Prediction under label shift becomes nonstandard when responses are censored. In a two-sided censored Gaussian model, latent values below $L$ and above $U$ are recorded at the boundary values, so the observed predictive distribution is mixed, with atoms at $L$ and $U$ and a continuous density on $(L,U)$. In this paper we develop conformal Bayes for this mixed-space setting by combining posterior predictive tilting with weighted conformal calibration. Under a two-sided Tobit Gaussian Bayesian prediction head with a Laplace posterior approximation, the tilted predictive distribution has left-atom, interior, and right-atom components, with a three-term closed-form normalizer. The resulting prediction set is a mixed highest density region that can combine boundary atoms with an interior interval and can reduce to atom-only sets under strong censoring. The main technical issue is that latent label shift does not directly give an ordinary density ratio on the observed censored scale. A latent exponential tilt induces tail-averaged atom weights at the censored boundaries, while the interior ratio remains density based. This yields a mixed observed-space calibration weight with two atom ratios and one interior density ratio. The weight corrects the calibration measure, while predictive tilting gives target-adapted mixed-HDR geometry. Synthetic experiments show that weighted tilted conformal Bayes restores marginal coverage with smaller sets than weighted source-score calibration, while revealing a trade-off between marginal coverage and component-wise behavior across atoms and interior observations.

Kornelius Raeth 1 Nicole Ludwig 1 2 Abstractscoring rules distribute the training gradient in proportion to Sample-based generative models are increasingly data density, with no awareness of the decision maker's cost structure. The model's limited capacity is allocated globused for probabilistic forecasting in high-stakes ally, leaving decision-critical regions of the output space decision settings, yet their training objectives are potentially underserved. These models are commonly trained with strictly proper Given a forecast, a decision maker with cost function c(a,y), scoring rules, such as the energy score, which al-of action aand outcome y, selects the action that minimises locate their training signal in proportion to dataexpected cost under the forecast distribution; a point forecast density, with no awareness of where forecast eris insufficient to evaluate this expectation. A good forecast rors are most costly for downstream decisions. Crucially, the energy score objective with a differentiable deci-observed cost of the optimal action is itself a proper scoring sion loss that directly penalises the cost incurredrule (Hartline et al., 2025; Kleinberg et al., 2023), placing by acting on the model's forecast. This combinedit in the same family as the energy score which licenses loss is theoretically grounded, as the decision losstheir combination as a theoretically well-founded training is itself a proper scoring rule. Introduction score acts as that anchor, preventing the model from collapsing outside cost-sensitive regions. Our method is theo-tion based on a temperature forecast, balancing asset loss against the cost of intervention. In the weather domain, retically grounded and leads to better downstream decisions state-of-the-art forecasting systems (Lang et al., 2024; Pricewhile retaining full probabilistic forecasts, as validated on et al., 2023) are trained with strictly proper scoring rulessynthetic and real-world forecasting tasks. A gradient analysis showing which regions benefitscore reduces to the continuous ranked probability score from the decision loss and why, based on the cost (CRPS), widely used in meteorological forecast verificafunction structure. Both model classes introduced above are commonly trained by minimising strictly proper sion calibration.

Accurate energy demand forecasting is essential for the reliable operation and planning of modern sustainable energy systems. Spatial-temporal graph neural networks (STGNNs) have recently achieved strong performance in point forecasting by jointly modeling temporal dynamics and relational dependencies across interconnected energy nodes. However, in real-world energy systems, accurate point forecasts alone are insufficient, as operators also require reliable uncertainty estimates to support risk-aware decision-making, grid stability, and operational planning under uncertainty. Conformal prediction provides a principled and model-agnostic framework for uncertainty quantification with statistical coverage guarantees, making it particularly attractive for safety-critical energy applications. However, existing conformal prediction approaches often fail to fully capture the complex spatial-temporal structure of energy systems. To address these limitations, we propose STOIC (Spatial-Temporal Graph Conformal Prediction with In-Context Learning), a novel framework that integrates graph-based forecasting with the zero-shot calibration capabilities of tabular foundation models. STOIC first generates point forecasts using an STGNN and subsequently reformulates spatial-temporal residuals into a tabular representation suitable for in-context learning. Leveraging a tabular foundation model, STOIC calibrates prediction intervals without task-specific retraining, effectively capturing both sequential and relational dependencies. We evaluate STOIC on five diverse benchmarks, including synthetic simulations as well as real-world electricity and district heating networks. Across all datasets, STOIC consistently outperforms existing conformal prediction baselines, delivering more reliable and robust uncertainty estimates for complex graph-structured energy time series.

Conformal prediction and its variants, including the split conformal prediction, provide a distribution-free framework for uncertainty quantification by constructing prediction intervals or sets with finite-sample coverage guarantees. The statistical efficiency of these intervals depends critically on how the data are split into training and calibration samples. Despite its practical importance, a principled characterization of the training-calibration split that minimizes prediction interval length while maintaining coverage has remained largely unresolved. In this paper, we develop a theoretical framework for optimal data splitting in split conformal prediction. We first analyze the problem in a general setting and derive analytical characterizations of the length-optimal split ratio under both symmetric and asymmetric regimes. We then show how the general results specialize to several commonly used regression settings, including linear regression, nonparametric regression, and neural networks, thereby demonstrating the scope of the framework. We also describe a data-based method for selecting the optimal proportion. Our analysis clarifies how model-related features govern the optimal allocation of samples between training and calibration and provides principled guidance for constructing shorter prediction intervals. Experiments on both synthetic and real-world datasets demonstrate the applicability of the proposed methodology across a variety of practical scenarios.

