Objective interpretation of the penalised complexity prior in copula models

In many statistical models it is natural to have a nested structure. Consider a model of a given complexity, one way to obtain a more flexible model is to include an extra component so that the simpler model would be nested in the more complex one. One may think, for instance, of a situation where one wants to model the joint distribution of several random variables through a copula function. In the case of dependence among variables, the joint density can be expressed as the product of the marginal distributions times a copula function, on the contrary, the joint density boils down to the only product of the marginals when the variables are independent. For the Gaussian copula model we derive a Penalised Complexity prior. We also show that for any copula function and for any dimension the elicitation of the PC prior is invariant with respect to the marginals and their parameters, indeed only the copula structure matters. PC priors are constructed by means of a user-defined scaling, but in order to be objective we assign an intrinsic prior to the scale parameter of the PC prior. Then, objectivity is attained by maximizing the variance of the PC prior where an intrinsic prior is put on the scale parameter. The PC prior can be used in Bayesian hypothesis testing both in the calculation of the Bayes factor and in the calibration of objective prior distributions on the models. To check the goodness of the prior, a comparison with other existing priors is carried out.