Bayesian Conformal Prediction as a Decision Risk Problem
Department of Computer Science, University of Manchester（曼彻斯特大学计算机科学系）；UKRI AI Centre for Doctoral Training in Decision Making for Complex Systems（UKRI人工智能博士培训中心（复杂系统决策））；Department of Computer Science, Aalto University（艾洛大学计算机科学系）；ELLIS Institute, Finland（芬兰ELLIS研究所）
Fanyi Wu, Veronika Lohmanova, Samuel Kaski, Michele Caprio
备注：
22 pages, 8 figures. A previous version was accepted at the EIML Workshop at NeurIPS 2025
We propose Bayesian Conformal Prediction (BCP), a framework that combines Bayesian posterior predictive distributions with PAC-style conformal risk control to produce prediction sets with finite-sample coverage guarantees. Standard quantile-threshold conformal methods often construct prediction sets using a single fixed threshold, which typically yields connected prediction sets. While valid, such sets can be inefficient when the posterior predictive distribution is multimodal, since they may span low-density regions between separated modes. The main contribution of BCP is to formulate conformal prediction as a decision-risk optimisation problem, extending standard fixed quantile-threshold sets to optimised highest posterior density (HPD) prediction sets. These sets can be disjoint, concentrating probability mass on separated high-density regions. Validity is enforced using a PAC-style risk constraint, which provides coverage control even when the Bayesian model is misspecified. In standard nested-threshold settings, BCP recovers the smallest feasible threshold, aligning with existing PAC-based approaches. In the multimodal experiment, HPD geometry substantially improves efficiency, reducing mean prediction set size from $4.82$ to $2.07$ while satisfying the target PAC pass rate. Across regression, classification, and distribution-shift experiments, BCP maintains reliable coverage under model misspecification, whereas Bayesian credible intervals can fail to preserve nominal coverage.
Department of Computer Science, New Jersey Institute of Technology（计算机科学系，新泽西理工学院）；Department of Mathematical Sciences, New Jersey Institute of Technology（数学科学系，新泽西理工学院）
Division of Systems and Control, Department of Information Technology, Uppsala University, Sweden（信息科技系系统与控制 division，乌普萨拉大学，瑞典）；Science for Life Laboratory, Uppsala, Sweden（生命科学实验室，乌普萨拉，瑞典）
Conformalized Generative Bayesian Imaging: An Uncertainty Quantification Framework for Computational Imaging
Department of Electrical and Computer Engineering, University of Rochester（电气与计算机工程系，罗切斯特大学）；Goergen Institute for Data Science & AI, University of Rochester（数据科学与人工智能研究院，罗切斯特大学）
Many black-box optimization tasks arising in high-stakes applications require risk-averse decisions. The standard Bayesian optimization (BO) paradigm, however, optimizes the expected value only. We generalize BO to trade mean and input-dependent variance of the objective, both of which we assume to be unknown a priori. In particular, we propose a novel risk-averse heteroscedastic Bayesian optimization algorithm (RAHBO) that aims to identify a solution with high return and low noise variance, while learning the noise distribution on the fly. To this end, we model both expectation and variance a…
Confident Object Detection via Conformal Prediction and Conformal Risk Control: an Application to Railway Signaling
Institut de Mathématiques de Toulouse（图卢兹数学研究所）；SNCF（法国国家铁路公司）；Brown University（布朗大学）；Thales AVS France SAS（泰雷兹航空航天系统法国股份有限公司）；IRT Saint Exupéry（圣埃克苏佩里技术研究所）
The Bayesian posterior minimizes the "inferential risk" which itself bounds the "predictive risk". This bound is tight when the likelihood and prior are well-specified. However since misspecification induces a gap, the Bayesian posterior predictive distribution may have poor generalization performance. This work develops a multi-sample loss (PAC$^m$) which can close the gap by spanning a trade-off between the two risks. The loss is computationally favorable and offers PAC generalization guarantees. Empirical study demonstrates improvement to the predictive distribution.
We present a Bayesian view of counterfactual risk minimization (CRM) for offline learning from logged bandit feedback. Using PAC-Bayesian analysis, we derive a new generalization bound for the truncated inverse propensity score estimator. We apply the bound to a class of Bayesian policies, which motivates a novel, potentially data-dependent, regularization technique for CRM. Experimental results indicate that this technique outperforms standard $L_2$ regularization, and that it is competitive with variance regularization while being both simpler to implement and more computationally efficient.
Heinz College of Information Systems and Public Policy, Carnegie Mellon University（卡内基梅隆大学信息系统与公共政策学院）；Saïd Business School, University of Oxford（牛津大学said商学院）
The Australian National University（澳大利亚国立大学）；Data61；Inria（法国国家信息与自动化研究所）；University College London（伦敦大学学院）
Conditional Value at Risk (CVaR) is a family of "coherent risk measures" which generalize the traditional mathematical expectation. Widely used in mathematical finance, it is garnering increasing interest in machine learning, e.g., as an alternate approach to regularization, and as a means for ensuring fairness. This paper presents a generalization bound for learning algorithms that minimize the CVaR of the empirical loss. The bound is of PAC-Bayesian type and is guaranteed to be small when the empirical CVaR is small. We achieve this by reducing the problem of estimating CVaR to that of merel…
Siemens AG（西门子股份公司）；TU Munich（慕尼黑工业大学）；University of Cambridge（剑桥大学）；Harvard University（哈佛大学）
Bayesian neural networks with latent variables are scalable and flexible probabilistic models: They account for uncertainty in the estimation of the network weights and, by making use of latent variables, can capture complex noise patterns in the data. We show how to extract and decompose uncertainty into epistemic and aleatoric components for decision-making purposes. This allows us to successfully identify informative points for active learning of functions with heteroscedastic and bimodal noise. Using the decomposition we further define a novel risk-sensitive criterion for reinforcement lea…
Machine unlearning refers to mechanisms that can remove the influence of a subset of training data upon request from a trained model without incurring the cost of re-training from scratch. This paper develops a unified PAC-Bayesian framework for machine unlearning that recovers the two recent design principles - variational unlearning (Nguyen et.al., 2020) and forgetting Lagrangian (Golatkar et.al., 2020) - as information risk minimization problems (Zhang,2006). Accordingly, both criteria can be interpreted as PAC-Bayesian upper bounds on the test loss of the unlearned model that take the form…
A Data-Driven Probabilistic Framework for Cascading Urban Risk Analysis Using Bayesian Networks
Department of Computer Science and Engineering, School of Computing, Coimbatore, Amrita Vishwa Vidyapeetham, Tamil Nadu, India（计算机科学与工程系，计算学院，科瓦尔，阿米塔大学文理学院，泰米尔纳德邦，印度）
General predictive models do not provide a measure of confidence in predictions without Bayesian assumptions. A way to circumvent potential restrictions is to use conformal methods for constructing non-parametric confidence regions, that offer guarantees regarding validity. In this paper we provide a detailed description of a computationally efficient conformal procedure for Kernel Ridge Regression (KRR), and conduct a comparative numerical study to see how well conformal regions perform against the Bayesian confidence sets. The results suggest that conformalized KRR can yield predictive confi…
Reliable Real-Time Value at Risk Estimation via Quantile Regression Forest with Conformal Calibration
Institute of Statistics and Big Data, Renmin University of China（统计与大数据研究院，中国人民大学）；Department of Decision Analytics and Operations, City University of Hong Kong（决策分析与运营部门，香港城市大学）；School of Information, Renmin University of China（信息学院，中国人民大学）
