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 work, 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 the Bayesian hypothesis test for skewness. In the last chapter, we propose two ways to extend the univariate PC prior to the multivariate case.