Bayesian Inference for Joint Tail Risk in Paired Biomarkers via Archimedean Copulas with Restricted Jeffreys Priors

2026-02-16

AI Summary

This study addresses the quantification of clinically interpretable joint tail risk for paired continuous biomarkers and the accurate assessment of its uncertainty. To this end, the authors propose a rank-based pseudo-observation approach that models the dependence structure using a one-parameter Archimedean copula, focusing on three probability-scale metrics: lower tail, upper tail, and conditional tail probabilities. By incorporating a constrained Jeffreys prior and a grid-based posterior approximation, the method enables dependence-aware Bayesian inference, yielding precise posterior distributions for tail risk functionals. Simulation studies demonstrate that the resulting posterior credible intervals achieve coverage close to nominal levels. Application to NHANES data reveals that the upper-tail joint risk of glucose and HbA1c is substantially elevated relative to the independence assumption, reaching 11.46 times the benchmark level.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Calibration & Uncertainty Quantification
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

We propose a Bayesian copula-based framework to quantify clinically interpretable joint tail risks from paired continuous biomarkers. After converting each biomarker margin to rank-based pseudo-observations, we model dependence using one-parameter Archimedean copulas and focus on three probability-scale summaries at tail level $α$: the lower-tail joint risk $R_L(θ)=C_θ(α,α)$, the upper-tail joint risk $R_U(θ)=2α-1+C_θ(1-α,1-α)$, and the conditional lower-tail risk $R_C(θ)=R_L(θ)/α$. Uncertainty is quantified via a restricted Jeffreys prior on the copula parameter and grid-based posterior approximation, which induces an exact posterior for each tail-risk functional. In simulations from Clayton and Gumbel copulas across multiple dependence strengths, posterior credible intervals achieve near-nominal coverage for $R_L$, $R_U$, and $R_C$. We then analyze NHANES 2017--2018 fasting glucose (GLU) and HbA1c (GHB) ($n=2887$) at $α=0.05$, obtaining tight posterior credible intervals for both the dependence parameter and induced tail risks. The results reveal markedly elevated extremal co-movement relative to independence; under the Gumbel model, the posterior mean joint upper-tail risk is $R_U(α)=0.0286$, approximately $11.46\times$ the independence benchmark $α^2=0.0025$. Overall, the proposed approach provides a principled, dependence-aware method for reporting joint and conditional extremal-risk summaries with Bayesian uncertainty quantification in biomedical applications.

Problem

• joint tail risk
• paired biomarkers
• Bayesian inference
• extremal co-movement
• copula

Innovation

• Bayesian copula
• tail risk
• restricted Jeffreys prior
• Archimedean copulas
• joint extremal dependence