Calibrated Bayesian Nonparametric Tolerance Intervals

2026-03-11

AI Summary

This study addresses the challenge that traditional tolerance interval methods often fail to simultaneously achieve accurate coverage and efficiency when parametric assumptions are violated or sample sizes are small. The authors propose a novel approach by introducing calibrated Gibbs posteriors for constructing tolerance intervals, leveraging quantile regression based on the asymmetric Laplace (check) loss function. A carefully chosen learning rate ensures nominal frequentist coverage under arbitrary data-generating distributions. By integrating Bayesian flexibility with guaranteed frequentist coverage, the method overcomes the reliance of nonparametric approaches on large samples. Extensive simulations and real-world applications in ecology, biopharmaceuticals, and environmental monitoring demonstrate that the proposed intervals consistently attain reliable coverage while typically being shorter than those produced by classical nonparametric methods.

Technology Category

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

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Tolerance intervals provide bounds that contain a specified proportion of a population with a given confidence level, yet their construction remains challenging when parametric assumptions fail or sample sizes are small. Traditional nonparametric methods, such as Wilks' intervals, lack flexibility and often require large samples to be valid. We propose a fully nonparametric approach for constructing one-sided and two-sided tolerance intervals using a calibrated Gibbs posterior. Leveraging the connection between tolerance limits and population quantiles, we employ a Gibbs posterior based on the asymmetric Laplace (check) loss function. A key feature of our method is the calibration of the learning rate, which ensures nominal frequentist coverage across diverse distributional shapes. Simulation studies show that the proposed approach often yields shorter intervals than classical nonparametric benchmarks while maintaining reliable coverage. The framework's practical utility is illustrated through applications in ecology, biopharmaceutical manufacturing, and environmental monitoring, demonstrating its flexibility and robustness across diverse applications.

Problem

• tolerance intervals
• nonparametric
• small sample
• coverage
• calibration

Innovation

• calibrated Gibbs posterior
• nonparametric tolerance intervals
• asymmetric Laplace loss
• learning rate calibration
• frequentist coverage