  Parametric conditional copula models allow the copula parameters to vary with
a set of covariates according to an unknown calibration function. Flexible
Bayesian inference for the calibration function of a bivariate conditional
copula is proposed via a sparse Gaussian process (GP) prior distribution over
the set of smooth calibration functions for the single index model (SIM). The
estimation of parameters from the marginal distributions and the calibration
function is done jointly via Markov Chain Monte Carlo sampling from the full
posterior distribution. A new Conditional Cross Validated Pseudo-Marginal
(CCVML) criterion is introduced in order to perform copula selection and is
modified using a permutation-based procedure to assess data support for the
simplifying assumption. The performance of the estimation method and model
selection criteria is studied via a series of simulations using correct and
misspecified models with Clayton, Frank and Gaussian copulas and a numerical
application involving red wine features.
