E} [ u (| \hat {\tau} (x) - (Y (1) - \hat {\mu} _ {0} (x)) |) ] + \\ (1 - \pi (x)) \cdot \mathbb {E} [ u (| \widehat {\tau} (x) - (\widehat {\mu} _ {1} (x) - Y (0)) |) ]. \tag {A.21} \\ \end{array}
$$
We start by showing that there exists nuisance and CATE estimates and a data distribution for which  $V_{\varphi}$  does not have FOSD on  $V^{*}$ . Assume that  $u$  is non-decreasing concave. By Jensen inequality, we have:
$$
\mathbb {E} [ u (V _ {\varphi} (x)) ] \leq \mathbb {E} [ u (\pi (x) \cdot | \widehat {\tau} (x) - (Y (1) - \widehat {\mu} _ {0} (x)) | + (1 - \pi (x)) \cdot | \widehat {\tau} (x) - (\widehat {\mu} _ {1} (x) - Y (0)) |) ].
$$
Now consider a nuisance estimates of  $\widehat{\mu}_0 = 0$  and  $\widehat{\mu}_1 = 0$ , a data distribution for which  $Y(0) > 0$  and  $Y(1) < 0$  almost surely, and a CATE estimator  $\widehat{\tau}(x) > 0$ ,  $\forall x \in \mathcal{X}$ , then we have
$$
\begin{array}{l} \mathbb {E} [ u (V _ {\varphi} (x)) ] \leq \mathbb {E} [ u (\pi (x) \cdot | \widehat {\tau} (x) - Y (1) | + (1 - \pi (x)) \cdot | \widehat {\tau} (x) + Y (0) |) ], \\ = \mathbb {E} [ u (\pi (x) \cdot (\widehat {\tau} (x) - Y (1)) + (1 - \pi (x)) \cdot (\widehat {\tau} (x) + Y (0))) ], \\ \leq \mathbb {E} [ u (\widehat {\tau} (x) - (Y (1) - Y (0))) ], \\ \leq \mathbb {E} [ u (| \widehat {\tau} (x) - (Y (1) - Y (0)) |) ] = \mathbb {E} [ u (V ^ {*} (x)) ], \tag {A.22} \\ \end{array}
$$
and hence  $V^{*}(x) \succeq_{(2)} V_{\varphi}(x)$  for all  $x$ , and by Lemma A1, we have  $V^{*} \succeq_{(2)} V_{\varphi}$ . The last inequality in (A.22) holds because  $u$  is non-decreasing and  $(1 - \pi(x))Y(1) - \pi(x)Y(0)$  is always non-positive when  $Y(0) > 0$  and  $Y(1) < 0$  (almost surely). Since SOSD is a necessary condition for FOSD, then it follows that  $V_{\varphi}$  does not have FOSD over  $V^{*}$  in this counter-example.
Now we show that there exists nuisance and CATE estimates and a data distribution for which  $V_{\varphi}$  does not have MCX over  $V^{*}$ . This follows directly from the previous counter-example. Since  $V^{*} \succeq_{(2)} V_{\varphi}$  implies that  $-V_{\varphi} \succeq_{mcx} -V^{*}$  (Theorem 4.A.1. in [28]), then  $V_{\varphi}$  does not have MCX over  $V^{*}$ .
