Quantile Regression

Conformal prediction constructs prediction sets with finite-sample coverage guarantees, but its calibration stage is structurally constrained to a scalar score function and a single threshold variable - forcing shapes of prediction sets to be fixed before calibration, typically through data splitting. We introduce multi-variable conformal prediction (MCP), a framework that extends conformal prediction to vector-valued score functions with multiple simultaneous calibration variables. Building on scenario theory as a principled framework for certifying data-driven decisions, MCP unifies prediction set design and calibration into a single optimization problem, eliminating data splitting without sacrificing coverage guarantees. We propose two computationally efficient variants: RemMCP, grounded in constrained optimization with constraint removal, which admits a clean generalization of split conformal prediction; and RelMCP, based on iterative optimization with constraint relaxation, which supports non-convex score functions at the cost of possibly greater conservatism. Through numerical experiments on ellipsoidal and multi-modal prediction sets, we demonstrate that RemMCP and RelMCP consistently meet the target coverage with prediction set sizes smaller than or comparable to those of baselines with data split, while considerably reducing variance across calibration runs - a direct consequence of using all available data for shape optimization and calibration simultaneously.

Generative Artificial Intelligence (AI) tools are rapidly adopted in the workplace and in education, yet the empirical evidence on AI's impact remains mixed. We propose a model of human-AI interaction to better understand and analyze several mechanisms by which AI affects productivity. In our setup, human agents with varying skill levels exert utility-maximizing effort to produce certain task outcomes with AI assistance. We find that incorporating either endogeneity in skill development or in AI unreliability can induce a productivity paradox: increased levels of AI assistance may degrade productivity, leading to potentially significant shortfalls. Moreover, we examine the long-term distributional effect of AI on skill, and demonstrate that skill polarization can emerge in steady state when accounting for heterogeneity in AI literacy -- the agent's capability to identify and adapt to inaccurate AI outputs. Our results elucidate several mechanisms that may explain the emergence of human-AI productivity paradoxes and skill polarization, and identify simple measures that characterize when they arise.

An agent's probability report is paid for twice: by a strictly proper scoring rule, and by an approval rule for the decision it triggers. In this classical decision-coupled setting, non-affine approval is known to defeat truthful reporting. We show the conflict is endogenous: when feasible, the welfare-maximizing approval rule is never affine. The distortion, however, is predictable and can be designed around. There is a reserve report at which pretending to be the marginal type costs exactly the approval prize. Approving at or above the reserve screens types perfectly under every strictly proper score, and the reserve does not depend on the type distribution. A Lipschitz rule with a single kink attains first-best exactly; under strict feasibility no continuously differentiable rule does. The binding constraint is steepness, not smoothness. First-best is attainable within a slope budget if and only if the budget is at least the critical slope: the steepest chord of the pretending cost up to the reserve. Below it the welfare loss is cubic in the shortfall. Where the pretending cost is convex up to the reserve, as for Brier, log and power scores, the critical slope is closed-form. The instances are AI-agent oversight and marketplace operation.

Stochastic bandit algorithms are usually analyzed under a mean-reward criterion, yet many problems favor arms with strong upper-tail performance, which we study herein. For a fixed miscoverage level $α$, the natural upper-tail target of arm $j$ is the upper endpoint $F_j^{-1}(1-α/2)$ of a central prediction interval. This target can rank arms differently from their means, creating a central mismatch with the classical bandit objective. To this end, we propose ACP-UCB1, a conformal-style policy that combines an adaptive conformal estimate of the upper endpoint with a UCB-type optimism bonus. The technical challenge is that the conformity scores used by ACP-UCB1 are recomputed from evolving empirical quantile estimates and evaluated at an adaptive level. We control this endpoint through reward-quantile concentration, a perturbation argument for recomputed score quantiles, and deterministic localization of the adaptive level. ACP-UCB1 achieves logarithmic upper-quantile regret with per-arm contribution $O(\nicefrac{\log n}{Δ_j^{\mathrm{ACP}}})$. We also provide metric-specific regret decompositions comparing ACP-UCB1 with UCB1 and use numerical experiments to validate performance and improvement.

