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

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 RL(θ)=Cθ(α,α), the upper-tail joint risk RU(θ)=2α-1+Cθ(1-α,1-α), and the conditional lower-tail risk RC(θ)=RL(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 RL, RU, and RC. 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 RU(α)=0.0286, approximately 11.46× the independence benchmark α2=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.