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

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 $\alpha$: the lower-tail joint risk $R_L(\theta )=C_{\theta }(\alpha ,\alpha )$, the upper-tail joint risk $R_U(\theta )=2\alpha -1+C_{\theta }(1-\alpha ,1-\alpha )$, and the conditional lower-tail risk $R_C(\theta )=R_L(\theta )/\alpha$. 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 $\alpha =0.05$, 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 $R_U(\alpha )=0.0286$, approximately $11.46\times$ the independence benchmark ${\alpha }^2=0.0025$. 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.