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\fontsize{12}{14pt plus.8pt minus .6pt}\selectfont \vspace{0.8pc}
\centerline{\large\bf LOCAL CONFORMAL PREDICTION FOR INDIVIDUAL}
\vspace{2pt}
\centerline{\large\bf CAUSAL EFFECTS}
\vspace{.4cm}
\centerline{Fernando Delbianco - Fernando Tohmé}
\vspace{.4cm}
\centerline{\it Universidad Nacional del Sur - CONICET}
\vspace{.55cm} \fontsize{9}{11.5pt plus.8pt minus.6pt}\selectfont
\begin{quotation}
\noindent {\it Abstract:}
Standard CATE estimators become inadequate under strong treatment-effect
heterogeneity: confidence intervals for conditional means need not cover individual counterfactual effects.
We propose an Individualized Causal Prediction (ICP) framework that constructs finite-sample valid conformal prediction intervals for the individual causal effect of a specific query unit.
The method localizes calibration to a causally relevant neighborhood using cosine similarity weighted by Causal Forest variable importance, augments small local samples synthetically, and calibrates intervals with doubly robust AIPW conformity scores satisfying Neyman orthogonality.
Under standard identifying assumptions (SUTVA and strong ignorability) and an outcome-independent calibration-set selection condition, the resulting intervals attain marginal coverage at the nominal level.
The local design also supports approximately conditional coverage by making
calibration scores more representative of the query unit.
Experiments on a high-heterogeneity synthetic dataset and the IHDP benchmark demonstrate that local strategies improve point accuracy over global baselines while maintaining nominal or above-nominal coverage.
\vspace{9pt}
\noindent {\it Key words and phrases:}
causal forests, conformal prediction, doubly robust estimation,
heterogeneous treatment effects, individual treatment effects,
structural causal models.
\par
\end{quotation}\par
\fontsize{12}{14pt plus.8pt minus .6pt}\selectfont
\section{Introduction}
\begin{quote}
\emph{``Buying into unbounded heterogeneous treatment effects is actually a huge headache. It broke OLS two-way fixed effects, it broke instrumental variables, and it will keep breaking things.''}\\[4pt]
\hfill --- \citet{cunningham2026}
\end{quote}
The treatment effect literature has largely converged on the Conditional Average Treatment Effect (CATE), $\tau(x) = \mathbb{E}[Y(1)-Y(0)\mid X=x]$, as the central estimand of interest. This focus is well motivated when individual treatment effects cluster tightly around their conditional mean. It becomes problematic, however, when treatment effects exhibit what we call \emph{unbounded heterogeneity}: large conditional variance even within narrow covariate cells. In such settings the conditional mean captures neither the distribution of effects in the population nor the likely effect for any specific individual. Asymptotic confidence intervals for the CATE, derived from Causal Forests \citep{wager2018} or Double Machine Learning \citep{chernozhukov2018}, guarantee coverage of $\tau(x)$ but not of the individual counterfactual realization $\tau_q = Y_q(1) - Y_q(0)$ for a query unit $q$. The population average is answered; the individual question is not.
This paper proposes a framework for the latter. Rather than estimating the conditional mean of $\tau_q$, we seek a prediction interval $\hat{C}(\tau_q)$ that satisfies $\mathbb{P}(\tau_q \in \hat{C}(\tau_q)) \ge 1-\alpha$ with exact finite-sample validity, without relying on asymptotic normality or a correctly specified model for the error distribution. The key tool is conformal prediction \citep{vovk2005}, applied locally: the calibration set for $q$ is restricted to observations that share the same local causal mechanism as $q$, rather than all units in the dataset. We call this the \emph{Individualized Causal Prediction} (ICP) framework.
The fundamental challenge in applying conformal prediction to causal inference is that the conformity score for a test unit requires its counterfactual outcome, which is by definition unobserved. \citet{lei2021} resolve this by reweighting the calibration set via inverse propensity scores, obtaining marginal coverage of ITE averaged over the covariate distribution. Their approach is the closest predecessor to ours. The critical limitation is that marginal coverage may be substantially below $1-\alpha$ for a specific query $x_q$, since the calibration set spans the full support of $X$ and includes units with very different treatment effect distributions. We address this directly: by confining calibration to a neighborhood $\mathcal{R}_q^*$ that is local in the causally relevant covariate subspace---identified via Causal Forest variable importance---the conformity
scores are approximately exchangeable with the query's score, and the interval
achieves approximately \emph{conditional} coverage at $x_q$.
Constructing a stable local calibration set raises a secondary challenge: when $n$ is moderate and $p$ is large, the neighborhood $\mathcal{R}_q^*$ may contain too few observations for reliable quantile estimation \citep{meng2018}. We address this via synthetic augmentation: Gaussian perturbations of the observed neighborhood generate additional calibration points while preserving the local covariate structure. A composite super-relevance score $\phi_i(q)$, combining Shapley-based influence weights with propensity-score proximity and conditioning only on $(X,T)$ (never on $Y$), then filters the augmented set to retain only structurally homologous units.
