A semiparametric Bayesian estimator of copula density

We propose a semiparametric Bayesian estimator for copula density, formulated as a generalized exponential family that combines a parametric baseline copula with a flexible adjustment modeled by a Gaussian Process (GP). The baseline captures the main features of dependence and accommodates possibly unbounded densities, while the GP component introduces nonparametric refinement. To modulate the influence of the baseline, we further introduce a tuning parameter that adjusts its weight in the final estimate. Both the GP hyperparameters and this tuning parameter are inferred jointly within a Bayesian framework. We also present a posterior sampler based on importance sampling, with the baseline copula serving as a natural approximating distribution. Simulations demonstrate the efficacy of the proposed estimator and the efficiency of the sampler. The method is further illustrated with an application to commodity futures markets.