Finally, we show that there exists nuisance and CATE estimates and a data distribution for which  $V^{*}$  does not have SOSD over  $V_{\varphi}$ . Note that  $\mathbb{E}[V^{*}] \geq \mathbb{E}[V_{\phi}]$  is a necessary condition for  $V^{*} \succeq_{(2)} V_{\varphi}$  (See Eq. (4.A.6) in [28]). To compare  $\mathbb{E}[V^{*}]$  and  $\mathbb{E}[V_{\phi}]$ , we rewrite  $\mathbb{E}[V_{\phi}(x)]$  as:
$$
\begin{array}{l} \mathbb {E} [ V _ {\phi} (x) ] = \mathbb {E} [ | \widehat {\tau} (x) - W (Y - \widehat {\mu_ {0}} (x)) - (1 - W) (\widehat {\mu_ {1}} (x) - Y) | ], \\ = \pi (x) \cdot \mathbb {E} [ | \widehat {\tau} (x) - (Y (1) - \widehat {\mu} _ {0} (x)) | ] + (1 - \pi (x)) \cdot \mathbb {E} [ | \widehat {\tau} (x) - (\widehat {\mu} _ {1} (x) - Y (0)) | ], \\ = \mathbb {E} [ \pi (x) \cdot | \hat {\tau} (x) - (Y (1) - \hat {\mu} _ {0} (x)) | + (1 - \pi (x)) \cdot | \hat {\tau} (x) - (\hat {\mu} _ {1} (x) - Y (0)) | ], \\ \geq \mathbb {E} [ | \widehat {\tau} (x) - \pi (x) (Y (1) - \widehat {\mu_ {0}} (x)) - (1 - \pi (x)) (\widehat {\mu_ {1}} (x) - Y (0)) | ], \\ = \mathbb {E} [ | \widehat {\tau} (x) - (Y (1) - Y (0)) + (1 - \pi (x)) (Y (1) - \widehat {\mu} _ {1} (x)) + \pi (x) (\widehat {\mu} _ {0} (x) - Y (0)) | ]. \\ \end{array}
$$
For nuisance estimates that satisfy  $\widehat{\mu}_1(x) < Y(1)$  and  $\widehat{\mu}_0(x) > Y(0)$  almost surely, it follows that  $(1 - \pi (x))(Y(1) - \widehat{\mu}_1(x)) + \pi (x)(\widehat{\mu}_0(x) - Y(0))$  is always positive and hence we have:
$$
\begin{array}{l} \mathbb {E} \left[ V _ {\phi} (x) \right] \geq \mathbb {E} \left[ | \widehat {\tau} (x) - (Y (1) - Y (0)) + (1 - \pi (x)) (Y (1) - \widehat {\mu} _ {1} (x)) + \pi (x) (\widehat {\mu} _ {0} (x) - Y (0)) | \right], \\ \geq \mathbb {E} [ | \widehat {\tau} (x) - (Y (1) - Y (0)) | ] = \mathbb {E} [ V ^ {*} (x) ], \tag {A.23} \\ \end{array}
$$
which implies that  $V^{*}$  does not have SOSD over  $V_{\varphi}$ .
Proof of statement (ii):
From Table 1, the pseudo-outcomes for the IPW- and DR-learners can be written as:
$$
V _ {\varphi} (x) = W \left| \hat {\tau} (x) - Y _ {\pi} (x) \right| + (1 - W) \left| \hat {\tau} (x) - Y _ {1 - \pi} (x) \right|, \tag {A.24}
$$
where  $Y_{\pi}$  and  $Y_{1 - \pi}$  are defined as follows:
$$
\begin{array}{l} Y _ {\pi} (x) = \left\{ \begin{array}{l l} \frac {1}{\pi (x)}   Y (1), & \text {f o r I P W - l e a r n e r}, \\ \frac {1}{\pi (x)}   (Y (1) - \widehat {\mu} _ {1} (x)) + (\widehat {\mu} _ {1} (x) - \widehat {\mu} _ {0} (x)), & \text {f o r D R - l e a r n e r , a n d} \end{array} \right. \\ Y _ {1 - \pi} (x) = \left\{ \begin{array}{l l} \frac {- 1}{1 - \pi (x)} Y (0), & \text {f o r I P W - l e a r n e r}, \\ \frac {- 1}{1 - \pi (x)} \left(Y (0) - \widehat {\mu} _ {0} (x)\right) + \left(\widehat {\mu} _ {1} (x) - \widehat {\mu} _ {0} (x)\right), & \text {f o r D R - l e a r n e r}. \end{array} \right. \tag {A.25} \\ \end{array}
$$
Note that, following the definition in (A.25), the following holds for both the IPW- and DR-learners:
$$
\pi (x) Y _ {\pi} (x) + (1 - \pi (x)) Y _ {1 - \pi} (x) = Y (1) - Y (0). \tag {A.26}
$$
Hence, applying (A.26) to the definition of the oracle score, we can write  $V^{*}(x)$  as follows:
$$
\begin{array}{l} V ^ {*} (x) = | \widehat {\tau} (x) - (Y (1) - Y (0)) | = | \widehat {\tau} (x) - \pi (x) Y _ {\pi} (x) - (1 - \pi (x)) Y _ {1 - \pi} (x) |, \\ = | \pi (x) (\widehat {\tau} (x) - Y _ {\pi} (x)) + (1 - \pi (x)) (\widehat {\tau} (x) - Y _ {1 - \pi} (x)) |. \tag {A.27} \\ \end{array}
$$
From the expressions in (A.24) and (A.27), we can see that  $V_{\varphi}(x)$  is a mixture of the absolute values of the two random variables  $(\widehat{\tau}(x) - Y_{\pi}(x))$  and  $(\widehat{\tau}(x) - Y_{1 - \pi}(x))$  with mixing proportions  $\pi(x)$  and  $1 - \pi(x)$ , and  $V^{*}(x)$  is the absolute value of the convex combination of the same random variables,  $(\widehat{\tau}(x) - Y_{\pi}(x))$  and  $(\widehat{\tau}(x) - Y_{1 - \pi}(x))$ , with weights  $\pi(x)$  and  $1 - \pi(x)$ . Applying Lemma A2, it follows that  $V_{\varphi}(x) \succeq_{mcx} V^{*}(x), \forall x \in \mathcal{X}$ , and from Lemma A1 we have that  $V_{\varphi} \succeq_{mcx} V^{*}$ .