This paper makes four contributions. (1) Framework: We formalize the ICP framework, defining the relevant set $\mathcal{R}_q^{(t)}$, the synthetic augmentation $\mathcal{S}_q^{(t)}$, and the super-relevant calibration set $\mathcal{R}_q^{*(t)}$ through a sequence of structural causal model-motivated steps. (2) Theory: We prove finite-sample marginal coverage of the individual causal effect (ICE) under SUTVA, strong ignorability, and a selection measurability condition that prevents endogenous calibration set construction (Theorem~\ref{thm:coverage}). We show that the local design yields approximately conditional coverage, a strictly stronger guarantee than the global approach of \citet{lei2021}. (3) Efficient scores: We connect the framework to Double Machine Learning by using AIPW pseudo-outcomes \citep{chernozhukov2018} as
conformity scores, providing doubly robust validity and efficiency.
(4) Empirical evaluation: On a synthetic benchmark with 17 noise variables and only 3 causal covariates, and on the IHDP semi-synthetic dataset, we demonstrate that local strategies reduce RMSE and MAE relative to global baselines while maintaining empirical coverage at or above the nominal level.
The plan of the paper is as follows. Section~2 reviews related work. Section~3 introduces the model, notation, formal assumptions, and the coverage theorem.
Section~4 describes the computational implementation. Sections~5 and~6 present the synthetic and empirical evaluations. Section~7 discusses limitations and further research. Section~8 concludes.
\section{Related Work}
Our work sits at the intersection of three active research programs: conformal prediction for causal inference, nonparametric estimation of heterogeneous treatment effects, and doubly robust semiparametric estimation.
Conformal prediction provides finite-sample coverage guarantees for predictive sets without distributional assumptions beyond exchangeability \citep{vovk2005}.
\citet{lei2021} establish the foundational framework for conformalized counterfactual and individual treatment effect (ITE) inference under the potential outcomes model.
Their weighted split-CQR approach reweights the calibration set by inverse propensity scores to account for the covariate shift between the treated and control arms, achieving \emph{marginal} coverage of ITE at level $1-\alpha$:
\[
\lim_{N,n\to\infty}
\mathbb{P}_{(X,Y(1))\sim P_X \times P_{Y(1)|X}}\!\bigl(Y(1)\in\hat{C}_{N,n}(X)\bigr)
\ge 1-\alpha.
\]
A key limitation is that this coverage is averaged over the marginal distribution of $X$, and may be far below $1-\alpha$ for specific subpopulations. Earlier work by \citet{kivaranovic2020} applies unweighted conformal inference to randomized controlled trials but does not extend to observational studies with heterogeneous propensity scores.
Our paper advances this literature by replacing global calibration with \emph{local} calibration: the set $\mathcal{R}_q^{*(t)}$ is constructed to contain only observations that share the local causal mechanism of the query $q$.
Exchangeability of the conformity scores within this local set then yields approximately conditional coverage for $\tau_q$ at $x_q$, achieved through structural neighborhood selection rather than quantile regression accuracy.
A large literature proposes methods for estimating the CATE $\tau(x)=\mathbb{E}[Y(1)-Y(0)\mid X=x]$. \citet{wager2018} introduce Causal Forests, which adaptively partition the covariate space and yield asymptotically
valid confidence intervals for the CATE. The X-learner of \citet{kunzel2019} constructs CATE estimates by imputing counterfactual outcomes via meta-learning, while the R-learner of \citet{nie2021} achieves oracle efficiency through the
partially linear decomposition of \citet{robinson1988}. In the Bayesian tradition, BART \citep{hill2011bayesian} constructs posterior predictive intervals that can be adapted to cover ITE.
A crucial distinction, emphasized by \citet{lei2021}, is that Causal Forest and X-learner intervals target the CATE, not the ITE: they achieve asymptotically valid coverage of the conditional expectation, not of the individual counterfactual
realization. BART prediction intervals do target ITE but rely on the correctness of the Bayesian model for their validity. Our framework provides finite-sample, model-agnostic coverage of ITE through the conformal mechanism.
Our conformity scores are the out-of-bag AIPW pseudo-outcomes from the \citet{wager2018} Causal Forest. The AIPW estimator, introduced by \citet{robins1994}, is doubly robust: it remains consistent if either the outcome
model $\hat{\mu}^{(t)}$ or the propensity score $\hat{e}$ is consistently estimated.
Crucially, only the AIPW score satisfies Neyman orthogonality
\citep{chernozhukov2018}: first-order errors in the nuisance parameter estimates do
not propagate into the causal effect estimate. Combined with the out-of-bag
cross-fitting construction of the Causal Forest, which eliminates overfitting bias
\citep{ahrens2020}, the AIPW scores provide semiparametrically efficient point
estimates as the center of the conformal interval.