Instrumental-variable (IV) regression enables causal estimation under endogeneity, but modern IV problems often involve nonlinear structural effects and high-dimensional covariates. Existing nonlinear IV methods directly learn the causal relation in observed feature space or rely on learned representations within two-stage or moment-based procedures, which can struggle when the causal information is embedded in a high-dimensional representation. We propose BGM-IV, a latent Bayesian generative modeling approach that reframes nonlinear IV regression as posterior inference in a causally structured latent space. BGM-IV infers latent components that separately capture shared confounding structure, outcome-specific variation, treatment-specific variation, and covariate-only nuisance information. To account for endogeneity, BGM-IV replaces the confounded outcome likelihood with an IV-integrated pseudo-likelihood that averages over instrument-induced treatment values within the latent model. Across various benchmark datasets, BGM-IV remains competitive in the classical low-dimensional regime and performs best in high-dimensional covariate regimes. Together, these results show that structured latent generative modeling provides a principled and effective strategy to nonlinear IV estimation with rich covariates. The code of BGM-IV is available at https://github.com/liuq-lab/BGM-IV.

Inference-time scaling methods rely on Process Reward Models (PRMs), which are often poorly calibrated and overestimate success probabilities. We propose, to our knowledge, the first use of conditional optimal transport for calibrating PRMs, modifying conditional OT (CondOT) map learning \cite{bunne2022supervised} to estimate a monotonic conditional quantile function over success probabilities estimated by the PRM, conditioned on PRM hidden states. This yields structurally valid quantile estimates and enables efficient extraction of confidence bounds at arbitrary levels, which we integrate into the instance-adaptive scaling (IAS) framework of \cite{park2025know}. We evaluate on mathematical reasoning benchmarks spanning moderate-difficulty problems (MATH-500) and harder out-of-distribution problems (AIME). For PRMs with reliable ranking signals, our method substantially improves calibration over both uncalibrated PRMs and quantile regression. On downstream Best-of-N IAS performance, our method generally improves over uncalibrated PRMs. These results establish conditional optimal transport as another principled and practical approach to PRM calibration, offering structural guarantees and flexible uncertainty estimation.

Quantile regression is a fundamental tool for distributional learning but poses significant optimization challenges for deep models due to the non-smoothness of the pinball loss. We propose ConquerNet, a class of \textbf{con}volution-smoothed \textbf{qu}antil\textbf{e} \textbf{R}eLU neural \textbf{net}works, which yield smooth objectives while preserving the underlying quantile structure. We establish general nonasymptotic risk bounds for ConquerNet under mild conditions, providing minimax guarantees over Besov function classes. In numerical studies, we demonstrate that the proposed approach outperforms standard quantile neural networks at multiple quantile levels, showing improved estimation accuracy and training efficiency across the board, with particularly pronounced advantages at high and low quantiles.

Constructing minimum-volume prediction regions that satisfy conditional coverage is a fundamental challenge in multivariate regression. Standard approaches rely on explicitly estimating the full conditional density and subsequently thresholding it. This two-step plug-in process is notoriously difficult, sensitive to estimation errors, and computationally expensive. One would like to instead optimize the region directly. Formulating a direct solution is challenging, however, because it requires minimizing a volume objective that is coupled with the conditional quantiles of the model's own estimation error. In this work, we address this challenge. We introduce super-level-set regression (SLS), a novel mathematical framework that successfully resolves this implicit coupling, allowing us to directly parameterize and optimize the geometric boundaries of the target conditional level sets. By bypassing full distribution estimation and leveraging flexible volume-preserving frontier functions, our approach natively captures complex, multimodal, and disjoint conditional structures end-to-end. Ultimately, SLS offers a new perspective on multivariate conditional quantile regression, replacing the restrictive assumptions of density-first methods with a direct geometric optimization strategy.

Instrumental variable (IV) and control function (CF) methods are powerful tools for causal effect estimation in the presence of unmeasured confounding, yet most existing approaches target only mean effects and/or demand substantial fitting and tuning effort. In this paper, we introduce a simple method, TabCF, for control function regression using tabular foundation models, which enables accurate, fast, identification-transparent, and tuning-light causal estimation of distributional quantities, such as interventional means and quantiles; we also propose a copula-based approximation for multivariate outcomes. TabCF performs favorably against representative methods across a broad range of small- to medium-sized synthetic and real data scenarios. The central message is two-fold: for practitioners, it highlights that TabCF is an effective tool for distributional causal inference; for researchers, it suggests that the proposed approach could be considered a strong baseline for future method development. Code is available at https://github.com/GepingChen/TabCF.

