Penalising model complexity

We consider a multivariate model with independent marginals as a benchmark for a generic multivariate model where the marginals are not independent. The Penalised Complexity (PC) prior takes natural place in such a context, as we can include in the simpler model an extra-component taking into account for dependence. In this paper, the additional component is represented by the parameter of the Gaussian copula density function. We show that the PC prior for a generic copula parameter can be derived regardless of the parameters of the marginal densities. Then, we propose a hierarchical PC prior for the Gaussian copula model. We also derive the PC prior for the shape parameter of the skew-normal distribution and we use it for Bayesian hypothesis test for skewness. In the last chapter, we propose two ways to extend the univariate PC prior to the multivariate case.