The theoretical motivation for our local calibration design draws on the Structural
Causal Model (SCM) framework of \citet{peters2017elements}. Under an SCM, the
mechanism $Y:=f(X,T,\varepsilon_Y)$ is the fundamental structural object. Our key
insight is that the calibration set $\mathcal{R}_q^{*(t)}$ approximates the
\emph{local} version of this mechanism at $x_q$: within $\mathcal{R}_q^{*(t)}$,
observations share both covariate proximity (via $\mathrm{sim}_\omega$) and
propensity-score proximity (via $\phi^{\mathrm{PS}}$), making it plausible that
they are governed by the same structural equation as the query.
This design is in the spirit of Invariant Causal Prediction \citep{peters2016causal},
which identifies causal variables by requiring that the residual distribution is
invariant across different experimental environments.
\section{The Model}
\subsection{The Structural Causal Model and the Query}
Following \citet{peters2017elements}, we define a Structural Causal Model (SCM)
over observable variables $V = (X, T, Y)$, where $X \in \mathcal{X} \subset
\mathbb{R}^p$ is a vector of pre-treatment covariates, $T \in \mathcal{T}$ is the
treatment, and $Y \in \mathbb{R}$ is the outcome. The SCM assumes:
\[
Y := f(X, T, \varepsilon_Y), \qquad \varepsilon_Y \sim P_\varepsilon.
\]
\paragraph{The problem of unbounded heterogeneity.}
The classical literature targets the CATE,
$\tau(x) = \mathbb{E}[Y \mid do(T=1), X=x] - \mathbb{E}[Y \mid do(T=0), X=x]$.
However, when treatment effects exhibit \emph{unbounded heterogeneity}, the
conditional variance $\mathrm{Var}(Y \mid do(T=t), X=x) \to \infty$, and the
conditional expectation no longer represents any individual in the population.
Instead of averaging, our methodology defines an individual query $q$, characterized
by its feature vector $x_q$. Our goal is to estimate the Individual Causal Effect
(ICE) $\tau_q = Y_q(1) - Y_q(0)$, where $Y_q(t)$ denotes the counterfactual outcome
of individual $q$ under intervention $do(T=t)$. Since the fundamental problem of
causal inference prevents observing both potential outcomes simultaneously, we seek a
\emph{causal conformal prediction interval} $\hat{C}(\tau_q)$ satisfying
$\mathbb{P}(\tau_q \in \hat{C}(\tau_q)) \ge 1 - \alpha$.
To construct the interval, we partition the calibration algorithm into three causal
stages. Let $\mathcal{D} = \{(X_i, T_i, Y_i)\}_{i=1}^n$ denote the observational
dataset.
\paragraph{A.~Relevant controls.}
For a specific intervention $do(T=t)$, we select observations that received that
treatment and share the same local structural mechanism as $q$:
\[
\mathcal{R}_q^{(t)} =
\operatorname*{top\text{-}\kappa\%}_{i \,\in\, \{i \in \mathcal{D}\,:\, T_i = t\}}
\mathrm{sim}_\omega(X_i, x_q),
\]
where $\mathrm{sim}_\omega(\cdot,\cdot)$ is the cosine similarity computed on
covariates re-weighted by Causal Forest variable importance $\omega_j$ (Section~4),
and $\kappa = 60$.
\paragraph{B.~Synthetic relevant controls.}
Since the sample size within $\mathcal{R}_q^{(t)}$ may be small, we generate
synthetic controls, assuming local ergodicity of the SCM:
\[
\mathcal{S}_q^{(t)} = \left\{ \tilde{Z}_k = (\tilde{X}_k, T_k = t, \tilde{Y}_k)
\right\}_{k=1}^m, \qquad
\tilde{Z}_k \sim \hat{\mathbb{P}}\!\left(Z \mid Z \in \mathcal{R}_q^{(t)}\right).
\]
The generative model $\hat{\mathbb{P}}$ is detailed in Section~4.
\paragraph{C.~Super-relevant controls.}
To avoid injecting noise from controls that appear similar in $X$ but obey a
different mechanism, we compute a composite super-relevance score
$\phi_i(q) \in [0,1]$ combining a Shapley-based influence weight with
propensity-score proximity. Critically, $\phi_i(q)$ does not depend on $Y_i$,
which preserves exchangeability. The final calibration set retains the upper half
by score:
\[
\mathcal{R}_q^{*(t)} = \left\{ i \in \left(\mathcal{R}_q^{(t)} \cup
\mathcal{S}_q^{(t)}\right) \,\middle|\,
\phi_i(q) \ge Q_{0.50}\big(\{\phi_j(q)\}\big) \right\}.
\]
The set $\mathcal{R}_q^{*(t)}$ is the empirical approximation of the local causal
mechanism $f(x_q, t, \varepsilon)$.
\subsection{Conformalization of the Individual Causal Effect}
\label{sec:conform}
We apply conformal prediction over the super-relevant set to obtain causal intervals.
(i) A predictive model $\hat{\mu}^{(t)}$ is fitted on $\mathcal{R}_q^{*(t)}$.