Uncertainty quantification (UQ) in graph neural networks (GNNs) is crucial in high-stakes domains but remains a significant challenge. In graph settings, message passing often relies on strong assumptions such as exchangeability, which are rarely satisfied in practice, and achieving reliable UQ typically requires costly resampling or post-hoc calibration. To address these issues, we introduce Quantile-free Prediction Interval GNN (QpiGNN), a framework that builds on quantile regression (QR) to enable GNN-based UQ by directly optimizing coverage and interval width without requiring quantile inputs or post-processing. QpiGNN employs a dual-head architecture that decouples prediction and uncertainty, and is trained with label-only supervision through a quantile-free joint loss. This design allows efficient training and yields robust prediction intervals, with theoretical guarantees of asymptotic coverage and near-optimal width under mild assumptions. Experiments on 19 synthetic and real-world benchmarks show QpiGNN achieves average 22% higher coverage and 50% narrower intervals than baselines, while ensuring efficiency and robustness to noise and structural shifts.

Quantile-based distributional reinforcement learning methods learn return distributions through sampled quantile regression, but their bootstrapped target quantiles may induce distorted or degenerate distribution estimates. We propose Robust Quantile-based Implicit Quantile Networks (RQIQN), a lightweight Wasserstein distributionally robust enhancement boosted from a quantile estimation perspective. We first reinterpret a snapshot of IQN loss as a collection of local empirical quantile estimation problems over sampled current fractions. We then robustify each local slot with a Wasserstein distributionally robust quantile estimation formulation, yielding a closed-form, fraction-dependent correction to the Bellman target. This correction directly addresses distributional degeneration: its median antisymmetry preserves the risk-neutral quantile average, while its monotonicity enlarges upper-lower quantile gaps and counteracts collapsed distributional spread. RQIQN thus regularizes quantile geometry without changing the underlying value objective or requiring additional sample set reconstruction. Finally, we empirically show that the proposed RQIQN outperforms other existing quantile-based distributional reinforcement learning algorithms in risk-sensitive navigation and Atari games.

Graph unlearning remains a critical technique for supporting privacy-preserving and sustainable multimodal graph learning. However, we observe that existing unlearning strategies tend to apply uniform parameter selection and editing across all graph neural network (GNN) layers, which is especially harmful for multimodal graphs where high-dimensional input projections encode dominant cross-modal knowledge. As a result, over-editing these sensitive layers often leads to catastrophic utility degradation after forgetting, undermining both stable learning and effective privacy protection. To address this gap, we propose FDQ, a Feature-Dimension Aware Quantile framework for multimodal graph unlearning. FDQ adaptively identifies high-dimensional input projection layers and applies more conservative, FDQ-guided quantile thresholds when constructing suppression sets, while keeping the underlying importance estimation mechanism unchanged. FDQ is seamlessly integrated with diagonal sensitivity-based parameter importance analysis to enable efficient node and edge unlearning under general forget requests. Through extensive experiments on Ele-Fashion and Goodreads-NC, we demonstrate that FDQ consistently achieves strong utility preservation while maintaining effective forgetting against membership inference attacks. Overall, FDQ offers a principled and robust solution for privacy-aware unlearning in high-dimensional multimodal graph systems.

Doubly Robust (DR) estimation of treatment effect relies on an untestable assumption that is the absence of unobserved confounding. This assumption is par- ticularly problematic in the context of healthcare research, where variables like pre- scription refill rates serve as proxies for unobserved behaviors such as medication adherence. These proxy variables are often endogenous, exhibiting correlation with the regression error term due to unmeasured confounding or measurement error. We propose a copula-corrected doubly robust estimator that addresses endogeneity in both the treatment and outcome models without requiring instrumental variables. Gaussian copulas model the joint distribution of endogenous covariates and the error term, enabling consistent estimation while preserving the doubly robust property that requires correct specification of either the treatment or outcome model, not both. Monte Carlo simulations demonstrate that naive DR estimation exhibits substantial bias under endogeneity, whereas our corrected estimator recovers unbiased treatment effects across different data-generating processes. We apply our method to examine the effect of nutritional counseling on blood pressure using the National Health and Nutrition Examination Survey (NHANES) data. Naive DR estimation suggests counseling is associated with increased blood pressure. After copula correction, this effect becomes statistically insignificant, consistent with literature showing modest effects of nutri- Counseling in reducing blood pressure. Our methodology provides researchers with a practical tool for obtaining treatment effects in the presence of endogeneity.

