Bayesian inference for conditional copulas using Gaussian Process single index models

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 introduced. The prior distribution over the set of smooth calibration functions is built using a sparse Gaussian process (GP) prior 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 used 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.