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.

翻译：我们提出一个基于贝叶斯Copula的框架，用于从配对的连续生物标志物中量化具有临床可解释性的联合尾部风险。将每个生物标志物的边缘分布转换为基于秩的伪观测值后，我们使用单参数Archimedean Copula对相依性进行建模，并重点关注尾部水平$α$上的三个概率尺度摘要：下尾部联合风险$R_L(θ)=C_θ(α,α)$、上尾部联合风险$R_U(θ)=2α-1+C_θ(1-α,1-α)$以及条件性下尾部风险$R_C(θ)=R_L(θ)/α$。通过在Copula参数上设置限制Jeffreys先验并采用基于网格的后验近似，我们对不确定性进行量化，这为每个尾部风险泛函导出了精确后验。在Clayton和Gumbel Copula的多种相依强度模拟中，后验可信区间对$R_L$、$R_U$和$R_C$均实现了接近名义水平的覆盖。随后，我们在$α=0.05$水平下分析了NHANES 2017--2018的空腹血糖（GLU）与糖化血红蛋白（GHB）数据（$n=2887$），获得了关于相依参数及导出的尾部风险的紧凑后验可信区间。结果显示，相对于独立情形，极端协同运动显著增强；在Gumbel模型下，后验平均联合上尾部风险为$R_U(α)=0.0286$，约为独立基准$α^2=0.0025$的$11.46$倍。总体而言，所提出的方法为生物医学应用提供了一种原则性的、考虑相依性的途径，能够以贝叶斯不确定性量化的方式报告联合及条件性极端风险摘要。