Stochastic orders for the signed distance conformity score:
We have shown that for the absolute residual conformity score,  $V_{\varphi}(\widehat{\tau}) = |\widehat{\tau}(X) - \widetilde{Y}_{\varphi}|$ , the DR- and IPW-learners guarantee  $V_{\varphi} \succeq_{mcx} V^{*}$  irrespective of the data distribution and underlying models. We now study the stochastic orders achieved by the signed distance conformity score proposed in [32], defined as:  $V_{\varphi}(\widehat{\tau}) = \widehat{\tau}(X) - \widetilde{Y}_{\varphi}$ . Let  $u$  be a non-decreasing concave function, then we have:
$$
\begin{array}{l} \mathbb {E} [ u (V _ {\varphi} (\widehat {\tau})) ] = \mathbb {E} [ u (\widehat {\tau} (X) - \widetilde {Y} _ {\varphi}) ] \\ = \pi \mathbb {E} [ u (\widehat {\tau} (X) - \widetilde {Y} _ {\pi}) ] + (1 - \pi) \mathbb {E} [ u (\widehat {\tau} (X) - \widetilde {Y} _ {1 - \pi}) ] \\ \leq \mathbb {E} [ u (\pi \widehat {\tau} (X) - \pi \widetilde {Y} _ {\pi} + (1 - \pi) \widehat {\tau} (X) - (1 - \pi) \widetilde {Y} _ {1 - \pi}) ] \\ = \mathbb {E} [ u (\hat {\tau} (X) - (Y (1) - Y (0))) ] = \mathbb {E} [ u (V ^ {*} (\hat {\tau})) ], \tag {A.28} \\ \end{array}
$$
where the inequality follows from the concavity of  $u(.)$  and the last equality follows from (A.26). From Definition 1, (A.28) implies that  $V^{*} \succeq_{(2)} V_{\varphi}$  (Table 2). Note, however, that the construction of predictive intervals with the signed error distance does not follow (5) since the signed distance score does not sort absolute errors from largest to smallest, but it sorts conformity scores from maximum positive error to maximum negative error. Hence, Theorem 1 does not apply to the signed distance error and the SOSD condition  $V^{*} \succeq_{(2)} V_{\varphi}$  does not guarantee coverage.
Another more commonly used variant of the signed distance score was proposed in [25]. Given a quantile regression model  $\left[\widehat{\tau}_l,\widehat{\tau}_h\right]$ , [25] uses a variant of the signed error score defined as:
$$
V _ {\varphi} (\widehat {\tau}) = \max  \left\{\widehat {\tau} _ {l} (X) - \widetilde {Y} _ {\varphi}, \widetilde {Y} _ {\varphi} - \widehat {\tau} _ {h} (X) \right\}. \tag {A.29}
$$
The construction of the predictive intervals in [25] follows (5). Note that  $\max \{x,y\}$  can be written as:
$$
\max  \{x, y \} = \frac {x + y}{2} + \frac {\left| x - y \right|}{2}. \tag {A.30}
$$
Applying (A.30) to (A.29), we can rewrite the conformity scores as:
$$
\begin{array}{l} V _ {\varphi} (\widehat {\tau}) = \frac {\widehat {\tau} _ {l} - \widehat {\tau} _ {h}}{2} + \frac {\left| \widehat {\tau} _ {l} + \widehat {\tau} _ {h} - 2 \widetilde {Y} _ {\varphi} \right|}{2}, \\ = \frac {\widehat {\tau} _ {l} - \widehat {\tau} _ {h}}{2} + \left| \frac {\widehat {\tau} _ {l} + \widehat {\tau} _ {h}}{2} - \widetilde {Y} _ {\varphi} \right|. \tag {A.31} \\ \end{array}
$$
Similarly, the oracle scores can be written as:
$$
V ^ {*} (\widehat {\tau}) = \frac {\widehat {\tau} _ {l} - \widehat {\tau} _ {h}}{2} + \left| \frac {\widehat {\tau} _ {l} + \widehat {\tau} _ {h}}{2} - (Y (1) - Y (0)) \right|. \tag {A.32}
$$