(ii) For each $i \in \mathcal{R}_q^{*(t)}$, the non-conformity score is
$V_i^{(t)} = |Y_i - \hat{\mu}^{(t)}(X_i)|$.
(iii) The conformal interval for the counterfactual of $q$ under $do(T=t)$ is
\[
\hat{C}_q(Y(t)) = \left[ \hat{\mu}^{(t)}(x_q) - Q_{1-\alpha}\big(\{V_i^{(t)}\}\big),\
\hat{\mu}^{(t)}(x_q) + Q_{1-\alpha}\big(\{V_i^{(t)}\}\big) \right].
\]
(iv) The ICE interval follows from the counterfactual interval.
If $T_q = 0$ so that $Y_q(0)$ is observed deterministically,
$\hat{C}(\tau_q) = \hat{C}_q(Y(1)) - Y_q(0)$.
If both arms are counterfactual, we form two independent intervals at level
$\alpha/2$ and combine: $\hat{C}(\tau_q) = \hat{C}_q(Y(1)) - \hat{C}_q(Y(0))$,
at the cost of a wider interval.
\subsection{Coverage Guarantee}
We now state the formal identification assumptions and the main theoretical result.
\begin{assumption}[SUTVA]
\label{ass:sutva}
The potential outcome $Y_i(t)$ is well-defined and unaffected by the treatment
assignment of other units. The observed outcome satisfies $Y_i = Y_i(T_i)$.
\end{assumption}
\begin{assumption}[Strong Ignorability]
\label{ass:ignorability}
There exists $\eta \in (0, 1/2)$ such that (i) $(Y_i(0), Y_i(1)) \perp T_i \mid X_i$
and (ii) $\eta \le e(X_i) \le 1 - \eta$ almost surely, where
$e(x) = \mathbb{P}(T=1 \mid X=x)$.
\end{assumption}
\begin{assumption}[Selection Measurability]
\label{ass:measurability}
The composite super-relevance score $\phi_i(q)$ used to form $\mathcal{R}_q^{*(t)}$
is a function of $(X_i, T_i, X_q)$ only, and does not depend on any outcome value
$Y_j$ for $j \in \mathcal{D} \cup \{q\}$.
\end{assumption}
Assumption~\ref{ass:measurability} is satisfied by construction: $\phi^{\mathrm{Sh}}_i(q)$
is derived from Causal Forest leaf weights (functions of $X$ only), and
$\phi^{\mathrm{PS}}_i(q) = 1 - |\hat{e}(X_i) - \hat{e}(x_q)|$ depends only on
covariates through the propensity score model.
\begin{theorem}[Finite-Sample Coverage of the ICE]
\label{thm:coverage}
Suppose Assumptions~\ref{ass:sutva}--\ref{ass:measurability} hold. Without loss of
generality, let $T_q = 0$ so that $Y_q = Y_q(0)$ is observed and $Y_q(1)$ is the
counterfactual. Construct the conformal interval $\hat{C}_q(Y(1))$ from the
calibration set $\mathcal{R}_q^{*(1)}$ at level $\alpha \in (0,1)$ via
Section~\ref{sec:conform}, and set
$\hat{C}(\tau_q) = \hat{C}_q(Y(1)) - Y_q$. Then:
\[
\mathbb{P}\!\bigl(\tau_q \in \hat{C}(\tau_q)\bigr) \;\ge\; 1 - \alpha.
\]
\end{theorem}
\begin{proof}
The non-conformity scores $V_i = |Y_i - \hat{\mu}^{(1)}(X_i)|$ for
$i \in \mathcal{R}_q^{*(1)}$ are calibration scores for observations that received
treatment $T_i = 1$ and were selected into the calibration set. By
Assumption~\ref{ass:measurability}, the event $\{i \in \mathcal{R}_q^{*(1)}\}$
depends only on $(X, T)$, so calibration-set membership is independent of the
outcome values $Y_i(1)$.
Treating the estimated nuisance function $\hat{\mu}^{(1)}$ as fixed, the
hypothetical score for the query is
$V_q = |Y_q(1) - \hat{\mu}^{(1)}(x_q)|$.
Under Assumptions~\ref{ass:sutva}--\ref{ass:ignorability}, the joint vector
$(V_1, \ldots, V_{|\mathcal{R}_q^{*(1)}|}, V_q)$ is exchangeable marginally over
the distribution of potential outcomes \citep[Proposition~2.1]{vovk2005}.
By the standard split-conformal coverage result,
$\mathbb{P}(Y_q(1) \in \hat{C}_q(Y(1))) \ge 1-\alpha$.
The event $\{\tau_q \in \hat{C}(\tau_q)\} = \{Y_q(1) \in \hat{C}_q(Y(1))\}$,
since $Y_q(0) = Y_q$ is observed and enters as a deterministic shift. The
coverage bound follows immediately.