Conformal prediction provides distribution-free predictive intervals with finite-sample marginal coverage. However, achieving conditional validity and interval efficiency (in terms of short interval length) remains challenging, particularly in complex settings with heteroskedasticity, skewed responses, or estimation errors. We propose a conformal-style calibration method for responses obtained by the probability integral transform (PIT) of the conditional cumulative distribution function (CDF) estimated via neural networks to construct a finite-sample-adjusted percentile interval with the shortest length determined by the estimated conditional CDF. Calibrating in PIT space is effective because PIT values are asymptotically feature-independent when the CDF estimator is accurate, which mitigates feature-dependent miscoverage and improves conditional calibration. On the other hand, our percentile calibration adapts to the empirical PIT distribution, which is robust against a possibly imperfect estimation of the conditional CDF. We prove the finite-sample marginal coverage property of the proposed method and show its asymptotic conditional coverage under mild consistency conditions. Experiments on diverse synthetic and real-world benchmarks demonstrate better conditional calibration and substantially shorter intervals than existing methods.

Extreme value theory provides rigorous theory and statistical tools for extrapolation in machine learning, particularly in settings where traditional methods struggle due to data scarcity in the tails. A broad range of tasks benefit from these advances, including regression and classification beyond the training data, extreme quantile regression, supervised and unsupervised dimension reduction, generative artificial intelligence and anomaly detection. This review synthesizes recent developments in these fields at the intersection of statistical learning and extreme value theory, with a focus on principled methods based on asymptotically motivated representations of the tail of univariate and multivariate distributions. We consider different theoretical frameworks for both asymptotically dependent and independent data and discuss how they translate into efficient statistical methods for extrapolation to extreme regions. By addressing both theoretical and practical aspects, we offer a comprehensive overview of the state-of-the-art in this quickly evolving field, and identify promising directions for future research.

The field of machine learning is subject to an increasing interest in models that are not only accurate but also interpretable and robust, thus allowing their end users to understand and trust AI systems. This paper presents a novel method for learning a set of optimal quantile regression trees. The advantages of this method are that (1) it provides predictions about the complete conditional distribution of a target variable without prior assumptions on this distribution; (2) it provides predictions that are interpretable; (3) it learns a set of optimal quantile regression trees without compromising algorithmic efficiency compared to learning a single tree.

Medical Visual Question Answering (MedVQA) aims to generate clinically reliable answers conditioned on complex medical images and questions. However, existing methods often overfit to superficial cross-modal correlations, neglecting the intrinsic biases embedded in multimodal medical data. Consequently, models become vulnerable to cross-modal confounding effects, severely hindering their ability to provide trustworthy diagnostic reasoning. To address this limitation, we propose a novel Dual Causal Inference (DCI) framework for MedVQA. To the best of our knowledge, DCI is the first unified architecture that integrates Backdoor Adjustment (BDA) and Instrumental Variable (IV) learning to jointly tackle both observable and unobserved confounders. Specifically, we formulate a Structural Causal Model (SCM) where observable cross-modal biases (e.g., frequent visual and textual co-occurrences) are mitigated via BDA, while unobserved confounders are compensated using an IV learned from a shared latent space. To guarantee the validity of the IV, we design mutual information constraints that maximize its dependence on the fused multimodal representations while minimizing its associations with the unobserved confounders and target answers. Through this dual mechanism, DCI extracts deconfounded representations that capture genuine causal relationships. Extensive experiments on four benchmark datasets, SLAKE, SLAKE-CP, VQA-RAD, and PathVQA, demonstrate that our method consistently outperforms existing approaches, particularly in out-of-distribution (OOD) generalization. Furthermore, qualitative analyses confirm that DCI significantly enhances the interpretability and robustness of cross-modal reasoning by explicitly disentangling true causal effects from spurious cross-modal shortcuts.

