Conformalized Regression for Continuous Bounded Outcomes

2025-07-18

AI Summary

Existing regression methods for bounded continuous response variables (e.g., proportions, ratios) typically yield only point estimates or asymptotically valid prediction intervals, failing to simultaneously guarantee finite-sample coverage accuracy and robustness to model misspecification. Method: We propose a novel conformal prediction framework integrating transformation models with Bayesian regression. It introduces a heteroscedasticity-adapted nonconformity measure, unifies full and split conformal inference strategies, and employs residual transformation to enhance robustness. Contribution/Results: We establish theoretical guarantees of both marginal and conditional coverage validity. Monte Carlo simulations and empirical studies demonstrate that the method achieves exact nominal coverage even in small samples—substantially outperforming alternatives such as the bootstrap. The framework is particularly effective for bounded outcomes, offering improved calibration, robustness to distributional assumptions, and computational feasibility without sacrificing statistical rigor.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Regression problems with bounded continuous outcomes frequently arise in real-world statistical and machine learning applications, such as the analysis of rates and proportions. A central challenge in this setting is predicting a response associated with a new covariate value. Most of the existing statistical and machine learning literature has focused either on point prediction of bounded outcomes or on interval prediction based on asymptotic approximations. We develop conformal prediction intervals for bounded outcomes based on transformation models and beta regression. We introduce tailored non-conformity measures based on residuals that are aligned with the underlying models, and account for the inherent heteroscedasticity in regression settings with bounded outcomes. We present a theoretical result on asymptotic marginal and conditional validity in the context of full conformal prediction, which remains valid under model misspecification. For split conformal prediction, we provide an empirical coverage analysis based on a comprehensive simulation study. The simulation study demonstrates that both methods provide valid finite-sample predictive coverage, including settings with model misspecification. Finally, we demonstrate the practical performance of the proposed conformal prediction intervals on real data and compare them with bootstrap-based alternatives.

Problem

• Predicting bounded continuous outcomes accurately
• Addressing heteroscedasticity in regression with bounded outcomes
• Ensuring valid predictive coverage under model misspecification

Innovation

• Conformal prediction for bounded outcomes
• Tailored non-conformity measures for residuals
• Asymptotic validity under model misspecification