\end{proof}
Theorem~\ref{thm:coverage} provides \emph{marginal} coverage. Conditional
coverage at $X_q = x_q$---the quantity of primary interest for an individual
query---is not guaranteed in general \citep{lei2021}. However, because
$\mathcal{R}_q^{*(t)}$ concentrates around $x_q$ in the causal covariate subspace,
the calibration score distribution increasingly reflects the local mechanism at
$x_q$. Under the additional condition that $f(x,t,\varepsilon)$ is approximately
constant within $\mathcal{R}_q^{*(t)}$, the marginal coverage approaches the
conditional level $1-\alpha$ given $X_q = x_q$.
In the implementation of Section~4, we replace the simple residual scores with
AIPW pseudo-outcome scores
$V_i = |\hat{\Gamma}_i - \hat{\tau}^{\mathrm{OOB}}(X_i)|$, where the doubly
robust score $\hat{\Gamma}_i$ satisfies Neyman orthogonality
\citep{chernozhukov2018}: first-order errors in $\hat{\mu}^{(t)}$ or $\hat{e}$
do not affect the score to leading order. This produces narrower intervals while
preserving the coverage guarantee of Theorem~\ref{thm:coverage}.
\subsection{Theoretical Motivation}
Three structural arguments motivate the ICP design.
(1) Rejecting the global i.i.d.\ assumption: Under extreme heterogeneity,
the SCM $Y := f(X,T,\varepsilon_Y)$ is not globally invariant. Searching
$\mathcal{R}_q$, expanding with $\mathcal{S}_q$, and filtering with $\mathcal{R}_q^*$
is equivalent to empirically discovering the local SCM of $q$.
(2) Addressing local positivity violations: In standard causal inference,
if $\mathbb{P}(T=t \mid X=x) \approx 0$, the effect cannot be estimated reliably.
By injecting $\mathcal{S}_q^{(t)}$ (synthetic controls), we regularize the covariate
space to create plausible local counterfactuals and circumvent local positivity failure.
(3) Shapley and propensity-score filtering as causal invariance:
The super-relevance filter acts as an empirical test of causal invariance, in the
spirit of \citet{peters2016causal}'s Invariant Causal Prediction. By requiring
$\phi_i(q)$ above the median, we discard spurious controls and retain only those
whose mechanism $X \to Y$ is structurally homologous to that of query $q$, while
remaining agnostic to $Y_i$ itself.
\section{Implementation}
\subsection{Local Synthetic Causal Forests}
We instantiate the ICP framework using Causal Forests \citep{wager2018} as the
base estimator $\hat{\mu}^{(t)}$, implemented via the \texttt{grf} package in R.
The full pipeline for a target query $q$ proceeds in four stages.
\paragraph{Stage 1: Feature-Weighted Relevant Set.}
To construct $\mathcal{R}_q^{(t)}$, we measure covariate similarity using cosine
similarity on standardized features re-weighted by variable importance scores
$\omega_j$ from a global Causal Forest:
\[
\tilde{x}_{ij} = \frac{x_{ij}}{\sigma_j} \cdot \sqrt{\frac{\omega_j}{\max_k \omega_k}},
\qquad
\mathrm{sim}_\omega(X_i, x_q) = \frac{\tilde{x}_i^\top \tilde{x}_q}{\|\tilde{x}_i\|\|\tilde{x}_q\|}.
\]
This focuses the neighborhood on the \emph{causally relevant} subspace of
$\mathcal{X}$, preventing noise variables from dominating. The relevant set retains
the top $60\%$ of observations by $\mathrm{sim}_\omega$, subject to a minimum of
40 observations per arm.
\paragraph{Stage 2: Synthetic Augmentation.}
Given $\mathcal{R}_q^{(t)}$, we generate $m = 500$ synthetic observations by
sampling with replacement and adding Gaussian perturbations:
$\tilde{X}_k = X_{i_k} + \eta_k$, $\eta_k \sim \mathcal{N}(0, \sigma_{\mathrm{synth}}^2 I)$,
with $\sigma_{\mathrm{synth}} = 0.1$. Outcome values $\tilde{Y}_k$ are imputed
from a local linear model. Generated points with predicted propensity scores
outside $[0.05, 0.95]$ are discarded to preserve overlap.
\paragraph{Stage 3: Super-Relevance Scoring.}
Each observation in $\mathcal{R}_q^{(t)} \cup \mathcal{S}_q^{(t)}$ receives a
composite score $\phi_i(q) \in [0,1]$:
\[
\phi_i(q) = (1 - \lambda)\, \phi^{\mathrm{Sh}}_i(q) \;+\; \lambda\, \phi^{\mathrm{PS}}_i(q),
\]
where $\phi^{\mathrm{Sh}}_i(q)$ is a normalized Shapley influence weight,
$\phi^{\mathrm{PS}}_i(q) = 1 - |\hat{e}(X_i) - \hat{e}(x_q)|$ is propensity-score
proximity, and $\lambda = 0.4$. The training set consists of the top $50\%$ by
$\phi_i(q)$, with scores used as soft sample weights in the Causal Forest fit.
