  Copula models are flexible tools to represent complex structures of
dependence for multivariate random variables. According to Sklar's theorem
(Sklar, 1959), any d-dimensional absolutely continuous density can be uniquely
represented as the product of the marginal distributions and a copula function
which captures the dependence structure among the vector components. In real
data applications, the interest of the analyses often lies on specific
functionals of the dependence, which quantify aspects of it in a few numerical
values. A broad literature exists on such functionals, however extensions to
include covariates are still limited. This is mainly due to the lack of
unbiased estimators of the copula function, especially when one does not have
enough information to select the copula model. Recent advances in computational
methodologies and algorithms have allowed inference in the presence of
complicated likelihood functions, especially in the Bayesian approach, whose
methods, despite being computationally intensive, allow us to better evaluate
the uncertainty of the estimates. In this work, we present several Bayesian
methods to approximate the posterior distribution of functionals of the
dependence, using nonparametric models which avoid the selection of the copula
function. These methods are compared in simulation studies and in two realistic
applications, from civil engineering and astrophysics.
