We push each class to its supernode, then apply majority vote to determine the final class.

Full conformal prediction is a framework that implicitly formulates distribution-free confidence prediction regions for a wide range of estimators. However, a classical limitation of the full conformal framework is the computation of the confidence prediction regions, which is usually impossible since it requires training infinitely many estimators (for real-valued prediction for instance). The main purpose of the present work is to describe a generic strategy for designing a tight approximation to the full conformal prediction region that can be efficiently computed. Along with this approximate confidence region, a theoretical quantification of the tightness of this approximation is developed, depending on the smoothness assumptions on the loss and score functions. The new notion of thickness is introduced for quantifying the discrepancy between the approximate confidence region and the full conformal one.

In this work, we propose a set of conformal prediction procedures tailored to compositional responses, where outcomes are proportions that must be positive and sum to one. Building on Dirichlet regression, we introduce a split conformal approach based on quantile residuals and a highest-density region strategy that combines a fast coordinate-floor approximation with an internal grid refinement to restore sharpness. Both constructions are model-agnostic at the conformal layer and guarantee finite-sample marginal coverage under exchangeability, while respecting the geometry of the simplex. A comprehensive Monte Carlo study spanning homoscedastic and heteroscedastic designs shows that the quantile residual and grid-refined HDR methods achieve empirical coverage close to the nominal 90\% level and produce substantially narrower regions than the coordinate-floor approximation, which tends to be conservative. We further demonstrate the methods on household budget shares from the BudgetItaly dataset, using standardized socioeconomic and price covariates with a train, calibration, and test split. In this application, the grid-refined HDR attains coverage closest to the target with the smallest average widths, closely followed by the quantile residual approach, while the simple triangular HDR yields wider, less informative sets. Overall, the results indicate that conformal prediction on the simplex can be both calibrated and efficient, providing practical uncertainty quantification for compositional prediction tasks.

Conformal Prediction (CP) is a powerful statistical machine learning tool to construct uncertainty sets with coverage guarantees, which has fueled its extensive adoption in generating prediction regions for decision-making tasks, e.g., Trajectory Optimization (TO) in uncertain environments. However, existing methods predominantly employ a sequential scheme, where decisions rely unidirectionally on the prediction regions, and consequently the information from decision-making fails to be fed back to instruct CP. In this paper, we propose a novel Feedback-Based CP (Fb-CP) framework for shrinking-horizon TO with a joint risk constraint over the entire mission time. Specifically, a CP-based posterior risk calculation method is developed by fully leveraging the realized trajectories to adjust the posterior allowable risk, which is then allocated to future times to update prediction regions. In this way, the information in the realized trajectories is continuously fed back to the CP, enabling attractive feedback-based adjustments of the prediction regions and a provable online improvement in trajectory performance. Furthermore, we theoretically prove that such adjustments consistently maintain the coverage guarantees of the prediction regions, thereby ensuring provable safety. Additionally, we develop a decision-focused iterative risk allocation algorithm with theoretical convergence analysis for allocating the posterior allowable risk which closely aligns with Fb-CP. Furthermore, we extend the proposed method to handle distribution shift. The effectiveness and superiority of the proposed method are demonstrated through benchmark experiments.

Trustworthy decision making in networked, dynamic environments calls for innovative uncertainty quantification substrates in predictive models for graph time series. Existing conformal prediction (CP) methods have been applied separately to multivariate time series and static graphs, but they either ignore the underlying graph topology or neglect temporal dynamics. To bridge this gap, here we develop a CP-based sequential prediction region framework tailored for graph time series. A key technical innovation is to leverage the graph structure and thus capture pairwise dependencies across nodes, while providing user-specified coverage guarantees on the predictive outcomes. We formally establish that our scheme yields an exponential shrinkage in the volume of the ellipsoidal prediction set relative to its graph-agnostic counterpart. Using real-world datasets, we demonstrate that the novel uncertainty quantification framework maintains desired empirical coverage while achieving markedly smaller (up to 80% reduction) prediction regions than existing approaches.

Conformal prediction is a model-free machine learning method for creating prediction regions with a guaranteed coverage probability level. However, a data scientist often faces three challenges in practice: (i) the determination of a conformal prediction region is only approximate, jeopardizing the finite-sample validity of prediction, (ii) the computation required could be prohibitively expensive, and (iii) the shape of a conformal prediction region is hard to control. This article offers new insights into the relationship among the monotonicity of the non-conformity measure, the monotonicity of the plausibility function, and the exact determination of a conformal prediction region. Based on these new insights, we propose a simple strategy to alleviate the three challenges simultaneously.

Treatment effect estimation is essential for informed decision-making in many fields such as healthcare, economics, and public policy. While flexible machine learning models have been widely applied for estimating heterogeneous treatment effects, quantifying the inherent uncertainty of their point predictions remains an issue. Recent advancements in conformal prediction address this limitation by allowing for inexpensive computation, as well as distribution shifts, while still providing frequentist, finite-sample coverage guarantees under minimal assumptions for any point-predictor model. This advancement holds significant potential for improving decision-making in especially high-stakes environments. In this work, we perform a systematic review regarding conformal prediction methods for treatment effect estimation and provide for both the necessary theoretical background. Through a systematic filtering process, we select and analyze eleven key papers, identifying and describing current state-of-the-art methods in this area. Based on our findings, we propose directions for future research.

This paper introduces a framework for uncertainty quantification in regression models defined in metric spaces. Leveraging a newly defined notion of homoscedasticity, we develop a conformal prediction algorithm that offers finite-sample coverage guarantees and fast convergence rates of the oracle estimator. In heteroscedastic settings, we forgo these non-asymptotic guarantees to gain statistical efficiency, proposing a local $k$--nearest--neighbor method without conformal calibration that is adaptive to the geometry of each particular nonlinear space. Both procedures work with any regression algorithm and are scalable to large data sets, allowing practitioners to plug in their preferred models and incorporate domain expertise. We prove consistency for the proposed estimators under minimal conditions. Finally, we demonstrate the practical utility of our approach in personalized--medicine applications involving random response objects such as probability distributions and graph Laplacians.

Depth measures are powerful tools for defining level sets in emerging, non--standard, and complex random objects such as high-dimensional multivariate data, functional data, and random graphs. Despite their favorable theoretical properties, the integration of depth measures into regression modeling to provide prediction regions remains a largely underexplored area of research. To address this gap, we propose a novel, model-free uncertainty quantification algorithm based on conditional depth measures--specifically, conditional kernel mean embeddings and an integrated depth measure. These new algorithms can be used to define prediction and tolerance regions when predictors and responses are defined in separable Hilbert spaces. The use of kernel mean embeddings ensures faster convergence rates in prediction region estimation. To enhance the practical utility of the algorithms with finite samples, we also introduce a conformal prediction variant that provides marginal, non-asymptotic guarantees for the derived prediction regions. Additionally, we establish both conditional and unconditional consistency results, as well as fast convergence rates in certain homoscedastic settings. We evaluate the finite--sample performance of our model in extensive simulation studies involving various types of functional data and traditional Euclidean scenarios. Finally, we demonstrate the practical relevance of our approach through a digital health application related to physical activity, aiming to provide personalized recommendations