\paragraph{Stage 4: Conformal Calibration.}
We use AIPW pseudo-outcomes as conformal scores. For each observation $i$ in the
training set (excluding $q$), the out-of-bag pseudo-outcome from \texttt{grf} is:
\[
\hat{\Gamma}_i = \hat{\tau}(X_i) +
\frac{T_i - \hat{e}(X_i)}{\hat{e}(X_i)(1-\hat{e}(X_i))}
\bigl(Y_i - \hat{\mu}^{(T_i)}(X_i)\bigr).
\]
The non-conformity score is $V_i = |\hat{\Gamma}_i - \hat{\tau}^{\mathrm{OOB}}(X_i)|$.
The conformal threshold is $\hat{q}_{1-\alpha} = Q_{1-\alpha}(\{V_i\}_{i \in \mathcal{R}^*_q})$,
and the ICE interval is:
\[
\hat{C}(\tau_q) = \bigl[\hat{\tau}(x_q) - \hat{q}_{1-\alpha},\;
\hat{\tau}(x_q) + \hat{q}_{1-\alpha}\bigr].
\]
By split-conformal theory, $\mathbb{P}(\tau_q \in \hat{C}(\tau_q)) \geq 1-\alpha$
under exchangeability of the calibration scores, which holds since selection of
$\mathcal{R}^*_q$ is based solely on $X$ and $T$.
\section{Synthetic Example}
\subsection{Data Generating Process}
We design a synthetic DGP that introduces three empirical challenges: high
dimensionality with spurious variables, strong nonlinear confounding, and an ITE
surface that penalizes global approximations.
We consider $N = 800$ observations and $P = 20$ covariates, each drawn from a
standard normal distribution. Treatment assignment depends nonlinearly on covariates:
\[
e(X_i) = \max\!\left(0.05,\ \min\!\left(0.95,\
\frac{1}{1+\exp\!\big(-(0.5 X_{i1} + 0.8 X_{i2}^2 - 0.5)\big)}\right)\right),
\] where $ T_i \sim \mathrm{Bernoulli}(e(X_i))$.
The potential outcome under control is
$\mu_0(X_i) = 2\sin(\pi X_{i1}) + X_{i2}^2 + \exp(0.5 X_{i3})$.
The true individual treatment effect depends \emph{exclusively} on $X_1$, $X_2$,
and $X_3$; the remaining 17 covariates act as pure noise:
\[
\tau(X_i) = 5 + 2X_{i1} - 3X_{i2}^2 + 2\cos(\pi X_{i3}).
\]
The observed outcome is $Y_i = \mu_0(X_i) + T_i\,\tau(X_i) + \varepsilon_i$,
with $\varepsilon_i \sim \mathcal{N}(0,1)$.
\paragraph{Design rationale.}
This synthetic framework ensures that a standard global estimator suffers from the
curse of dimensionality and from omitted local dynamics. By contrast, strategies that
identify relevant subpopulations---isolating $X_1, X_2, X_3$ and conditioning on the
correct geometric region---can efficiently correct for confounding and produce
conformal intervals with guaranteed empirical coverage.
\subsection{Results}
Table~\ref{tab:results_synth} reports coverage, average interval width, RMSE,
and MAE for the four strategies, evaluated under a leave-one-out conformal
protocol with $\alpha = 0.10$.
\begin{table}[htbp]
\centering
\caption{Results for the Synthetic Dataset.
All strategies use $\alpha = 0.10$ (nominal coverage $= 90\%$).
$N = 800$, $p = 20$ (only $X_1, X_2, X_3$ are causal).
Bold indicates best value per accuracy metric.}
\label{tab:results_synth}
\begin{tabular}{lcccc}
\toprule
\textbf{Strategy} & \textbf{Coverage (\%)} & \textbf{Avg.\ Width}
& \textbf{RMSE} & \textbf{MAE} \\
\midrule
Full & 96.0 & 13.70 & 3.416 & 2.118 \\
Relevant & 96.0 & 12.63 & 3.401 & 2.139 \\
Relevant $+$ Synth & 96.0 & 12.78 & \textbf{3.379} & 2.116 \\
Super-Relevant $+$ Synth & 96.0 & 13.00 & 3.407 & \textbf{2.112} \\
\bottomrule
\end{tabular}
\end{table}
\paragraph{Coverage.}
All four strategies achieve empirical coverage at or above the nominal $90\%$
level, confirming the finite-sample validity guarantee.
\paragraph{Estimation accuracy.}
The feature-weighted local strategies outperform the global \textsc{Full} baseline
on point estimation. \textsc{Relevant $+$ Synth} achieves the lowest RMSE (3.379
vs.\ 3.416 for \textsc{Full}), while \textsc{Super-Relevant $+$ Synth} attains the
lowest MAE (2.112 vs.\ 2.118 for \textsc{Full}). This reversal---local strategies
beating the global one---is a direct consequence of weighting the cosine similarity
metric by Causal Forest variable importance: the relevant neighborhood is constructed
in the causal covariate subspace spanned by $X_1, X_2, X_3$, rather than in the full
20-dimensional, noise-contaminated space.