Let  $\delta \tau_{-} = \frac{\widehat{\tau}_{l} - \widehat{\tau}_{h}}{2}$ ,  $\delta \tau_{+} = \frac{\widehat{\tau}_{l} + \widehat{\tau}_{h}}{2}$ , and let  $u$  be a non-decreasing convex function, then we have:
$$
\begin{array}{l} \mathbb {E} [ u (V _ {\varphi} (\widehat {\tau})) ] = \mathbb {E} \left[ u \left(\delta \tau_ {-} + | \delta \tau_ {+} - \widetilde {Y} _ {\varphi} |\right) \right] \\ = \pi \mathbb {E} [ u (\delta \tau_ {-} + | \delta \tau_ {+} - \widetilde {Y} _ {\pi} |) ] + (1 - \pi) \mathbb {E} [ u (\delta \tau_ {-} + | \delta \tau_ {+} - \widetilde {Y} _ {1 - \pi} |) ] \\ \geq \mathbb {E} [ u (\pi \delta \tau_ {-} + \pi | \delta \tau_ {+} - \widetilde {Y} _ {\pi} | + (1 - \pi) \delta \tau_ {-} + (1 - \pi) | \delta \tau_ {+} - \widetilde {Y} _ {1 - \pi}) ] \\ \geq \mathbb {E} [ u (\delta \tau_ {-} + | \delta \tau_ {+} - \pi \tilde {Y} _ {\pi} - (1 - \pi) \tilde {Y} _ {1 - \pi}) ] \\ = \mathbb {E} [ u (\delta \tau_ {-} + | \delta \tau_ {+} - (Y (1) - Y (0)) |) ] = \mathbb {E} [ u (V ^ {*} (\hat {\tau})) ], \tag {A.33} \\ \end{array}
$$
where the first inequality follows from the convexity of  $u$ , the second inequality is an application of the triangle inequality, and the last equality follows from (A.26). Following Definition 2, (A.33) implies that  $V_{\varphi} \succeq_{mcx} V^{*}$  when the base models apply quantile regression and the conformity scores follow the signed distance score in [25]. Hence, the coverage guarantee in Theorem 1 applies to conformal meta-learners based on conditional quantile regression.
Appendix B: Literature Review
In this Section, we provide a comprehensive overview of the relevant literature. We categorize the previous literature into three distinct strands: (1) Bayesian methods for modeling individualized causal effects, (2) frequentist methods for inference on parameters pertaining to individualized causal effects, and (3) conformal methods for predictive inference of ITEs.
(1) Bayesian methods. Predictive inference on ITEs has been traditionally conducted using Bayesian methods. These methods place a prior distribution over the nuisance parameters  $\mu_0$  and  $\mu_{1}$ , and then estimate the CATE function through the posterior distribution computed conditional on the observational dataset  $\mathcal{D} = \{(X_i,W_i,Y_i)\}_{i}$ . More formally, Bayesian procedures operate as follows:
$$
Y (w) = \mu_ {w} + \epsilon , \mu_ {0} \sim \Pi_ {0}, \mu_ {1} \sim \Pi_ {1}, \tag {A.34}
$$
where  $\Pi_0$  and  $\Pi_1$  are priors over function spaces, and  $\epsilon \sim \mathcal{N}(0,\sigma^2)$ . Given the dataset  $\mathcal{D}$ , the posterior distributions over the two nuisance functions are used to estimate CATE as follows:
$$
\widehat {\tau} (x) = \mathbb {E} _ {\mu_ {1} \sim \Pi_ {1} | \mathcal {D}} [ \mu_ {1} (x) ] - \mathbb {E} _ {\mu_ {0} \sim \Pi_ {0} | \mathcal {D}} [ \mu_ {0} (x) ]. \tag {A.35}
$$
Furthermore, the posterior distributions  $\Pi_0|\mathcal{D}$  and  $\Pi_1|\mathcal{D}$  can be used to construct predictive intervals on ITEs through the posterior credible intervals of  $P_{\Pi_0}(\mu_1(x)|\mathcal{D})$  and  $P_{\Pi_1}(\mu_1(x)|\mathcal{D})$ , i.e.,
$$
P _ {\mu_ {0}, \mu_ {1} \sim \Pi_ {0}, \Pi_ {1} | \mathcal {D}} (Y (1) - Y (0) \in \widehat {C} (X) \mid X = x) = 1 - \alpha . \tag {A.36}
$$