Previous work has focused primarily on developing ML-based "meta-learners" that can provide point estimates of the conditional

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, 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.

Ensemble learning is widely used in applications to make predictions in complex decision problems---for example, averaging models fitted to a sequence of samples bootstrapped from the available training data. While such methods offer more accurate, stable, and robust predictions and model estimates, much less is known about how to perform valid, assumption-lean inference on the output of these types of procedures. In this paper, we propose the jackknife+-after-bootstrap (J+aB), a procedure for constructing a predictive interval, which uses only the available bootstrapped samples and their corresponding fitted models, and is therefore free in terms of the cost of model fitting. The J+aB offers a predictive coverage guarantee that holds with no assumptions on the distribution of the data, the nature of the fitted model, or the way in which the ensemble of models are aggregated---at worst, the failure rate of the predictive interval is inflated by a factor of 2. Our numerical experiments verify the coverage and accuracy of the resulting predictive intervals on real data.

Large language models have achieved remarkable progress in complex problem-solving, but suffer from high computational costs during deployment (Kwon et al., 2023). To address this, various approaches have been proposed, including model routing (Ong et al., 2025; Dekoninck et al., 2025), speculative decoding (Leviathan et al., 2023), and adaptive reasoning strategies (Snell et al., 2024). Zeng et al. (2025) proposed PAC reasoning, which constructs a composite model ˆ f that selectively switches between an expensive expert model f and a cheaper fast model f while providing statistical guarantees on performance loss. A typical example is the thinking-nonthinking paradigm, where the expert model performs extended chain-of-thought reasoning while the fast model generates direct responses. The original PAC reasoning provides marginal guarantees, controlling the expected risk over the input distribution. A natural extension is whether we can achieve a stronger, conditional guarantee that controls the risk for each input point individually. This is analogous to the notion of object-conditional validity in conformal prediction (Vovk, 2012; Lei and Wasserman, 2014; Lei et al., 2018). 1

Previous work has focused primarily on developing ML-based "meta-learners" that can provide point estimates of the conditional

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.

Ensemble learning is widely used in applications to make predictions in complex decision problems---for example, averaging models fitted to a sequence of samples bootstrapped from the available training data. While such methods offer more accurate, stable, and robust predictions and model estimates, much less is known about how to perform valid, assumption-lean inference on the output of these types of procedures. In this paper, we propose the jackknife -after-bootstrap (J aB), a procedure for constructing a predictive interval, which uses only the available bootstrapped samples and their corresponding fitted models, and is therefore "free" in terms of the cost of model fitting. The J aB offers a predictive coverage guarantee that holds with no assumptions on the distribution of the data, the nature of the fitted model, or the way in which the ensemble of models are aggregated---at worst, the failure rate of the predictive interval is inflated by a factor of 2. Our numerical experiments verify the coverage and accuracy of the resulting predictive intervals on real data.

Due to their complex structures, these models are generally accessed as black boxes. To assess their reliability and safeguard against potential errors, it is important to quantify the uncertainty in their outputs. Predictive inference is a popular methodology for this purpose. It takes as input a prediction algorithm and calibration data, and outputs a prediction set that contains the true outcome with a prescribed probability. The validity of the prediction set hinges on the assumption that the calibration data truthfully represents the underlying environment. However, this assumption is frequently violated in practice, where the data distribution may drift over time. Integrating data from both current and historical periods to construct faithful prediction sets remains a significant challenge. Despite a large body of literature on learning under distribution drift over the past two decades (Hazan and Seshadhri, 2009; Mohri and Muñoz Medina, 2012; Besbes et al., 2015; Hanneke et al., 2015; Mazzetto and Upfal, 2023; Huang and Wang, 2023), statistical inference within this context is much less explored.

Knowledge of the effect of interventions, called the treatment effect, is paramount for decision-making. Approaches to estimating this treatment effect, e.g. by using Conditional Average Treatment Effect (CATE) estimators, often only provide a point estimate of this treatment effect, while additional uncertainty quantification is frequently desired instead. Therefore, we present a novel method, the Conformal Monte Carlo (CMC) meta-learners, leveraging conformal predictive systems, Monte Carlo sampling, and CATE meta-learners, to instead produce a predictive distribution usable in individualized decision-making. Furthermore, we show how specific assumptions on the noise distribution of the outcome heavily affect these uncertainty predictions. Nonetheless, the CMC framework shows strong experimental coverage while retaining small interval widths to provide estimates of the true individual treatment effect.

Discrete data are abundant and often arise as counts or rounded data. However, even for linear regression models, conjugate priors and closed-form posteriors are typically unavailable, thereby necessitating approximations or Markov chain Monte Carlo for posterior inference. For a broad class of count and rounded data regression models, we introduce conjugate priors that enable closed-form posterior inference. Key posterior and predictive functionals are computable analytically or via direct Monte Carlo simulation. Crucially, the predictive distributions are discrete to match the support of the data and can be evaluated or simulated jointly across multiple covariate values. These tools are broadly useful for linear regression, nonlinear models via basis expansions, and model and variable selection. Multiple simulation studies demonstrate significant advantages in computing, predictive modeling, and selection relative to existing alternatives.