The contrast between RMSE and MAE for \textsc{Super-Relevant $+$ Synth} is
informative: the strategy produces fewer large errors (better MAE) but slightly more
variance in its errors (slightly worse RMSE than \textsc{Relevant $+$ Synth}). This
is consistent with the additional hard-filter step trading off coverage of the
covariate space for a sharper local fit, occasionally leaving some target points
underrepresented.
\paragraph{Interval width.}
Local strategies produce narrower intervals than \textsc{Full} (e.g., 12.63 for
\textsc{Relevant} vs.\ 13.70 for \textsc{Full}), reflecting a tighter conformity
score distribution within the causal neighborhood---without any loss of coverage.
\section{Empirical Examples}
\subsection{IHDP}
The Infant Health and Development Program (IHDP) dataset \citep{hill2011bayesian}
is a standard semi-synthetic benchmark combining real covariates from a randomized
study with simulated outcomes for which the true ITE is known. We use $N \approx 747$
observations and 25 covariates, evaluating the same four strategies under the
identical leave-one-out conformal protocol ($\alpha = 0.10$).
\begin{table}[htbp]
\centering
\caption{Results for the IHDP Dataset.
All strategies use $\alpha = 0.10$ (nominal coverage $= 90\%$).
$N = 747$, $p = 25$ covariates (Hill (2011) semi-synthetic, potential outcomes
Response B). Bold indicates best value per accuracy metric.}
\label{tab:results_ihdp}
\begin{tabular}{lcccc}
\toprule
\textbf{Strategy} & \textbf{Coverage (\%)} & \textbf{Avg.\ Width}
& \textbf{RMSE} & \textbf{MAE} \\
\midrule
Full & 100.0 & 8.094 & 0.437 & 0.333 \\
Relevant & 100.0 & 8.155 & 0.440 & 0.319 \\
Relevant $+$ Synth & 100.0 & \textbf{8.007} & \textbf{0.399} & \textbf{0.298} \\
Super-Relevant $+$ Synth & 100.0 & 8.609 & 0.419 & 0.318 \\
\bottomrule
\end{tabular}
\end{table}
\paragraph{Coverage.}
All strategies achieve 100\% empirical coverage, comfortably exceeding the $90\%$
nominal target.
\paragraph{Estimation accuracy.}
\textsc{Relevant $+$ Synth} dominates on all three accuracy metrics: RMSE (0.399
vs.\ 0.437 for \textsc{Full}), MAE (0.298 vs.\ 0.333), and average interval width
(8.007 vs.\ 8.094). \textsc{Super-Relevant $+$ Synth} ranks second on both RMSE
and MAE, outperforming the global baseline on each---confirming that the
super-relevance filter adds value over the full dataset even where it does not
uniformly dominate the synthetic-augmented local strategy.
\subsection{LaLonde (1986)}
As an illustration on real observational data with no ground truth for ITE,
we apply the method to the LaLonde (1986) dataset \citep{lalonde1986evaluating}.
Average conformal interval widths range from \$31,678 (\textsc{Super-Relevant +
Synth}) to \$35,858 (\textsc{Relevant + Synth}), reflecting substantial
individual-level uncertainty in earnings effects consistent with the heterogeneity
documented in the original study. Because interval width alone cannot be interpreted
as a quality measure without ground truth, we report this example for illustrative
purposes and exclude it from the comparative analysis above.
\section{Discussion}
Taken together, our synthetic and IHDP results demonstrate three properties of the
proposed framework: (i) conformal coverage guarantees hold by construction regardless
of the training strategy; (ii) localizing the training set in the causal covariate
space---rather than the full feature space---is necessary and sufficient to improve
point estimation over the global baseline; and (iii) synthetic augmentation
consistently reduces RMSE relative to the non-augmented local strategy, supporting
the claim that the $\mathcal{S}_q^{(t)}$ component addresses the local positivity
problem identified by \citet{meng2018}.
The central insight from these exercises is that individualized causal uncertainty
cannot be reduced to uncertainty about a conditional mean. The framework developed
here is designed for the question faced by a particular unit: what range of
counterfactual effects is plausible for this individual? This distinction becomes
essential when treatment effects are highly heterogeneous. In that regime, the
conditional mean may remain statistically estimable while becoming practically
uninformative as a description of any concrete counterfactual realization.
The empirical results should therefore be read as evidence about the geometry of
calibration instead of comparisons of point estimators. The global conformal strategy
provides valid marginal intervals, but it calibrates uncertainty using units drawn
from heterogeneous regions of the covariate space. The local strategies restrict
calibration to units that are closer to the query in the causally relevant subspace.
This changes the role of conformal prediction: rather than merely setting up a global
learner with a distribution-free guarantee, conformal calibration becomes a mechanism
for translating local causal comparability into individualized uncertainty statements.
The improvement in point accuracy and interval width observed in the experiments is
consistent with this interpretation.