Predicting the aerodynamic performance (e.g. lift, drag, and moment coefficients) of an aircraft is challenging -- computational models are biased and direct simulations are prohibitive. A pragmatic way to overcome this limitation is by calibrating low-fidelity computational predictions with experimental measurements. This, however, requires calibrating against \emph{sparse} measurements contaminated with \emph{uncertainty} in both the control inputs and the measured aerodynamic response. We develop a methodology to address this problem based on Gaussian process surrogates and the classical Kennedy-O'Hagan calibration. A surrogate model learned on abundant-but-cheap low-fidelity data is calibrated with a sparse set of measurement data. Crucialy, we develop a Bayesian latent Gaussian process based approach that marginalizes the calibrated surrogate model over the input uncertainty, while also matching the marginal mean and variance of the measured output uncertainty. Once calibrated, our surrogate model predicts the uncertainty in aerodynamic coefficients with very high accuracy, including at extrapolative input settings. We validate our calibrated surrogate model predictions against measurement data with \emph{true} uncertainty intervals to demonstrate that the model places $94.2-95.8\%$ of its predictive samples inside the released $95\%$ truth intervals, with endpoint cumulative probabilities very close to the nominal 0.025 and 0.975 levels.

Conformal prediction guarantees marginal coverage, but pooled calibration averages over heterogeneous regions and can mask regional undercoverage in safety-critical subgroups. We introduce Self-Organized Conformal Prediction (SOCP), a calibration scheme that discovers input-space groups with a Self-Organizing Map (SOM) and, at test time, draws a local calibration buffer from the query's best-matching unit (BMU) cell or a fixed grid neighborhood. The same retrieval rule applies to regression and classification tasks across tabular features and image embeddings, leaving the predictor and nonconformity score untouched. SOCP gives exact validity for BMU-cell retrieval and fixed retrieved-set validity for neighborhood buffers; central-cell validity for neighborhood retrieval holds up to a Kolmogorov-Smirnov (KS) bias term. A split-routed extension recovers fixed retrieved-set validity conditional on the routing split. On eight regression and classification benchmarks, SO-SCP reduces the weighted regional coverage gap on $7/8$ datasets (mean paired change $-7.1\%$) for a mean prediction-set size increase of $6.2\%$, with negligible overhead on the largest six datasets; SO-CQR yields smaller gains, since quantile regression already absorbs much of the heterogeneity. By learning groups directly from the input geometry, SOCP provides group-local calibration with exact fixed-group guarantees and approximate central-cell guarantees, without supervised partitions or predictor retraining.

We propose doubly robust adaptive conformal inference (DR-ACI), which constructs prediction intervals for doubly robust pseudo-outcomes under temporal dependence. Calibration targets the pseudo-outcome ψDRt; under estimator consistency, this yields asymptotically conservative CATE containment (Corollary 6). Temporal block cross-fitting preserves switch-coefficient mixing bounds and the DML product-bias rate up to an explicit coupling remainder.

Conformal Bayes combines Bayesian posterior predictives with conformal calibration to produce prediction sets that are both statistically valid and geometrically efficient. We study conformal Bayes under label shift from a unified perspective, identifying two complementary approaches that restore nominal target-domain coverage through importance-weighted conformal calibration but operate through independent mechanisms. \emph{Post-hoc calibration} tilts the posterior predictive toward the target domain and corrects the conformal threshold via an importance-weighted quantile, leaving the parameter posterior unchanged. \emph{In-training adaptation} tilts the parameter posterior itself to the target domain, producing a corrected predictive whose highest predictive density region serves as the highest predictive density (HPD)-based prediction set under the fitted target predictive; efficiency is model-dependent and does not imply finite-sample conditional optimality. Two controlled experiments isolate the regime-dependence of each strategy: in the low-dimensional, well-estimated regime Strategy~A produces the narrowest valid intervals, while in the high-dimensional, underdetermined regime Strategy~B achieves up to $43\%$ width reduction at unchanged coverage, under the stated source-sampling and label-shift assumptions.

Reliably quantifying predictive uncertainty is difficult for complex, high-dimensional, or misspecified models. Both fully Bayesian and bootstrap resampling methods provide principled uncertainty estimates but are often too expensive for modern machine-learning models because they require posterior sampling or repeated model refitting. We introduce Ribbon, a scalable approximation to Dirichlet-reweighted bootstrap uncertainty. Ribbon replaces repeated refitting with an influence-function linearization around a single fitted model, preserving the first-order data-reweighting structure of the Bayesian bootstrap while requiring only post-hoc linear algebra. Ribbon approximates the Bayesian-bootstrap or weighted-likelihood-bootstrap refitting target. With a general concentration parameter, Ribbon gives a calibrated Dirichlet-reweighting family whose uncertainty scale can be tuned on validation data. We show that Ribbon is asymptotically equivalent to a flat-prior Laplace approximation under correct likelihood specification and recovers the robust sandwich covariance under misspecification. Across synthetic regression, MNIST classification, and California Housing benchmarks, Ribbon provides competitive predictive performance and improved calibration in several settings while avoiding repeated model retraining.

Graph few-shot learning has attracted increasing attention due to its ability to rapidly adapt models to new tasks with only limited labeled nodes. Despite the remarkable progress made by existing graph few-shot learning methods, several key limitations remain. First, most current approaches rely on predefined and unified graph filters (e.g., low-pass or high-pass filters) to globally enhance or suppress node frequency signals. Such fixed spectral operations fail to account for the heterogeneity of local topological structures inherent in real-world graphs. Moreover, these methods often assume that the support and query sets are drawn from the same distribution. However, under few-shot conditions, the limited labeled data in the support set may not sufficiently capture the complex distribution of the query set, leading to suboptimal generalization.