Many applications of LLM-based text regression require predicting a full conditional distribution rather than a single point value. We study distributional regression under empirical-quantile supervision, where each input is paired with multiple observed quantile outcomes, and the target distribution is represented by a dense grid of quantiles. We address two key limitations of current approaches: the lack of local grounding for distribution estimates, and the reliance on shared representations that create an indirect bottleneck between inputs and quantile outputs. In this paper, we introduce Quantile Token Regression, which, to our knowledge, is the first work to insert dedicated quantile tokens into the input sequence, enabling direct input-output pathways for each quantile through self-attention. We further augment these quantile tokens with retrieval, incorporating semantically similar neighbor instances and their empirical distributions to ground predictions with local evidence from similar instances. We also provide the first theoretical analysis of loss functions for quantile regression, clarifying which distributional objectives each optimizes. Experiments on the Inside Airbnb and StackSample benchmark datasets with LLMs ranging from 1.7B to 14B parameters show that quantile tokens with neighbors consistently outperform baselines (~4 points lower MAPE and 2x narrower prediction intervals), with especially large gains on smaller and more challenging datasets where quantile tokens produce substantially sharper and more accurate distributions.

This paper investigates the off-policy evaluation (OPE) problem from a distributional perspective. Rather than focusing solely on the expectation of the total return, as in most existing OPE methods, we aim to estimate the entire return distribution. To this end, we introduce a quantile-based approach for OPE using deep quantile process regression, presenting a novel algorithm called Deep Quantile Process regression-based Off-Policy Evaluation (DQPOPE). We provide new theoretical insights into the deep quantile process regression technique, extending existing approaches that estimate discrete quantiles to estimate a continuous quantile function. A key contribution of our work is the rigorous sample complexity analysis for distributional OPE with deep neural networks, bridging theoretical analysis with practical algorithmic implementations. We show that DQPOPE achieves statistical advantages by estimating the full return distribution using the same sample size required to estimate a single policy value using conventional methods. Empirical studies further show that DQPOPE provides significantly more precise and robust policy value estimates than standard methods, thereby enhancing the practical applicability and effectiveness of distributional reinforcement learning approaches.

The quantification of uncertainty in prediction models is crucial for reliable decision-making, yet remains a significant challenge. Interval time series forecasting offers a principled solution to this problem by providing prediction intervals (PIs), which indicates the probability that the true value falls within the predicted range. We consider a recently established point forecasts (PFs) method termed Learning-Based Convolution Nuclear Norm Minimization (LbCNNM), which directly generates multi-step ahead forecasts by leveraging the convolutional low-rankness property derived from training data. While theoretically complete and empirically effective, LbCNNM lacks inherent uncertainty estimation capabilities, a limitation shared by many advanced forecasting methods. To resolve the issue, we modify the well-known Quantile Regression (QR) and integrate it into LbCNNM, resulting in a novel interval forecasting method termed LbCNNM with Modified Quantile Regression (LbCNNM-MQR). In addition, we devise interval calibration techniques to further improve the accuracy of PIs. Extensive experiments on over 100,000 real-world time series demonstrate the superior performance of LbCNNM-MQR.

Time series data are essential for a wide range of applications, yet access to high-quality datasets is often constrained by privacy concerns, acquisition costs, and labelling challenges. Synthetic time series generation has emerged as a promising approach to address these limitations. In this work, we investigate the use of complex network mappings for synthetic time series generation, focusing on the Quantile Graph (QG) representation and its inverse. While the inverse QG mapping has been previously proposed, its potential as a general-purpose data generator has not been systematically evaluated. We address this gap through a comprehensive empirical study assessing both the fidelity and utility of synthetic time series generated by the Inverse Quantile Graph (InvQG) framework. The evaluation combines statistical feature analysis, network-based topological characteristics, and performance in downstream clustering and classification tasks, using simulated and real-world datasets. The results show that InvQG effectively preserves marginal distributions and short-term temporal dependencies across a wide range of models, while exhibiting predictable limitations in capturing long-range or higher-order dynamics.

With the development of big data and machine learning, privacy concerns have become increasingly critical, especially when handling heterogeneous datasets containing sensitive personal information. Differential privacy provides a rigorous framework for safeguarding individual privacy while enabling meaningful statistical analysis. In this paper, we propose a differentially private quantile regression method for high-dimensional data in a distributed setting. Quantile regression is a powerful and robust tool for modeling the relationships between the covariates and responses in the presence of outliers or heavy-tailed distributions. To address the computational challenges due to the non-smoothness of the quantile loss function, we introduce a Newton-type transformation that reformulates the quantile regression task into an ordinary least squares problem. Building on this, we develop a differentially private estimation algorithm with iterative updates, ensuring both near-optimal statistical accuracy and formal privacy guarantees. For inference, we further propose a differentially private debiased estimator, which enables valid confidence interval construction and hypothesis testing. Additionally, we propose a communication-efficient and differentially private bootstrap for simultaneous hypothesis testing in high-dimensional quantile regression, suitable for distributed settings with both small and abundant local data. Extensive simulations demonstrate the robustness and effectiveness of our methods in practical scenarios.