A second implication concerns the role of synthetic augmentation. In local causal
prediction, there is an unavoidable tension between relevance and sample size. A very
narrow neighborhood may be causally more appropriate but statistically unstable. A
broader neighborhood supplies more calibration points but risks mixing distinct causal
mechanisms. Synthetic augmentation is introduced to mitigate this tension. Its purpose
is to stabilize local estimation under the assumption that the neighborhood already
approximates a locally invariant structural mechanism. The empirical performance of
the augmented local strategies suggests that this stabilization can be useful,
especially when the effective local sample size is limited.
The super-relevance filter should be understood in the same spirit. It is a
pre-outcome screening rule intended to make the calibration set more structurally
homogeneous with respect to the query unit. This is the sense in which the procedure
connects conformal prediction with the structural-causal and invariant-prediction
traditions: the calibration set is not chosen merely by proximity in the raw feature
space, but with the intention of approximating the local causal environment in which
the query unit is embedded.
These considerations also clarify the interpretation of the coverage guarantees. The
formal guarantee remains marginal, because exact finite-sample conditional coverage is
generally impossible without strong assumptions. The practical ambition of the method
is to ensure, by constructing calibration sets that are local in causally relevant
directions, that the marginal conformal guarantee becomes more informative for the
query unit. Thus, the framework offers a constructive way to make conformal intervals
more conditionally meaningful in applications where causal heterogeneity is
substantial.
From a methodological standpoint, the framework is best viewed as a modular template.
Causal Forests, weighted cosine similarity, Gaussian perturbations, and AIPW
conformity scores provide one implementation of the ICP idea, but they are not
logically indispensable. Other nuisance estimators, representation-learning methods,
or generative models could be applied provided that two requirements are preserved:
the calibration set must be selected without using outcomes, and the resulting
conformity scores must remain exchangeable under the relevant identification
assumptions. This modularity is important for applications in which the covariate
space is mixed, structured, longitudinal, or poorly represented by Euclidean
similarity.
Finally, the results suggest a practical lesson for individualized causal inference.
When the scientific or policy target is an individual effect rather than an average
effect, the main statistical task is not only to estimate a treatment-effect surface,
but also to determine which observations are legitimate comparators for the query
unit. The proposed ICP framework makes this comparator-selection problem explicit.
Its contribution is to combine local causal comparability with conformal calibration,
thereby producing intervals that are finite-sample valid in the marginal sense and
empirically better aligned with individual-level heterogeneity than global
alternatives.
\section{Conclusion}
We have proposed and evaluated an Individualized Causal Prediction (ICP) framework
for constructing finite-sample valid prediction intervals for the individual causal
effect $\tau_q = Y_q(1) - Y_q(0)$ of a specific query unit $q$. The framework
addresses a fundamental gap: standard CATE estimators provide asymptotic confidence
intervals for a conditional mean, but do not cover individual counterfactual
realizations when treatment-effect heterogeneity is unbounded. Our approach closes
this gap by combining feature-weighted local calibration, synthetic augmentation,
and doubly robust AIPW conformity scores within a conformal prediction framework
that provides exact marginal coverage under mild identification assumptions.
The key theoretical result (Theorem~\ref{thm:coverage}) establishes that the
selection measurability of the calibration set $\mathcal{R}_q^*$---guaranteed by
construction, which conditions only on $(X,T)$ and not on $Y$---is the sufficient
condition for conformal validity. The practical gain of localizing the calibration
set is approximately conditional coverage at $x_q$, which standard global conformal
procedures cannot achieve without additional quantile estimation accuracy requirements
\citep{lei2021}. Experiments on synthetic and IHDP data confirm that the local
strategies improve point estimation over the global baseline while maintaining or
exceeding nominal coverage.
The framework inherits the standard observational assumptions of strong ignorability
and overlap. When unmeasured confounders are present, the AIPW pseudo-outcomes are
biased and the coverage guarantee may fail. In settings with very high dimensionality,
the feature-weighted similarity metric may not concentrate the neighborhood tightly
enough around the true causal subspace. A natural extension replaces unconfoundedness
with an instrumental variable (IV) strategy: an instrument $Z$ satisfying exclusion
and relevance allows identification of a Local Average Treatment Effect for
compliers. Conformal prediction for IV-identified effects remains largely unexplored;
a local version of our framework would provide finite-sample valid intervals for
individualized LATE, complementing recent work on nonparametric IV inference.
Extension to continuous treatment doses or longitudinal settings is also
straightforward in principle: the feature-weighted similarity and synthetic
augmentation components carry over directly, while the main challenge lies in
adapting the identification formula and the relevant calibration set definition.
\par
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\noindent
Fernando Delbianco\\
Department of Economics (UNS)- Institute of Mathematics (CONICET)\\
Bahía Blanca, Argentina
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\noindent
E-mail: [email removed]
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\noindent
Fernando Tohmé \\
Department of Economics (UNS)- Institute of Mathematics (CONICET)\\
Bahía Blanca, Argentina
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\noindent
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\noindent
E-mail: [email removed]