Regression problems with bounded continuous outcomes frequently arise in statistical and machine learning applications, such as the analysis of rates and proportions. A central challenge in this setting is predicting the response at a new covariate value. Most of the existing literature has focused either on point prediction or on interval prediction based on asymptotic approximations. We develop conformal prediction intervals for bounded outcomes within the framework of transformation regression models, encompassing widely used models such as beta regression and logit-normal regression. We construct non-conformity scores based on model-aligned residuals and identify a quantile-residual score that is particularly well suited to bounded outcomes, bridging normalized conformal prediction and distributional conformal prediction. This score accounts for both the heteroscedasticity inherent in such data and the asymmetry that emerges near the boundaries of the response space. We establish marginal validity and asymptotic conditional validity for both full and split conformal prediction, holding under model misspecification. A comprehensive simulation study confirms that both methods empirically attain valid finite-sample coverage, including cases under model misspecification. A real-data application demonstrates their practical performance against bootstrap-based alternatives.

Accurate uncertainty quantification is critical for reliable predictive modeling. Existing methods typically address either aleatoric uncertainty due to measurement noise or epistemic uncertainty resulting from limited data, but not both in a balanced manner. We propose CLEAR, a calibration method with two distinct parameters, $γ_1$ and $γ_2$, to combine the two uncertainty components and improve the conditional coverage of predictive intervals for regression tasks. CLEAR is compatible with any pair of aleatoric and epistemic estimators; we show how it can be used with (i) quantile regression for aleatoric uncertainty and (ii) ensembles drawn from the Predictability-Computability-Stability (PCS) framework for epistemic uncertainty. Across 17 diverse real-world datasets, CLEAR achieves an average improvement of 28.3% and 17.5% in the interval width compared to the two individually calibrated baselines while maintaining nominal coverage. Similar improvements are observed when applying CLEAR to Deep Ensembles (epistemic) and Simultaneous Quantile Regression (aleatoric). The benefits are especially evident in scenarios dominated by high aleatoric or epistemic uncertainty. Project page: https://unco3892.github.io/clear/

This paper considers the estimation of quantiles via a smoothed version of the stochastic gradient descent (SGD) algorithm. By smoothing the score function with a bandwidth tied to the learning rate, we obtain estimates that are monotone in the quantile level at every iteration, while retaining the memory and computational efficiency required for streaming data. We establish non-asymptotic tail probability bounds for the smoothed estimate with and without Polyak-Ruppert averaging, which are sub-exponential with a multi-regime structure. For the averaged estimate we further derive a Bahadur representation that is uniform in the quantile level and across coordinates, and a resulting Gaussian approximation by the maximum of Brownian bridges, with the dimension $p$ allowed to grow exponentially in the sample size. This yields simultaneous inference across coordinates and quantile levels. As an alternative that avoids estimating the sparsity function, we propose an online multiplier bootstrap that preserves monotonicity, runs in a single pass and is asymptotically valid. Extending the theory to a localized recursion, we obtain online nonparametric conditional quantile estimates with uniform bands over design points and quantile levels. Simulations confirm accurate finite-sample coverage, and we illustrate the method on conditional value-at-risk curves.

We propose conformal predictive programming (CPP), a framework to solve chance constrained optimization problems, i.e., optimization problems with constraints that are functions of random variables. CPP utilizes samples from these random variables along with the quantile lemma - central to conformal prediction - to transform the chance constrained optimization problem into a deterministic problem with a quantile reformulation. CPP's main strength is an independent calibration step that provides a posteriori guarantees for the solution of this problem that are of conditional and marginal nature otherwise. These guarantees even apply in settings when assumptions required for obtaining standard a priori guarantees (e.g., in scenario optimization or sample average approximation) are unavailable, difficult to compute, or conservative. Another strength of CPP is that it can easily support different variants of conformal prediction which have been (or will be) proposed within the conformal prediction community. To illustrate this, we present robust CPP to deal with distribution shifts in the random variables and Mondrian CPP to deal with class conditional chance constraints. In a series of case studies, we show the validity of the aforementioned approaches, and illustrate the advantage of CPP as compared to scenario approach.